Establish the dynamical origin of a possible log-periodic signature

Investigate the dynamical origin of the possible log-periodic modulation in high-redshift compact-object abundances generated by repeated collapse and recondensation of the $10^{-22}$ eV axion condensate.

Background

The paper suggests that approximately self-similar recurrent collapse events could generate a modulation periodic in  ln⁡(1+z)\,\ln(1+z). This proposed signature is motivated by reported phenomenological evidence but is not derived from a complete dynamical treatment of the axion condensate.

The unresolved task is to determine whether the nonlinear collapse--recondensation dynamics naturally produces the required recurrence, self-similarity, amplitude, and phase, and whether the resulting modulation is compatible with the observed abundance of high-redshift compact objects.

References

Several issues remain open. Most importantly, the present analysis does not derive the required alignment and multi-harmonic structure from a specific heterotic compactification. Establishing such a construction would provide a microscopic test of the scenario. Likewise, the $10{-28}$ eV sector should ultimately be confronted with the full set of CMB, BAO, supernova, and structure-formation constraints through a numerical cosmological analysis. Such an analysis is necessary to determine quantitatively how large a reduction of the sound horizon can be obtained while remaining consistent with the other cosmological observables. On the $10{-22}$ eV side, a quantitative treatment of the nonlinear condensation--collapse--recondensation cycle will be necessary to determine the resulting seed-mass distribution and to test its possible relation to the observed high-redshift compact-object population. The possible log-periodic signature discussed in Sec.~3 should likewise be regarded at present as a phenomenological indication whose dynamical origin requires further investigation.

— Toward a Unified Axion Cosmology  (2608.22667 - Fukuyama, 24 Aug 2026) in Section 4, Discussion and conclusions; see also Section 3, BEC and Early supermassive black holes