Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Global Portraits of Nonminimal Inflation: Metric and Palatini (2401.12314v2)

Published 22 Jan 2024 in gr-qc and hep-ph

Abstract: In this paper, we study the global phase space dynamics of single nonminimally coupled scalar field inflation models in the metric and Palatini formalisms. Working in the Jordan frame, we derive the scalar-tensor general field equations and flat FLRW cosmological equations, and present the Palatini and metric equations in a common framework. We show that inflation is characterized by a "master" trajectory from a saddle-type de Sitter fixed point to a stable node fixed point, approximated by slow roll conditions (presented for the first time in the Palatini formalism). We show that, despite different underlying equations, the fixed point structure and properties of many models are congruent in metric and Palatini, which explains their qualitative similarities and their suitability for driving inflation. On the other hand, the global phase portraits reveal how even models which predict the same values for observable perturbations differ, both to the extent of the phase space physically available to their trajectories, as well as their past asymptotic states. We also note how the slow roll conditions tend to underestimate the end of inflationary accelerated expansion experienced by the true nonlinear "master" solution. The explicit examples we consider range from the metric and Palatini induced gravity quintic potential with a Coleman-Weinberg correction factor to Starobinsky, metric and Palatini nonminimal Higgs, second order pole, and several nontrivial Palatini models.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (102)
  1. B. Ratra and P. J. E. Peebles, “Cosmological Consequences of a Rolling Homogeneous Scalar Field,” Phys. Rev. D 37 (1988) 3406.
  2. V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger, “Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions,” Phys. Rept. 215 (1992) 203–333.
  3. D. H. Lyth and A. Riotto, “Particle physics models of inflation and the cosmological density perturbation,” Phys. Rept. 314 (1999) 1–146, arXiv:hep-ph/9807278.
  4. J. Martin, C. Ringeval, and V. Vennin, “Encyclopædia Inflationaris,” Phys. Dark Univ. 5-6 (2014) 75–235, arXiv:1303.3787 [astro-ph.CO].
  5. F. S. Accetta, D. J. Zoller, and M. S. Turner, “Induced Gravity Inflation,” Phys. Rev. D31 (1985) 3046.
  6. F. Lucchin, S. Matarrese, and M. D. Pollock, “Inflation With a Nonminimally Coupled Scalar Field,” Phys. Lett. B 167 (1986) 163–168.
  7. F. L. Bezrukov and M. Shaposhnikov, “The Standard Model Higgs boson as the inflaton,” Phys. Lett. B659 (2008) 703–706, arXiv:0710.3755 [hep-th].
  8. S. Capozziello and M. De Laurentis, “Extended Theories of Gravity,” Phys. Rept. 509 (2011) 167–321, arXiv:1108.6266 [gr-qc].
  9. E. Gunzig, A. Saa, L. Brenig, V. Faraoni, T. M. Rocha Filho, and A. Figueiredo, “Superinflation, quintessence, and nonsingular cosmologies,” Phys. Rev. D63 (2001) 067301, arXiv:gr-qc/0012085 [gr-qc].
  10. L. Jarv, P. Kuusk, and M. Saal, “Remarks on (super-)accelerating cosmological models in general scalar-tensor gravity,” Proc. Est. Acad. Sci. Phys. Math. 59 (2010) 306–312, arXiv:0903.4357 [gr-qc].
  11. L. Järv, K. Kannike, L. Marzola, A. Racioppi, M. Raidal, M. Rünkla, M. Saal, and H. Veermäe, “Frame-Independent Classification of Single-Field Inflationary Models,” Phys. Rev. Lett. 118 (2017) no. 15, 151302, arXiv:1612.06863 [hep-ph].
  12. P. Kuusk, M. Rünkla, M. Saal, and O. Vilson, “Invariant slow-roll parameters in scalar–tensor theories,” Class. Quant. Grav. 33 (2016) no. 19, 195008, arXiv:1605.07033 [gr-qc].
  13. D. Burns, S. Karamitsos, and A. Pilaftsis, “Frame-Covariant Formulation of Inflation in Scalar-Curvature Theories,” Nucl. Phys. B 907 (2016) 785–819, arXiv:1603.03730 [hep-ph].
  14. U. Lindstrom, “Comments on the Jordan-Brans-Dicke Scalar Field Theory of Gravitation,” Nuovo Cim. B 32 (1976) 298–302.
  15. U. Lindstrom, “The Palatini Variational Principle and a Class of Scalar-Tensor Theories,” Nuovo Cim. B 35 (1976) 130–136.
  16. N. van den Bergh, “The Palatini variational principle for the general Bergmann–Wagoner–Nordtvedt theory of gravitation,” J. Math. Phys. 22 (1981) no. 10, 2245–2248.
  17. H. Burton and R. B. Mann, “Palatini variational principle for N-dimensional dilaton gravity,” Class. Quant. Grav. 15 (1998) 1375–1385, arXiv:gr-qc/9710139.
  18. G. J. Olmo, “Palatini Approach to Modified Gravity: f(R) Theories and Beyond,” Int. J. Mod. Phys. D 20 (2011) 413–462, arXiv:1101.3864 [gr-qc].
  19. F. Bauer and D. A. Demir, “Inflation with Non-Minimal Coupling: Metric versus Palatini Formulations,” Phys. Lett. B 665 (2008) 222–226, arXiv:0803.2664 [hep-ph].
  20. T. Tenkanen, “Tracing the high energy theory of gravity: an introduction to Palatini inflation,” Gen. Rel. Grav. 52 (2020) no. 4, 33, arXiv:2001.10135 [astro-ph.CO].
  21. I. D. Gialamas, A. Karam, T. D. Pappas, and E. Tomberg, “Implications of Palatini gravity for inflation and beyond,” arXiv:2303.14148 [gr-qc].
  22. L. Järv, A. Karam, A. Kozak, A. Lykkas, A. Racioppi, and M. Saal, “Equivalence of inflationary models between the metric and Palatini formulation of scalar-tensor theories,” Phys. Rev. D 102 (2020) no. 4, 044029, arXiv:2005.14571 [gr-qc].
  23. V. Belinsky, I. Khalatnikov, L. Grishchuk, and Y. Zeldovich, “Inflationary stages in cosmological models with a scalar field,” Phys. Lett. B 155 (1985) 232–236.
  24. A. D. Linde, “Initial conditions for inflation,” Phys. Lett. B 162 (1985) 281–286.
  25. A. R. Liddle, P. Parsons, and J. D. Barrow, “Formalizing the slow roll approximation in inflation,” Phys. Rev. D 50 (1994) 7222–7232, arXiv:astro-ph/9408015.
  26. S. Bahamonde, C. G. Böhmer, S. Carloni, E. J. Copeland, W. Fang, and N. Tamanini, “Dynamical systems applied to cosmology: dark energy and modified gravity,” Phys. Rept. 775-777 (2018) 1–122, arXiv:1712.03107 [gr-qc].
  27. L. Amendola, M. Litterio, and F. Occhionero, “The Phase space view of inflation. 1: The nonminimally coupled scalar field,” Int. J. Mod. Phys. A5 (1990) 3861–3886.
  28. A. Burd and A. Coley, “Extended inflation and generalized scalar - tensor theories,” Phys. Lett. B 267 (1991) 330–336.
  29. E. Gunzig, V. Faraoni, A. Figueiredo, T. M. Rocha, and L. Brenig, “The dynamical system approach to scalar field cosmology,” Class. Quant. Grav. 17 (2000) 1783–1814.
  30. A. Alho and C. Uggla, “Global dynamics and inflationary center manifold and slow-roll approximants,” J. Math. Phys. 56 (2015) no. 1, 012502, arXiv:1406.0438 [gr-qc].
  31. A. Alho and C. Uggla, “Inflationary α𝛼\alphaitalic_α-attractor cosmology: A global dynamical systems perspective,” Phys. Rev. D95 (2017) no. 8, 083517, arXiv:1702.00306 [gr-qc].
  32. G. N. Felder, A. V. Frolov, L. Kofman, and A. D. Linde, “Cosmology with negative potentials,” Phys. Rev. D 66 (2002) 023507, arXiv:hep-th/0202017.
  33. L. A. Urena-Lopez and M. J. Reyes-Ibarra, “On the dynamics of a quadratic scalar field potential,” Int. J. Mod. Phys. D 18 (2009) 621–634, arXiv:0709.3996 [astro-ph].
  34. L. A. Urena-Lopez, “Unified description of the dynamics of quintessential scalar fields,” JCAP 03 (2012) 035, arXiv:1108.4712 [astro-ph.CO].
  35. G. Álvarez, L. Martínez Alonso, E. Medina, and J. L. Vázquez, “Separatrices in the Hamilton-Jacobi Formalism of Inflaton Models,” J. Math. Phys. 61 (2020) no. 4, 043501, arXiv:1911.04750 [math-ph].
  36. I. Quiros, T. Gonzalez, R. De Arcia, R. García-Salcedo, U. Nucamendi, and J. F. Saavedra, “Inflationary equilibrium configurations of scalar-tensor theories of gravity,” Phys. Rev. D 101 (2020) no. 10, 103518, arXiv:2003.07431 [gr-qc].
  37. O. Hrycyna, “The non-minimal coupling constant and the primordial de Sitter state,” Eur. Phys. J. C 80 (2020) no. 9, 817, arXiv:2008.00943 [gr-qc].
  38. L. Järv and A. Toporensky, “Global portraits of nonminimal inflation,” Eur. Phys. J. C 82 (2022) no. 2, 179, arXiv:2104.10183 [gr-qc].
  39. L. Järv and J. Lember, “Global Portraits of Nonminimal Teleparallel Inflation,” Universe 7 (2021) no. 6, 179, arXiv:2104.14258 [gr-qc].
  40. O. Hrycyna, “A new generic and structurally stable cosmological model without singularity,” Phys. Lett. B 820 (2021) 136511, arXiv:2105.02815 [gr-qc].
  41. O. Hrycyna, “On the structural stability of a simple cosmological model in R+α⁢R2𝑅𝛼superscript𝑅2R+\alpha R^{2}italic_R + italic_α italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT theory of gravity,” arXiv:2212.14843 [gr-qc].
  42. A. Alho and C. Uggla, “Quintessential α𝛼\alphaitalic_α-attractor inflation: a dynamical systems analysis,” JCAP 11 (2023) 083, arXiv:2306.15326 [gr-qc].
  43. S. Carloni, S. Capozziello, J. A. Leach, and P. K. S. Dunsby, “Cosmological dynamics of scalar-tensor gravity,” Class. Quant. Grav. 25 (2008) 035008, arXiv:gr-qc/0701009 [gr-qc].
  44. M. Sami, M. Shahalam, M. Skugoreva, and A. Toporensky, “Cosmological dynamics of non-minimally coupled scalar field system and its late time cosmic relevance,” Phys. Rev. D86 (2012) 103532, arXiv:1207.6691 [hep-th].
  45. M. A. Skugoreva, A. V. Toporensky, and S. Yu. Vernov, “Global stability analysis for cosmological models with nonminimally coupled scalar fields,” Phys. Rev. D90 (2014) no. 6, 064044, arXiv:1404.6226 [gr-qc].
  46. J. Grain and V. Vennin, “Stochastic inflation in phase space: Is slow roll a stochastic attractor?,” JCAP 05 (2017) 045, arXiv:1703.00447 [gr-qc].
  47. E. E. Flanagan, “The Conformal frame freedom in theories of gravitation,” Class. Quant. Grav. 21 (2004) 3817, arXiv:gr-qc/0403063 [gr-qc].
  48. L. Järv, P. Kuusk, M. Saal, and O. Vilson, “Invariant quantities in the scalar-tensor theories of gravitation,” Phys. Rev. D91 (2015) no. 2, 024041, arXiv:1411.1947 [gr-qc].
  49. L. Järv, P. Kuusk, M. Saal, and O. Vilson, “Transformation properties and general relativity regime in scalar–tensor theories,” Class. Quant. Grav. 32 (2015) 235013, arXiv:1504.02686 [gr-qc].
  50. A. Kozak and A. Borowiec, “Palatini frames in scalar–tensor theories of gravity,” Eur. Phys. J. C 79 (2019) no. 4, 335, arXiv:1808.05598 [hep-th].
  51. T. Koivisto, “Covariant conservation of energy momentum in modified gravities,” Class. Quant. Grav. 23 (2006) 4289–4296, arXiv:gr-qc/0505128.
  52. T. Helpin and M. S. Volkov, “Varying the Horndeski Lagrangian within the Palatini approach,” JCAP 01 (2020) 044, arXiv:1906.07607 [hep-th].
  53. P. Kuusk, L. Jarv, and O. Vilson, “Invariant quantities in the multiscalar-tensor theories of gravitation,” Int. J. Mod. Phys. A 31 (2016) no. 02n03, 1641003, arXiv:1509.02903 [gr-qc].
  54. M. Hohmann, L. Jarv, P. Kuusk, E. Randla, and O. Vilson, “Post-Newtonian parameter γ𝛾\gammaitalic_γ for multiscalar-tensor gravity with a general potential,” Phys. Rev. D 94 (2016) no. 12, 124015, arXiv:1607.02356 [gr-qc].
  55. L. Järv, P. Kuusk, and M. Saal, “Scalar-tensor cosmologies: Fixed points of the Jordan frame scalar field,” Phys. Rev. D78 (2008) 083530, arXiv:0807.2159 [gr-qc].
  56. L. Järv, P. Kuusk, and M. Saal, “Potential dominated scalar-tensor cosmologies in the general relativity limit: phase space view,” Phys. Rev. D81 (2010) 104007, arXiv:1003.1686 [gr-qc].
  57. T. Chiba and M. Yamaguchi, “Extended Slow-Roll Conditions and Rapid-Roll Conditions,” JCAP 10 (2008) 021, arXiv:0807.4965 [astro-ph].
  58. K. Akın, A. Savaş Arapoglu, and A. Emrah Yükselci, “Formalizing slow-roll inflation in scalar–tensor theories of gravitation,” Phys. Dark Univ. 30 (2020) 100691, arXiv:2007.10850 [gr-qc].
  59. M. Karčiauskas and J. J. T. Díaz, “Slow-roll inflation in the Jordan frame,” Phys. Rev. D 106 (2022) no. 8, 083526, arXiv:2206.08677 [gr-qc].
  60. V. Faraoni, “Phase space geometry in scalar-tensor cosmology,” Annals Phys. 317 (2005) 366–382, arXiv:gr-qc/0502015 [gr-qc].
  61. M. Postma and M. Volponi, “Equivalence of the Einstein and Jordan frames,” Phys. Rev. D 90 (2014) no. 10, 103516, arXiv:1407.6874 [astro-ph.CO].
  62. S. Karamitsos and A. Pilaftsis, “Frame Covariant Nonminimal Multifield Inflation,” Nucl. Phys. B 927 (2018) 219–254, arXiv:1706.07011 [hep-ph].
  63. A. Karam, T. Pappas, and K. Tamvakis, “Frame-dependence of higher-order inflationary observables in scalar-tensor theories,” Phys. Rev. D 96 (2017) no. 6, 064036, arXiv:1707.00984 [gr-qc].
  64. A. Racioppi and M. Vasar, “On the number of e-folds in the Jordan and Einstein frames,” Eur. Phys. J. Plus 137 (2022) no. 5, 637, arXiv:2111.09677 [gr-qc].
  65. S. Karamitsos, “Beyond the Poles in Attractor Models of Inflation,” JCAP 09 (2019) 022, arXiv:1903.03707 [hep-th].
  66. L. Järv, P. Kuusk, M. Saal, and O. Vilson, “Parametrizations in scalar-tensor theories of gravity and the limit of general relativity,” J. Phys. Conf. Ser. 532 (2014) 012011, arXiv:1501.07781 [gr-qc].
  67. J. Dutta, L. Järv, W. Khyllep, and S. Tõkke, “From inflation to dark energy in scalar-tensor cosmology,” arXiv:2007.06601 [gr-qc].
  68. P. Binetruy, E. Kiritsis, J. Mabillard, M. Pieroni, and C. Rosset, “Universality classes for models of inflation,” JCAP 04 (2015) 033, arXiv:1407.0820 [astro-ph.CO].
  69. M. Pieroni, “β𝛽\betaitalic_β-function formalism for inflationary models with a non minimal coupling with gravity,” JCAP 02 (2016) 012, arXiv:1510.03691 [hep-ph].
  70. A. Karam, S. Karamitsos, and M. Saal, “β𝛽\betaitalic_β-function reconstruction of Palatini inflationary attractors,” JCAP 10 (2021) 068, arXiv:2103.01182 [gr-qc].
  71. A. Racioppi, “Coleman-Weinberg linear inflation: metric vs. Palatini formulation,” JCAP 12 (2017) 041, arXiv:1710.04853 [astro-ph.CO].
  72. A. Racioppi, “New universal attractor in nonminimally coupled gravity: Linear inflation,” Phys. Rev. D 97 (2018) no. 12, 123514, arXiv:1801.08810 [astro-ph.CO].
  73. A. Racioppi, J. Rajasalu, and K. Selke, “Multiple point criticality principle and Coleman-Weinberg inflation,” JHEP 06 (2022) 107, arXiv:2109.03238 [astro-ph.CO].
  74. A. A. Starobinsky, “A New Type of Isotropic Cosmological Models Without Singularity,” Phys. Lett. B 91 (1980) 99–102.
  75. A. Vilenkin, “Classical and Quantum Cosmology of the Starobinsky Inflationary Model,” Phys. Rev. D 32 (1985) 2511.
  76. N. E. Mavromatos and J. Solà Peracaula, “Stringy-running-vacuum-model inflation: from primordial gravitational waves and stiff axion matter to dynamical dark energy,” Eur. Phys. J. ST 230 (2021) no. 9, 2077–2110, arXiv:2012.07971 [hep-ph].
  77. E. Di Valentino and L. Mersini-Houghton, “Testing Predictions of the Quantum Landscape Multiverse 1: The Starobinsky Inflationary Potential,” JCAP 03 (2017) 002, arXiv:1612.09588 [astro-ph.CO].
  78. D. A. Gomes, R. Briffa, A. Kozak, J. Levi Said, M. Saal, and A. Wojnar, “Cosmological constraints of Palatini f(ℛℛ\mathscr{R}script_R) gravity,” JCAP 01 (2024) 011, arXiv:2310.17339 [gr-qc].
  79. S. S. Mishra, V. Sahni, and A. V. Toporensky, “Initial conditions for Inflation in an FRW Universe,” Phys. Rev. D 98 (2018) no. 8, 083538, arXiv:1801.04948 [gr-qc].
  80. S. S. Mishra, D. Müller, and A. V. Toporensky, “Generality of Starobinsky and Higgs inflation in the Jordan frame,” Phys. Rev. D 102 (2020) no. 6, 063523, arXiv:1912.01654 [gr-qc].
  81. T. Terada, “Generalized Pole Inflation: Hilltop, Natural, and Chaotic Inflationary Attractors,” Phys. Lett. B 760 (2016) 674–680, arXiv:1602.07867 [hep-th].
  82. R. Kallosh, A. Linde, and D. Roest, “Superconformal Inflationary α𝛼\alphaitalic_α-Attractors,” JHEP 11 (2013) 198, arXiv:1311.0472 [hep-th].
  83. R. Kallosh and A. Linde, “Planck, LHC, and α𝛼\alphaitalic_α-attractors,” Phys. Rev. D 91 (2015) 083528, arXiv:1502.07733 [astro-ph.CO].
  84. J. J. M. Carrasco, R. Kallosh, and A. Linde, “α𝛼\alphaitalic_α-Attractors: Planck, LHC and Dark Energy,” JHEP 10 (2015) 147, arXiv:1506.01708 [hep-th].
  85. M. Galante, R. Kallosh, A. Linde, and D. Roest, “Unity of Cosmological Inflation Attractors,” Phys. Rev. Lett. 114 (2015) no. 14, 141302, arXiv:1412.3797 [hep-th].
  86. A. B. Goncharov and A. D. Linde, “Chaotic Inflation in Supergravity,” Phys. Lett. B 139 (1984) 27–30.
  87. R. Kallosh and A. Linde, “Universality Class in Conformal Inflation,” JCAP 07 (2013) 002, arXiv:1306.5220 [hep-th].
  88. D. Sloan, K. Dimopoulos, and S. Karamitsos, “T-Model Inflation and Bouncing Cosmology,” Phys. Rev. D 101 (2020) no. 4, 043521, arXiv:1912.00090 [gr-qc].
  89. I. D. Gialamas, A. Karam, A. Lykkas, and T. D. Pappas, “Palatini-Higgs inflation with nonminimal derivative coupling,” Phys. Rev. D 102 (2020) no. 6, 063522, arXiv:2008.06371 [gr-qc].
  90. A. D. Linde, “Fast roll inflation,” JHEP 11 (2001) 052, arXiv:hep-th/0110195.
  91. L. Kofman and S. Mukohyama, “Rapid roll Inflation with Conformal Coupling,” Phys. Rev. D 77 (2008) 043519, arXiv:0709.1952 [hep-th].
  92. J. Martin, H. Motohashi, and T. Suyama, “Ultra Slow-Roll Inflation and the non-Gaussianity Consistency Relation,” Phys. Rev. D 87 (2013) no. 2, 023514, arXiv:1211.0083 [astro-ph.CO].
  93. H. Motohashi, A. A. Starobinsky, and J. Yokoyama, “Inflation with a constant rate of roll,” JCAP 09 (2015) 018, arXiv:1411.5021 [astro-ph.CO].
  94. G. Tasinato, “An analytic approach to non-slow-roll inflation,” Phys. Rev. D 103 (2021) no. 2, 023535, arXiv:2012.02518 [hep-th].
  95. L. Järv, A. Racioppi, and T. Tenkanen, “Palatini side of inflationary attractors,” Phys. Rev. D 97 (2018) no. 8, 083513, arXiv:1712.08471 [gr-qc].
  96. A. De Simone, M. P. Hertzberg, and F. Wilczek, “Running Inflation in the Standard Model,” Phys. Lett. B 678 (2009) 1–8, arXiv:0812.4946 [hep-ph].
  97. K. Finn, S. Karamitsos, and A. Pilaftsis, “Frame Covariance in Quantum Gravity,” Phys. Rev. D 102 (2020) no. 4, 045014, arXiv:1910.06661 [hep-th].
  98. B. M. N. Carter and I. P. Neupane, “Towards inflation and dark energy cosmologies from modified Gauss-Bonnet theory,” JCAP 06 (2006) 004, arXiv:hep-th/0512262.
  99. M. Satoh and J. Soda, “Higher Curvature Corrections to Primordial Fluctuations in Slow-roll Inflation,” JCAP 09 (2008) 019, arXiv:0806.4594 [astro-ph].
  100. T. Kobayashi, M. Yamaguchi, and J. Yokoyama, “Generalized G-inflation: Inflation with the most general second-order field equations,” Prog. Theor. Phys. 126 (2011) 511–529, arXiv:1105.5723 [hep-th].
  101. M. Shaposhnikov, A. Shkerin, I. Timiryasov, and S. Zell, “Higgs inflation in Einstein-Cartan gravity,” JCAP 02 (2021) 008, arXiv:2007.14978 [hep-ph]. [Erratum: JCAP 10, E01 (2021)].
  102. C. Rigouzzo and S. Zell, “Coupling metric-affine gravity to a Higgs-like scalar field,” Phys. Rev. D 106 (2022) no. 2, 024015, arXiv:2204.03003 [hep-th].
Citations (2)

Summary

We haven't generated a summary for this paper yet.