2000 character limit reached
Self-similar growth of Bose stars
Published 1 May 2023 in astro-ph.CO, astro-ph.GA, cond-mat.stat-mech, and hep-ph | (2305.01005v4)
Abstract: We analytically solve the problem of Bose star growth in the bath of gravitationally interacting particles. We find that after nucleation of this object the bath is described by a self-similar solution of kinetic equation. Together with the conservation laws, this fixes mass evolution of the Bose star. Our theory explains, in particular, the slowdown of the star growth at a certain "core-halo" mass, but also predicts formation of heavier and lighter objects in magistral dark matter models. The developed "adiabatic" approach to self-similarity may be of interest for kinetic theory in general.
- R. Ruffini and S. Bonazzola, Phys. Rev. 187, 1767 (1969).
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- Self-similar solutions are well-known in kinetic theory with short-range interactions Semikoz and Tkachev (1995); *Micha:2002ey; *Micha:2004bv; *SEMISALOV2021105903 and in dynamical long-range problems like collapse Choptuik (1993); *Maeda:2004kw; *Gundlach:2007gc or infall Bertschinger (1985); *Sikivie:1996nn. But their relevance for kinetics caused by gravitational (long-range) scattering was not observed before.
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- E. W. Kolb and I. I. Tkachev, Phys. Rev. D49, 5040 (1994a), arXiv:astro-ph/9311037 .
- B. Eggemeier et al., Phys. Rev. Lett. 125, 041301 (2020), arXiv:1911.09417 .
- E. Lifshitz and L. Pitaevskii, Course of Theoretical Physics, Vol. 10: Physical Kinetics (Elsevier Science, 2012). Zakharov and Karas’ (2013) V. E. Zakharov and V. I. Karas’, Physics Uspekhi 56, 49 (2013). Skipp, L’vov, and Nazarenko (2020) J. Skipp, V. L’vov, and S. Nazarenko, Phys. Rev. A 102, 043318 (2020), arXiv:2003.05558 . Zakharov, L’vov, and Falkovich (2012) V. Zakharov, V. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer, 2012). Note (3) Our best-fit value τi=−0.1subscript𝜏𝑖0.1\tau_{i}=-0.1italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 0.1 from Figs. 2 and SM-S2 is quite small and does not affect the agreement in Fig. 1(c). Lee and Pang (1989) T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. E. Zakharov and V. I. Karas’, Physics Uspekhi 56, 49 (2013). Skipp, L’vov, and Nazarenko (2020) J. Skipp, V. L’vov, and S. Nazarenko, Phys. Rev. A 102, 043318 (2020), arXiv:2003.05558 . Zakharov, L’vov, and Falkovich (2012) V. Zakharov, V. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer, 2012). Note (3) Our best-fit value τi=−0.1subscript𝜏𝑖0.1\tau_{i}=-0.1italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 0.1 from Figs. 2 and SM-S2 is quite small and does not affect the agreement in Fig. 1(c). Lee and Pang (1989) T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Skipp, V. L’vov, and S. Nazarenko, Phys. Rev. A 102, 043318 (2020), arXiv:2003.05558 . Zakharov, L’vov, and Falkovich (2012) V. Zakharov, V. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer, 2012). Note (3) Our best-fit value τi=−0.1subscript𝜏𝑖0.1\tau_{i}=-0.1italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 0.1 from Figs. 2 and SM-S2 is quite small and does not affect the agreement in Fig. 1(c). Lee and Pang (1989) T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. Zakharov, V. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer, 2012). Note (3) Our best-fit value τi=−0.1subscript𝜏𝑖0.1\tau_{i}=-0.1italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 0.1 from Figs. 2 and SM-S2 is quite small and does not affect the agreement in Fig. 1(c). Lee and Pang (1989) T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Our best-fit value τi=−0.1subscript𝜏𝑖0.1\tau_{i}=-0.1italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 0.1 from Figs. 2 and SM-S2 is quite small and does not affect the agreement in Fig. 1(c). Lee and Pang (1989) T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- V. E. Zakharov and V. I. Karas’, Physics Uspekhi 56, 49 (2013). Skipp, L’vov, and Nazarenko (2020) J. Skipp, V. L’vov, and S. Nazarenko, Phys. Rev. A 102, 043318 (2020), arXiv:2003.05558 . Zakharov, L’vov, and Falkovich (2012) V. Zakharov, V. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer, 2012). Note (3) Our best-fit value τi=−0.1subscript𝜏𝑖0.1\tau_{i}=-0.1italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 0.1 from Figs. 2 and SM-S2 is quite small and does not affect the agreement in Fig. 1(c). Lee and Pang (1989) T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Skipp, V. L’vov, and S. Nazarenko, Phys. Rev. A 102, 043318 (2020), arXiv:2003.05558 . Zakharov, L’vov, and Falkovich (2012) V. Zakharov, V. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer, 2012). Note (3) Our best-fit value τi=−0.1subscript𝜏𝑖0.1\tau_{i}=-0.1italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 0.1 from Figs. 2 and SM-S2 is quite small and does not affect the agreement in Fig. 1(c). Lee and Pang (1989) T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. Zakharov, V. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer, 2012). Note (3) Our best-fit value τi=−0.1subscript𝜏𝑖0.1\tau_{i}=-0.1italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 0.1 from Figs. 2 and SM-S2 is quite small and does not affect the agreement in Fig. 1(c). Lee and Pang (1989) T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Our best-fit value τi=−0.1subscript𝜏𝑖0.1\tau_{i}=-0.1italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 0.1 from Figs. 2 and SM-S2 is quite small and does not affect the agreement in Fig. 1(c). Lee and Pang (1989) T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
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Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. Zakharov, V. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer, 2012). Note (3) Our best-fit value τi=−0.1subscript𝜏𝑖0.1\tau_{i}=-0.1italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 0.1 from Figs. 2 and SM-S2 is quite small and does not affect the agreement in Fig. 1(c). Lee and Pang (1989) T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Our best-fit value τi=−0.1subscript𝜏𝑖0.1\tau_{i}=-0.1italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 0.1 from Figs. 2 and SM-S2 is quite small and does not affect the agreement in Fig. 1(c). Lee and Pang (1989) T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- V. Zakharov, V. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer, 2012). Note (3) Our best-fit value τi=−0.1subscript𝜏𝑖0.1\tau_{i}=-0.1italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 0.1 from Figs. 2 and SM-S2 is quite small and does not affect the agreement in Fig. 1(c). Lee and Pang (1989) T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Our best-fit value τi=−0.1subscript𝜏𝑖0.1\tau_{i}=-0.1italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 0.1 from Figs. 2 and SM-S2 is quite small and does not affect the agreement in Fig. 1(c). Lee and Pang (1989) T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- Our best-fit value τi=−0.1subscript𝜏𝑖0.1\tau_{i}=-0.1italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 0.1 from Figs. 2 and SM-S2 is quite small and does not affect the agreement in Fig. 1(c). Lee and Pang (1989) T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. 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Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- T. D. Lee and Y. Pang, Nucl. Phys. B 315, 477 (1989). Dmitriev et al. (2021) A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. 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Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- A. S. Dmitriev et al., Phys. Rev. D 104, 023504 (2021), arXiv:2104.00962 . Note (4) More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- More precise expression follows from the adiabatic theorem: Ee=−ζMbs2subscript𝐸𝑒𝜁superscriptsubscript𝑀𝑏𝑠2E_{e}=-\zeta M_{bs}^{2}italic_E start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = - italic_ζ italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, where ζ𝜁\zetaitalic_ζ depends on the occupation numbers of the bound states. Schwabe, Niemeyer, and Engels (2016) B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151 . Mocz et al. (2017) P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- P. Mocz et al., MNRAS 471, 4559 (2017), arXiv:1705.05845 . Schive et al. (2014) H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H.-Y. Schive et al., Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762 . Bar et al. (2018) N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
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Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- N. Bar et al., Phys. Rev. D 98, 083027 (2018), arXiv:1805.00122 . Mina, Mota, and Winther (2022) M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- M. Mina, D. F. Mota, and H. A. Winther, Astron. Astrophys. 662, A29 (2022), arXiv:2007.04119 . (33) J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- J. L. Zagorac et al., arXiv:2212.09349 . Chan et al. (2022) H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- H. Y. J. Chan et al., MNRAS 511, 943 (2022), arXiv:2110.11882 . Nori and Baldi (2021) M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- M. Nori and M. Baldi, Mon. Not. Roy. Astron. Soc. 501, 1539 (2021), arXiv:2007.01316 . Schwabe and Niemeyer (2022) B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- B. Schwabe and J. C. Niemeyer, Phys. Rev. Lett. 128, 181301 (2022), arXiv:2110.09145 [astro-ph.CO] . (37) Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- Z. Li et al., ApJ 889, 88, arXiv:2001.00318 . Eggemeier et al. (2022) B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- B. Eggemeier et al., Phys. Rev. D 105, 023516 (2022), arXiv:2110.15109 . Ellis et al. (2022) D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- D. Ellis et al., Phys. Rev. D 106, 103514 (2022), arXiv:2204.13187 . (40) X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- X. Du et al., arXiv:2301.09769 . Kolb and Tkachev (1994b) E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994b), arXiv:astro-ph/9403011 . Klaer and Moore (2017) V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- V. B. Klaer and G. D. Moore, JCAP 1711, 049 (2017), arXiv:1708.07521 . Gorghetto, Hardy, and Villadoro (2018) M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- M. Gorghetto, E. Hardy, and G. Villadoro, JHEP 07, 151 (2018), arXiv:1806.04677 . Gorghetto, Hardy, and Villadoro (2021) M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- M. Gorghetto, E. Hardy, and G. Villadoro, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 . Levkov, Panin, and Tkachev (2020) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. D 102, 023501 (2020), arXiv:2004.05179 . Eby et al. (2022) J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- J. Eby et al., Phys. Lett. B 825, 136858 (2022), arXiv:2106.14893 . Visinelli (2021) L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 . (48) M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- M. Escudero et al., arXiv:2302.10206 . Rogers and Peiris (2021) K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- K. K. Rogers and H. V. Peiris, Phys. Rev. Lett. 126, 071302 (2021), arXiv:2007.12705 . Note (5) This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- This solution satisfies Cs(ωs)−F0s2/2ωs→0→subscript𝐶𝑠subscript𝜔𝑠superscriptsubscript𝐹0𝑠22subscript𝜔𝑠0C_{s}(\omega_{s})-F_{0s}^{2}/2\omega_{s}\to 0italic_C start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT 0 italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0 as ωs→0→subscript𝜔𝑠0\omega_{s}\to 0italic_ω start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT → 0. Note (6) We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- We thank the Referee for suggesting this check. Chen et al. (2022) J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- J. Chen, X. Du, E. W. Lentz, and D. J. E. Marsh, Phys. Rev. D 106, 023009 (2022), arXiv:2109.11474 . Note (7) This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- This ratio is even smaller in magistral cosmological models. For example, τgr/τλ∼10−12similar-tosubscript𝜏𝑔𝑟subscript𝜏𝜆superscript1012\tau_{gr}/\tau_{\lambda}\sim 10^{-12}italic_τ start_POSTSUBSCRIPT italic_g italic_r end_POSTSUBSCRIPT / italic_τ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT and λ~∼10−4similar-to~𝜆superscript104\tilde{\lambda}\sim 10^{-4}over~ start_ARG italic_λ end_ARG ∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Levkov, Panin, and Tkachev (2018); Chen et al. (2022) inside QCD axion miniclusters. At these values, the effect of self-interactions on growth of Bose stars is negligible. To make them relevant, one switches Chen et al. (2021, 2022) to general “axion–like” models with deliberately enlarged λ𝜆\lambdaitalic_λ. Chavanis and Delfini (2011) P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- P. H. Chavanis and L. Delfini, Phys. Rev. D 84, 043532 (2011), arXiv:1103.2054 . Eby et al. (2016) J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- J. Eby et al., JHEP 12, 066 (2016), arXiv:1608.06911 . Levkov, Panin, and Tkachev (2017) D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 118, 011301 (2017), arXiv:1609.03611 . Note (8) We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- We still extract xesubscript𝑥𝑒x_{e}italic_x start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT from the distribution function and compute ϵ2≡E/γM3superscriptitalic-ϵ2𝐸𝛾superscript𝑀3{\epsilon^{2}\equiv E/\gamma M^{3}}italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_E / italic_γ italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT using the initial data. The value of τ*subscript𝜏\tau_{*}italic_τ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT is fixed by the condition Mbs=0subscript𝑀𝑏𝑠0M_{bs}=0italic_M start_POSTSUBSCRIPT italic_b italic_s end_POSTSUBSCRIPT = 0 at τ=1𝜏1\tau=1italic_τ = 1, see Sec. 4. Note (9) This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- This is a necessary part of the procedure compensating for our voluntary choice of the minicluster parameters ρ¯¯𝜌\bar{\rho}over¯ start_ARG italic_ρ end_ARG, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and R𝑅Ritalic_R. Semikoz and Tkachev (1995) D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995), arXiv:hep-ph/9409202 . Micha and Tkachev (2003) R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202 . Micha and Tkachev (2004) R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101 . Semisalov et al. (2021) B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- B. Semisalov et al., Communications in Nonlinear Science and Numerical Simulation 102, 105903 (2021), arXiv:2104.14591 . Choptuik (1993) M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). Maeda and Harada (2005) H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- H. Maeda and T. Harada, “Kinematic self-similar solutions in general relativity,” in General Relativity Research Trends. Horizons in World Physics, Vol. 249 (Nova Science Publishers, New York, 2005) p. 123, arXiv:gr-qc/0405113 . Gundlach and Martin-Garcia (2007) C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. 10, 5 (2007), arXiv:0711.4620 . Bertschinger (1985) E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- E. Bertschinger, Astrophys. J. Suppl. 58, 39 (1985). Sikivie, Tkachev, and Wang (1997) P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 . P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
- P. Sikivie, I. I. Tkachev, and Y. Wang, Phys. Rev. D 56, 1863 (1997), arXiv:astro-ph/9609022 .
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