- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Adiabatic initial conditions for a particle gas in cosmology

Guillem Domènech*

  • *Contact author: guillem.domenech@itp.uni-hannover.de
PDF

Phys. Rev. D 113, 043505 – Published 6 February, 2026

DOI: https://doi.org/10.1103/tmdn-s877

Abstract

In view of recent interest in the role of “dark” radiation in cosmology, such as cosmic gravitational waves, sterile neutrinos, and dark photons, we clarify the definition of adiabatic initial conditions in the kinetic theory of gases in an expanding universe. Without assuming any form for the phase space distribution function, we identify two possibilities: a strong and a weak adiabatic initial condition. The strong one corresponds to the standard adiabatic initial conditions, while the weak one is related to the strong via internal isocurvature fluctuations. We show that both types of adiabatic initial conditions are consistent with the separate universe approach, although the latter requires initial internal isocurvature. In passing, we stress the importance of using the particle local momentum in the phase space to define the notion of adiabatic initial conditions. Doing so, we clarify that a gas of gravitons can have adiabatic initial conditions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (81)

  1. N. Aghanim et al. (Planck Collaboration), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020); 652, C4(E) (2021).
  2. A. A. Starobinsky, Spectrum of relic gravitational radiation and the early state of the universe, JETP Lett. 30, 682 (1979).
  3. K. Sato, First-order phase transition of a vacuum and the expansion of the Universe, Mon. Not. R. Astron. Soc. 195, 467 (1981).
  4. A. H. Guth, The inflationary universe: A possible solution to the horizon and flatness problems, Phys. Rev. D 23, 347 (1981).
  5. A. D. Linde, A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems, Phys. Lett. 108B, 389 (1982).
  6. A. Albrecht and P. J. Steinhardt, Cosmology for grand unified theories with radiatively induced symmetry breaking, Phys. Rev. Lett. 48, 1220 (1982).
  7. Y. Akrami et al. (Planck Collaboration), Planck 2018 results. X. Constraints on inflation, Astron. Astrophys. 641, A10 (2020).
  8. H. Kodama and M. Sasaki, Cosmological perturbation theory, Prog. Theor. Phys. Suppl. 78, 1 (1984).
  9. 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. Rep. 215, 203 (1992).
  10. D. Baumann, Inflation, in Theoretical Advanced Study Institute in Elementary Particle Physics: Physics of the Large and the Small (World Scientific Publisher, 2011), pp. 523–686.
  11. M. Sasaki, Inflation and birth of cosmological perturbations, Fundam. Theor. Phys. 177, 305 (2014).
  12. D. Langlois, Isocurvature cosmological perturbations and the CMB, C.R. Phys. 4, 953 (2003).
  13. K. A. Malik and D. Wands, Adiabatic and entropy perturbations with interacting fluids and fields, J. Cosmol. Astropart. Phys. 02 (2005) 007.
  14. K. A. Malik and D. Wands, Cosmological perturbations, Phys. Rep. 475, 1 (2009).
  15. M. C. Guzzetti, N. Bartolo, M. Liguori, and S. Matarrese, Gravitational waves from inflation, Riv. Nuovo Cimento 39, 399 (2016).
  16. R. H. Cyburt, B. D. Fields, K. A. Olive, and E. Skillman, New BBN limits on physics beyond the standard model from He4, Astropart. Phys. 23, 313 (2005).
  17. T. L. Smith, E. Pierpaoli, and M. Kamionkowski, A new cosmic microwave background constraint to primordial gravitational waves, Phys. Rev. Lett. 97, 021301 (2006).
  18. T. J. Clarke, E. J. Copeland, and A. Moss, Constraints on primordial gravitational waves from the cosmic microwave background, J. Cosmol. Astropart. Phys. 10 (2020) 002.
  19. L. Valbusa Dall’Armi, A. Mierna, S. Matarrese, and A. Ricciardone, Adiabatic or non-adiabatic? Unraveling the nature of initial conditions in the cosmological gravitational wave background, arXiv:2307.11043.
  20. L. Valbusa Dall’Armi, A. Mierna, S. Matarrese, and A. Ricciardone, Inflationary initial conditions for the cosmological gravitational wave background, J. Cosmol. Astropart. Phys. 07 (2024) 043.
  21. P. Binetruy, A. Bohe, C. Caprini, and J.-F. Dufaux, Cosmological backgrounds of gravitational waves and eLISA/NGO: Phase transitions, cosmic strings and other sources, J. Cosmol. Astropart. Phys. 06 (2012) 027.
  22. C. Caprini and D. G. Figueroa, Cosmological backgrounds of gravitational waves, Classical Quantum Gravity 35, 163001 (2018).
  23. G. Domènech, Scalar induced gravitational waves review, Universe 7, 398 (2021).
  24. R. Roshan and G. White, Using gravitational waves to see the first second of the Universe, Rev. Mod. Phys. 97, 015001 (2025).
  25. L. Bian et al., Gravitational wave cosmology, Sci. China Phys. Mech. Astron. 69, 210401 (2026).
  26. C. R. Contaldi, Anisotropies of gravitational wave backgrounds: A line of sight approach, Phys. Lett. B 771, 9 (2017).
  27. N. Bartolo, D. Bertacca, S. Matarrese, M. Peloso, A. Ricciardone, A. Riotto, and G. Tasinato, Anisotropies and non-Gaussianity of the cosmological gravitational wave background, Phys. Rev. D 100, 121501 (2019).
  28. N. Bartolo, D. Bertacca, S. Matarrese, M. Peloso, A. Ricciardone, A. Riotto, and G. Tasinato, Characterizing the cosmological gravitational wave background: Anisotropies and non-Gaussianity, Phys. Rev. D 102, 023527 (2020).
  29. J. R. Pritchard and M. Kamionkowski, Cosmic microwave background fluctuations from gravitational waves: An analytic approach, Ann. Phys. (Amsterdam) 318, 2 (2005).
  30. R. A. Isaacson, Gravitational radiation in the limit of high frequency. I. The linear approximation and geometrical optics, Phys. Rev. 166, 1263 (1968).
  31. M. Giovannini, Effective energy density of relic gravitons, Phys. Rev. D 100, 083531 (2019).
  32. A. Negro and S. P. Patil, Hadamard regularization of the graviton stress tensor, Phys. Rev. D 111, 045009 (2025).
  33. A. Ricciardone, L. V. Dall’Armi, N. Bartolo, D. Bertacca, M. Liguori, and S. Matarrese, Cross-correlating astrophysical and cosmological gravitational wave backgrounds with the cosmic microwave background, Phys. Rev. Lett. 127, 271301 (2021).
  34. N. Bartolo, D. Bertacca, V. De Luca, G. Franciolini, S. Matarrese, M. Peloso, A. Ricciardone, A. Riotto, and G. Tasinato, Gravitational wave anisotropies from primordial black holes, J. Cosmol. Astropart. Phys. 02 (2020) 028.
  35. L. Valbusa Dall’Armi, A. Ricciardone, N. Bartolo, D. Bertacca, and S. Matarrese, Imprint of relativistic particles on the anisotropies of the stochastic gravitational-wave background, Phys. Rev. D 103, 023522 (2021).
  36. A. Mierna, S. Matarrese, N. Bartolo, and A. Ricciardone, Non-linear effects on the cosmological gravitational wave background anisotropies, J. Cosmol. Astropart. Phys. 07 (2025) 068.
  37. E. Dimastrogiovanni, M. Fasiello, A. Malhotra, and G. Tasinato, Enhancing gravitational wave anisotropies with peaked scalar sources, J. Cosmol. Astropart. Phys. 01 (2023) 018.
  38. A. Malhotra, E. Dimastrogiovanni, G. Domènech, M. Fasiello, and G. Tasinato, New universal property of cosmological gravitational wave anisotropies, Phys. Rev. D 107, 103502 (2023).
  39. F. Schulze, L. Valbusa Dall’Armi, J. Lesgourgues, A. Ricciardone, N. Bartolo, D. Bertacca, C. Fidler, and S. Matarrese, GW_CLASS: Cosmological gravitational wave background in the cosmic linear anisotropy solving system, J. Cosmol. Astropart. Phys. 10 (2023) 025.
  40. A. D. Dolgov, Neutrinos in cosmology, Phys. Rep. 370, 333 (2002).
  41. S. F. King, Models of neutrino mass, mixing and CP violation, J. Phys. G 42, 123001 (2015).
  42. S. Davidson, E. Nardi, and Y. Nir, Leptogenesis, Phys. Rep. 466, 105 (2008).
  43. A. Riotto and M. Trodden, Recent progress in baryogenesis, Annu. Rev. Nucl. Part. Sci. 49, 35 (1999).
  44. L. Canetti, M. Drewes, and M. Shaposhnikov, Matter and antimatter in the universe, New J. Phys. 14, 095012 (2012).
  45. G. Bertone and D. Hooper, History of dark matter, Rev. Mod. Phys. 90, 045002 (2018).
  46. C. Balazs, T. Bringmann, F. Kahlhoefer, and M. White, A primer on dark matter, Astrophysics 5, 17 (2026).
  47. C. Gerlach, W. Ratzinger, and P. Schwaller, Superhorizon isocurvature as a window into dark matter production, arXiv:2510.21917.
  48. M. Wyman, D. H. Rudd, R. A. Vanderveld, and W. Hu, Neutrinos Help reconcile Planck measurements with the local universe, Phys. Rev. Lett. 112, 051302 (2014).
  49. C. Dvorkin, M. Wyman, D. H. Rudd, and W. Hu, Neutrinos help reconcile Planck measurements with both the early and local universe, Phys. Rev. D 90, 083503 (2014).
  50. B. Leistedt, H. V. Peiris, and L. Verde, No new cosmological concordance with massive sterile neutrinos, Phys. Rev. Lett. 113, 041301 (2014).
  51. L. L. Graef, M. Benetti, and J. S. Alcaniz, Primordial gravitational waves and the H0-tension problem, Phys. Rev. D 99, 043519 (2019).
  52. M. Archidiacono, D. C. Hooper, R. Murgia, S. Bohr, J. Lesgourgues, and M. Viel, Constraining dark matter-dark radiation interactions with CMB, BAO, and Lyman-α, J. Cosmol. Astropart. Phys. 10 (2019) 055.
  53. C. D. Kreisch, F.-Y. Cyr-Racine, and O. Doré, Neutrino puzzle: Anomalies, interactions, and cosmological tensions, Phys. Rev. D 101, 123505 (2020).
  54. N. Schöneberg, J. Lesgourgues, and D. C. Hooper, The BAO+BBN take on the Hubble tension, J. Cosmol. Astropart. Phys. 10 (2019) 029.
  55. N. Blinov and G. Marques-Tavares, Interacting radiation after Planck and its implications for the Hubble Tension, J. Cosmol. Astropart. Phys. 09 (2020) 029.
  56. S. Gariazzo and O. Mena, On the dark radiation role in the Hubble constant tension, arXiv:2306.15067.
  57. M. A. Buen-Abad, Z. Chacko, I. Flood, C. Kilic, G. Marques-Tavares, and T. Youn, Atomic dark matter, interacting dark radiation, and the Hubble tension, J. High Energy Phys. 07 (2025) 084.
  58. E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, In the realm of the Hubble tension—A review of solutions, Classical Quantum Gravity 38, 153001 (2021).
  59. S. Vagnozzi, Seven hints that early-time new physics alone is not sufficient to solve the Hubble tension, Universe 9, 393 (2023).
  60. J.-P. Hu and F.-Y. Wang, Hubble tension: The evidence of new physics, Universe 9, 94 (2023).
  61. E. G. M. Ferreira, Ultra-light dark matter, Astron. Astrophys. Rev. 29, 7 (2021).
  62. F. Chadha-Day, J. Ellis, and D. J. E. Marsh, Axion dark matter: What is it and why now?, Sci. Adv. 8, abj3618 (2022).
  63. M. Fabbrichesi, E. Gabrielli, and G. Lanfranchi, The dark photon, The Physics of the Dark Photon (Springe, Cham, 2020).
  64. D. Cyncynates and Z. J. Weiner, Experimental targets for dark photon dark matter, Phys. Rev. D 111, 103535 (2025).
  65. H. Andreasson, The Einstein-Vlasov System/Kinetic theory, Living Rev. Relativity 14, 4 (2011).
  66. Y. Choquet-Bruhat, Relativistic kinetic theory, in Introduction to General Relativity, Black Holes, and Cosmology (Oxford University Press, New York, 2014).
  67. C.-P. Ma and E. Bertschinger, Cosmological perturbation theory in the synchronous and conformal Newtonian gauges, Astrophys. J. 455, 7 (1995).
  68. M. Bruni, S. Matarrese, S. Mollerach, and S. Sonego, Perturbations of space-time: Gauge transformations and gauge invariance at second order and beyond, Classical Quantum Gravity 14, 2585 (1997).
  69. C. Pitrou, Gauge invariant Boltzmann equation and the fluid limit, Classical Quantum Gravity 24, 6127 (2007).
  70. A. Naruko, C. Pitrou, K. Koyama, and M. Sasaki, Second-order Boltzmann equation: Gauge dependence and gauge invariance, Classical Quantum Gravity 30, 165008 (2013).
  71. R. Saito, A. Naruko, T. Hiramatsu, and M. Sasaki, Geodesic curve-of-sight formulae for the cosmic microwave background: A unified treatment of redshift, time delay, and lensing, J. Cosmol. Astropart. Phys. 10 (2014) 051.
  72. T. Namikawa, A. Naruko, R. Saito, A. Taruya, and D. Yamauchi, Unified approach to secondary effects on the CMB B-mode polarization, J. Cosmol. Astropart. Phys. 10 (2024) 029.
  73. D. Wands, K. A. Malik, D. H. Lyth, and A. R. Liddle, A new approach to the evolution of cosmological perturbations on large scales, Phys. Rev. D 62, 043527 (2000).
  74. J. Yepez, Einstein’s vierbein field theory of curved space, arXiv:1106.2037.
  75. E. Bertschinger, Cosmological dynamics, in Les Houches Summer School on Cosmology (Elsevier Science & Technology Books, 1993), pp. 273–348.
  76. Misao Sasaki’s Lectures on Theory of Cosmological Perturbations can be found at https://www2.yukawa.kyoto-u.ac.jp/~misao.sasaki/PDF/CPT/index.html.
  77. G. Domènech and M. Sasaki, Hamiltonian approach to second order gauge invariant cosmological perturbations, Phys. Rev. D 97, 023521 (2018).
  78. S. Dodelson, Modern Cosmology (Academic Press, Amsterdam, 2003).
  79. S. Pi, M. Sasaki, A. Wang, and J. Wang, Revisiting the ultraviolet tail of the primordial gravitational wave, Phys. Rev. D 110, 103529 (2024).
  80. L. Bordin, P. Creminelli, M. Mirbabayi, and J. Noreña, Tensor squeezed limits and the Higuchi bound, J. Cosmol. Astropart. Phys. 09 (2016) 041.
  81. N. Bartolo, G. B. Carollo, S. Matarrese, L. Pilo, and R. Rollo, The physical content of long tensor modes in cosmology, J. Cosmol. Astropart. Phys. 11 (2022) 034.

Outline

Information

Phys. Rev. D 113, 043505– Published 6 February, 2026
  • Received 10 November 2025
  • Accepted 6 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation