- Open Access
Imprints of reheating dynamics on gravitational waves from phase transitions
Phys. Rev. D 114, 023511 – Published 6 July, 2026
DOI: https://doi.org/10.1103/v9zj-29xj
Abstract
We investigate how perturbative reheating after inflation modifies the primordial gravitational wave (GW) spectrum generated by cosmological phase transitions. Within a specific inflationary setup, we show that the thermodynamic quantities that control the phase transition depend on the effective equation of state of the cosmological background, which is itself set by the form of the inflaton potential during reheating. Assuming reheating proceeds via perturbative dissipation of the inflaton condensate into boson or fermion pairs, we find that phase transitions taking place in this epoch generally produce GW signals that are systematically suppressed compared with the standard radiation-dominated scenario. We also identify characteristic spectral features that may arise in this case, which could serve as distinctive signatures of the modified expansion history during reheating.
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References (226)
- LIGO Scientific, Virgo, and KAGRA Collaborations, Constraints on cosmic strings using data from the third Advanced LIGO–Virgo observing run, Phys. Rev. Lett. 126, 241102 (2021).
- KAGRA, VIRGO, and LIGO Scientific Collaborations, GWTC-3: Compact binary coalescences observed by LIGO and Virgo during the second part of the third observing run, Phys. Rev. X 13, 041039 (2023).
- NANOGrav Collaboration, The NANOGrav 15 yr data set: Evidence for a gravitational-wave background, Astrophys. J. Lett. 951, L8 (2023).
- EPTA and InPTA Collaborations, The second data release from the European pulsar timing array—III. Search for gravitational wave signals, Astron. Astrophys. 678, A50 (2023).
- D. J. Reardon et al., Search for an isotropic gravitational-wave background with the parkes pulsar timing array, Astrophys. J. Lett. 951, L6 (2023).
- H. Xu et al., Searching for the nano-hertz stochastic gravitational wave background with the Chinese pulsar timing array data release I, Res. Astron. Astrophys. 23, 075024 (2023).
- LISA Cosmology Working Group Collaboration, Cosmology with the laser interferometer space antenna, Living Rev. Relativity 26, 5 (2023).
- NANOGrav Collaboration, The NANOGrav 15 yr data set: Search for signals from new physics, Astrophys. J. Lett. 951, L11 (2023).
- J. Ellis, M. Fairbairn, G. Franciolini, G. Hütsi, A. Iovino, M. Lewicki et al., What is the source of the PTA GW signal?, Phys. Rev. D 109, 023522 (2024).
- K. Kajantie, M. Laine, K. Rummukainen, and M. E. Shaposhnikov, Is there a hot electroweak phase transition at ?, Phys. Rev. Lett. 77, 2887 (1996).
- M. Laine and A. Vuorinen, Basics of Thermal Field Theory (Springer, New York, 2016), Vol. 925, 10.1007/978-3-319-31933-9.
- C. Caprini et al., Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions, J. Cosmol. Astropart. Phys. 04 (2016) 001.
- C. Caprini and D. G. Figueroa, Cosmological backgrounds of gravitational waves, Classical Quantum Gravity 35, 163001 (2018).
- C. Caprini et al., Detecting gravitational waves from cosmological phase transitions with LISA: An update, J. Cosmol. Astropart. Phys. 03 (2020) 024.
- P. Athron, C. Balázs, A. Fowlie, L. Morris, and L. Wu, Cosmological phase transitions: From perturbative particle physics to gravitational waves, Prog. Part. Nucl. Phys. 135, 104094 (2024).
- R. Roshan and G. White, Using gravitational waves to see the first second of the Universe, Rev. Mod. Phys. 97, 015001 (2025).
- D. Croon and D. J. Weir, Gravitational waves from phase transitions, Contemp. Phys. 65, 75 (2024).
- M. Kamionkowski, A. Kosowsky, and M. S. Turner, Gravitational radiation from first order phase transitions, Phys. Rev. D 49, 2837 (1994).
- R. Apreda, M. Maggiore, A. Nicolis, and A. Riotto, Gravitational waves from electroweak phase transitions, Nucl. Phys. B631, 342 (2002).
- C. Grojean and G. Servant, Gravitational waves from phase transitions at the electroweak scale and beyond, Phys. Rev. D 75, 043507 (2007).
- A. Ashoorioon and T. Konstandin, Strong electroweak phase transitions without collider traces, J. High Energy Phys. 07 (2009) 086.
- M. Kakizaki, S. Kanemura, and T. Matsui, Gravitational waves as a probe of extended scalar sectors with the first order electroweak phase transition, Phys. Rev. D 92, 115007 (2015).
- V. Vaskonen, Electroweak baryogenesis and gravitational waves from a real scalar singlet, Phys. Rev. D 95, 123515 (2017).
- G. C. Dorsch, S. J. Huber, T. Konstandin, and J. M. No, A second Higgs doublet in the early universe: Baryogenesis and gravitational waves, J. Cosmol. Astropart. Phys. 05 (2017) 052.
- A. Beniwal, M. Lewicki, J. D. Wells, M. White, and A. G. Williams, Gravitational wave, collider and dark matter signals from a scalar singlet electroweak baryogenesis, J. High Energy Phys. 08 (2017) 108.
- J. Ellis, M. Lewicki, and J. M. No, On the maximal strength of a first-order electroweak phase transition and its gravitational wave signal, J. Cosmol. Astropart. Phys. 04 (2019) 003.
- M. Lewicki, M. Merchand, and M. Zych, Electroweak bubble wall expansion: Gravitational waves and baryogenesis in standard model-like thermal plasma, J. High Energy Phys. 02 (2022) 017.
- A. Chatterjee, A. Datta, and S. Roy, Electroweak phase transition in the -invariant NMSSM: Implications of LHC and dark matter searches and prospects of detecting the gravitational waves, J. High Energy Phys. 06 (2022) 108.
- J. Ellis, M. Lewicki, M. Merchand, J. M. No, and M. Zych, The scalar singlet extension of the standard model: Gravitational waves versus baryogenesis, J. High Energy Phys. 01 (2023) 093.
- P. Ghosh, T. Ghosh, and S. Roy, Interplay among gravitational waves, dark matter and collider signals in the singlet scalar extended type-II seesaw model, J. High Energy Phys. 10 (2023) 057.
- P. Borah, P. Ghosh, S. Roy, and A. K. Saha, Electroweak phase transition in a right-handed neutrino superfield extended NMSSM, J. High Energy Phys. 08 (2023) 029.
- J. E. Camargo-Molina, R. Enberg, and J. Löfgren, A new perspective on the electroweak phase transition in the standard model effective field theory, J. High Energy Phys. 10 (2021) 127.
- E. Camargo-Molina, R. Enberg, and J. Löfgren, A catalog of first-order electroweak phase transitions in the standard model effective field theory, J. High Energy Phys. 08 (2025) 113.
- P. Schwaller, Gravitational waves from a dark phase transition, Phys. Rev. Lett. 115, 181101 (2015).
- J. Jaeckel, V. V. Khoze, and M. Spannowsky, Hearing the signal of dark sectors with gravitational wave detectors, Phys. Rev. D 94, 103519 (2016).
- M. Breitbach, J. Kopp, E. Madge, T. Opferkuch, and P. Schwaller, Dark, cold, and noisy: Constraining secluded hidden sectors with gravitational waves, J. Cosmol. Astropart. Phys. 07 (2019) 007.
- P. S. B. Dev, F. Ferrer, Y. Zhang, and Y. Zhang, Gravitational waves from first-order phase transition in a simple axion-like particle model, J. Cosmol. Astropart. Phys. 11 (2019) 006.
- J. B. Dent, B. Dutta, S. Ghosh, J. Kumar, and J. Runburg, Sensitivity to dark sector scales from gravitational wave signatures, J. High Energy Phys. 08 (2022) 300.
- E. Morgante, N. Ramberg, and P. Schwaller, Gravitational waves from dark SU(3) Yang-Mills theory, Phys. Rev. D 107, 036010 (2023).
- R. Pasechnik, M. Reichert, F. Sannino, and Z.-W. Wang, Gravitational waves from composite dark sectors, J. High Energy Phys. 02 (2024) 159.
- F. Koutroulis, M. McCullough, M. Merchand, S. Pokorski, and K. Sakurai, Phases of Pseudo-Nambu-Goldstone bosons, J. High Energy Phys. 05 (2024) 095.
- P. Di Bari and M. H. Rahat, Split Majoron model confronts the NANOGrav signal and cosmological tensions, Phys. Rev. D 110, 055019 (2024).
- W.-Z. Feng, J. Li, and P. Nath, Cosmologically consistent analysis of gravitational waves from hidden sectors, Phys. Rev. D 110, 015020 (2024).
- A. Banik, Y. Cui, Y.-D. Tsai, and Y. Tsai, The sound of dark sectors in pulsar timing arrays, arXiv:2412.16282.
- S. Balan, T. Bringmann, F. Kahlhoefer, J. Matuszak, and C. Tasillo, Sub-GeV dark matter and nano-Hertz gravitational waves from a classically conformal dark sector, J. Cosmol. Astropart. Phys. 08 (2025) 062.
- M. Kierkla, A. Karam, and B. Swiezewska, Conformal model for gravitational waves and dark matter: A status update, J. High Energy Phys. 03 (2023) 007.
- T. Bringmann, T. Konstandin, J. Matuszak, K. Schmidt-Hoberg, and C. Tasillo, Tuning the violins: Dark sector phase transition models for the PTA signal, arXiv:2602.09092.
- M. Kierkla, B. Swiezewska, T. V. I. Tenkanen, and J. V. D. Vis, Gravitational waves from supercooled phase transitions: Dimensional transmutation meets dimensional reduction, J. High Energy Phys. 02 (2024) 234.
- L. Sagunski, P. Schicho, and D. Schmitt, Supercool exit: Gravitational waves from QCD-triggered conformal symmetry breaking, Phys. Rev. D 107, 123512 (2023).
- D. Schmitt and L. Sagunski, QCD-sourced tachyonic phase transition in a supercooled Universe, J. Cosmol. Astropart. Phys. 02 (2025) 075.
- G. Barenboim and W.-I. Park, Gravitational waves from first order phase transitions as a probe of an early matter domination era and its inverse problem, Phys. Lett. B 759, 430 (2016).
- H.-K. Guo, K. Sinha, D. Vagie, and G. White, Phase transitions in an expanding universe: Stochastic gravitational waves in standard and non-standard histories, J. Cosmol. Astropart. Phys. 01 (2021) 001.
- J. Ellis, M. Lewicki, and V. Vaskonen, Updated predictions for gravitational waves produced in a strongly supercooled phase transition, J. Cosmol. Astropart. Phys. 11 (2020) 020.
- A. Hook, G. Marques-Tavares, and D. Racco, Causal gravitational waves as a probe of free streaming particles and the expansion of the Universe, J. High Energy Phys. 02 (2021) 117.
- M. A. Buen-Abad, J. H. Chang, and A. Hook, Gravitational wave signatures from reheating, Phys. Rev. D 108, 036006 (2023).
- M. Kierkla, G. Laverda, M. Lewicki, A. Mantziris, M. Piani, J. Rubio, and M. Zych, From Hubble to bubble, J. High Energy Phys. 11 (2023) 077.
- J. B. Dent, B. Dutta, and M. Rai, Imprints of early universe cosmology on gravitational waves, J. High Energy Phys. 03 (2025) 098.
- D. Bettoni, G. Laverda, A. Lopez-Eiguren, and J. Rubio, Hubble-induced phase transitions: Gravitational-wave imprint of Ricci reheating from lattice simulations, J. Cosmol. Astropart. Phys. 03 (2025) 027.
- Y. Xiao, H.-K. Guo, J.-H. Hu, J. M. Yang, and Y. Zhang, Growth of the gravitational wave spectrum from sound waves in a universe with a generic expansion rate, Phys. Rev. D 112, L081302 (2025).
- A. Banik, N. Bernal, and F. Hajkarim, Primordial gravitational waves from phase transitions during reheating, arXiv:2506.02116.
- D. Racco and A. Stanzione, Impact of supercooling on direct searches for dark matter and gravitational wave backgrounds, arXiv:2512.16809.
- G. Laverda and J. Rubio, Higgs-induced gravitational waves: The interplay of non-minimal couplings, kination and top quark mass, J. High Energy Phys. 08 (2025) 203.
- M. A. G. Garcia, K. Kaneta, Y. Mambrini, and K. A. Olive, Inflaton oscillations and post-inflationary reheating, J. Cosmol. Astropart. Phys. 04 (2021) 012.
- B. Barman, N. Bernal, Y. Xu, and Ó. Zapata, Gravitational wave from graviton Bremsstrahlung during reheating, J. Cosmol. Astropart. Phys. 05 (2023) 019.
- N. Bernal, S. Cléry, Y. Mambrini, and Y. Xu, Probing reheating with graviton bremsstrahlung, J. Cosmol. Astropart. Phys. 01 (2024) 065.
- Y. Xu, Constraining axion and ALP dark matter from misalignment during reheating, Phys. Rev. D 108, 083536 (2023).
- M. Becker, E. Copello, J. Harz, J. Lang, and Y. Xu, Confronting dark matter freeze-in during reheating with constraints from inflation, J. Cosmol. Astropart. Phys. 01 (2024) 053.
- D. Hooper, G. Krnjaic, A. J. Long, and S. D. Mcdermott, Can the inflaton also be a weakly interacting massive particle?, Phys. Rev. Lett. 122, 091802 (2019).
- M. A. G. Garcia, Y. Mambrini, K. A. Olive, and S. Verner, On the realization of WIMPflation, J. Cosmol. Astropart. Phys. 10 (2021) 061.
- R. Kallosh and A. Linde, Universality class in conformal inflation, J. Cosmol. Astropart. Phys. 07 (2013) 002.
- R. Kallosh, A. Linde, and D. Roest, Superconformal inflationary -attractors, J. High Energy Phys. 11 (2013) 198.
- R. Kallosh and A. Linde, Non-minimal inflationary attractors, J. Cosmol. Astropart. Phys. 10 (2013) 033.
- A. A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. 91B, 99 (1980).
- A. A. Starobinsky, Nonsingular model of the universe with the quantum gravitational de Sitter stage and its observational consequences, in Second Seminar on Quantum Gravity (Union of Soviet Socialist Republics, Moscow, 1981).
- A. A. Starobinsky, The perturbation spectrum evolving from a nonsingular initially de-Sitter cosmology and the microwave background anisotropy, Sov. Astron. Lett. 9, 302 (1983).
- L. A. Kofman, A. D. Linde, and A. A. Starobinsky, Inflationary Universe generated by the combined action of a scalar field and gravitational vacuum polarization, Phys. Lett. 157B, 361 (1985).
- S. Bhattacharya, K. Dutta, M. R. Gangopadhyay, and A. Maharana, -attractor inflation: Models and predictions, Phys. Rev. D 107, 103530 (2023).
- A. Chakraborty, M. R. Haque, D. Maity, and R. Mondal, Inflaton phenomenology via reheating in light of primordial gravitational waves and the latest BICEP/Keck data, Phys. Rev. D 108, 023515 (2023).
- M. R. Haque, S. Pal, and D. Paul, ACT DR6 insights on the inflationary attractor models and reheating, arXiv:2505.01517.
- Atacama Cosmology Telescope Collaboration, The atacama cosmology telescope: DR6 power spectra, likelihoods and parameters, J. Cosmol. Astropart. Phys. 11 (2025) 062.
- Atacama Cosmology Telescope Collaboration, The atacama cosmology telescope: DR6 constraints on extended cosmological models, J. Cosmol. Astropart. Phys. 11 (2025) 063.
- G. German and J. C. Hidalgo, Viability of generalized a-inflation from Planck, ACT, and DESI data, arXiv:2508.01017.
- J. Ellis, M. A. G. Garcia, K. A. Olive, and S. Verner, Constraints on attractor models of inflation and reheating from Planck, BICEP/Keck, ACT DR6, and SPT-3G data, arXiv:2510.18656.
- M. S. Turner, Coherent scalar field oscillations in an expanding Universe, Phys. Rev. D 28, 1243 (1983).
- S. Sarkar, Big bang nucleosynthesis and physics beyond the standard model, Rep. Prog. Phys. 59, 1493 (1996).
- M. Kawasaki, K. Kohri, and N. Sugiyama, MeV scale reheating temperature and thermalization of neutrino background, Phys. Rev. D 62, 023506 (2000).
- S. Hannestad, What is the lowest possible reheating temperature?, Phys. Rev. D 70, 043506 (2004).
- F. De Bernardis, L. Pagano, and A. Melchiorri, New constraints on the reheating temperature of the universe after WMAP-5, Astropart. Phys. 30, 192 (2008).
- P. F. de Salas, M. Lattanzi, G. Mangano, G. Miele, S. Pastor, and O. Pisanti, Bounds on very low reheating scenarios after Planck, Phys. Rev. D 92, 123534 (2015).
- T. Hasegawa, N. Hiroshima, K. Kohri, R. S. L. Hansen, T. Tram, and S. Hannestad, MeV-scale reheating temperature and thermalization of oscillating neutrinos by radiative and hadronic decays of massive particles, J. Cosmol. Astropart. Phys. 12 (2019) 012.
- Y. Shtanov, J. H. Traschen, and R. H. Brandenberger, Universe reheating after inflation, Phys. Rev. D 51, 5438 (1995).
- K. Ichikawa, T. Suyama, T. Takahashi, and M. Yamaguchi, Primordial curvature fluctuation and its non-gaussianity in models with modulated reheating, Phys. Rev. D 78, 063545 (2008).
- K. D. Lozanov and M. A. Amin, Equation of state and duration to radiation domination after inflation, Phys. Rev. Lett. 119, 061301 (2017).
- K. D. Lozanov and M. A. Amin, Self-resonance after inflation: Oscillons, transients and radiation domination, Phys. Rev. D 97, 023533 (2018).
- L. Kofman, A. D. Linde, and A. A. Starobinsky, Towards the theory of reheating after inflation, Phys. Rev. D 56, 3258 (1997).
- J. F. Dufaux, G. N. Felder, L. Kofman, M. Peloso, and D. Podolsky, Preheating with trilinear interactions: Tachyonic resonance, J. Cosmol. Astropart. Phys. 07 (2006) 006.
- B. Barman, N. Bernal, and J. Rubio, Two or three things particle physicists (mis)understand about (pre)heating, Nucl. Phys. B1018, 116996 (2025).
- P. B. Greene, L. Kofman, A. D. Linde, and A. A. Starobinsky, Structure of resonance in preheating after inflation, Phys. Rev. D 56, 6175 (1997).
- A. M. Green, Supersymmetry and primordial black hole abundance constraints, Phys. Rev. D 60, 063516 (1999).
- M. A. Amin, R. Easther, H. Finkel, R. Flauger, and M. P. Hertzberg, Oscillons after inflation, Phys. Rev. Lett. 108, 241302 (2012).
- D. G. Figueroa and F. Torrenti, Parametric resonance in the early Universe—a fitting analysis, J. Cosmol. Astropart. Phys. 02 (2017) 001.
- M. A. G. Garcia and M. Pierre, Reheating after inflaton fragmentation, J. Cosmol. Astropart. Phys. 11 (2023) 004.
- M. A. G. Garcia, M. Gross, Y. Mambrini, K. A. Olive, M. Pierre, and J.-H. Yoon, Effects of fragmentation on post-inflationary reheating, J. Cosmol. Astropart. Phys. 12 (2023) 028.
- M. R. Haque and D. Maity, Gravitational reheating, arXiv:2201.02348.
- S. Clery, Y. Mambrini, K. A. Olive, A. Shkerin, and S. Verner, Gravitational portals with nonminimal couplings, Phys. Rev. D 105, 095042 (2022).
- R. T. Co, Y. Mambrini, and K. A. Olive, Inflationary gravitational leptogenesis, arXiv:2205.01689.
- B. Barman, S. Cléry, R. T. Co, Y. Mambrini, and K. A. Olive, Gravity as a portal to reheating, leptogenesis and dark matter, J. High Energy Phys. 12 (2022) 072.
- M. R. Haque, D. Maity, and R. Mondal, WIMPs, FIMPs, and inflaton phenomenology via reheating, CMB and , arXiv:2301.01641.
- O. Gould, Real scalar phase transitions: A nonperturbative analysis, J. High Energy Phys. 04 (2021) 057.
- A. Ekstedt, Convergence of the nucleation rate for first-order phase transitions, Phys. Rev. D 106, 095026 (2022).
- M. Kierkla, P. Schicho, B. Swiezewska, T. V. I. Tenkanen, and J. van de Vis, Finite-temperature bubble nucleation with shifting scale hierarchies, J. High Energy Phys. 07 (2025) 153.
- A. Ekstedt, P. Schicho, and T. V. I. Tenkanen, Cosmological phase transitions at three loops: The final verdict on perturbation theory, arXiv:2405.18349.
- F. Bernardo, P. Klose, P. Schicho, and T. V. I. Tenkanen, Higher-dimensional operators at finite temperature affect gravitational-wave predictions, J. High Energy Phys. 08 (2025) 109.
- F. Bernardo, M. Chala, L. Gil, and P. Schicho, Hard thermal contributions to phase transition observables at NNLO, arXiv:2602.06962.
- M. Chala, J. C. Criado, L. Gil, and J. L. Miras, Higher-order-operator corrections to phase-transition parameters in dimensional reduction, J. High Energy Phys. 10 (2024) 025.
- P. Navarrete, R. Paatelainen, K. Seppänen, and T. V. I. Tenkanen, Cosmological phase transitions without high-temperature expansions, J. High Energy Phys. 01 (2026) 113.
- O. Gould and T. V. I. Tenkanen, Perturbative effective field theory expansions for cosmological phase transitions, J. High Energy Phys. 01 (2024) 048.
- A. Ekstedt, P. Schicho, and T. V. I. Tenkanen, DRalgo: A package for effective field theory approach for thermal phase transitions, Comput. Phys. Commun. 288, 108725 (2023).
- M. Kierkla, N. Ramberg, P. Schicho, and D. Schmitt, Theoretical uncertainties for primordial black holes from cosmological phase transitions, arXiv:2506.15496.
- J. Löfgren, Stop comparing resummation methods, J. Phys. G 50, 125008 (2023).
- J. Hirvonen, J. Löfgren, M. J. Ramsey-Musolf, P. Schicho, and T. V. I. Tenkanen, Computing the gauge-invariant bubble nucleation rate in finite temperature effective field theory, J. High Energy Phys. 07 (2022) 135.
- J. Löfgren, M. J. Ramsey-Musolf, P. Schicho, and T. V. I. Tenkanen, Nucleation at finite temperature: A gauge-invariant perturbative framework, Phys. Rev. Lett. 130, 251801 (2023).
- D. Croon, O. Gould, P. Schicho, T. V. I. Tenkanen, and G. White, Theoretical uncertainties for cosmological first-order phase transitions, J. High Energy Phys. 04 (2021) 055.
- S. R. Coleman, The fate of the false vacuum. 1. Semiclassical theory, Phys. Rev. D 15, 2929 (1977).
- C. G. Callan, Jr. and S. R. Coleman, The fate of the false vacuum. 2. First quantum corrections, Phys. Rev. D 16, 1762 (1977).
- A. D. Linde, Fate of the false vacuum at finite temperature: Theory and applications, Phys. Lett. 100B, 37 (1981).
- A. D. Linde, Decay of the false vacuum at finite temperature, Nucl. Phys. B216, 421 (1983).
- A. Ekstedt, Higher-order corrections to the bubble-nucleation rate at finite temperature, Eur. Phys. J. C 82, 173 (2022).
- A. Ekstedt, Bubble nucleation to all orders, J. High Energy Phys. 08 (2022) 115.
- A. Ekstedt, O. Gould, and J. Hirvonen, BubbleDet: A python package to compute functional determinants for bubble nucleation, J. High Energy Phys. 12 (2023) 056.
- M. Carosi, On equivalent methods for functional determinants, arXiv:2601.08686.
- G. V. Dunne, Functional determinants in quantum field theory, J. Phys. A 41, 304006 (2008).
- L. Brown, S. Profumo, A. Gangadharan, and Z. S. Koç, Louder gravitational wave bang from a fast-expanding universe, Phys. Rev. D 112, 023509 (2025).
- J. R. Espinosa, T. Konstandin, J. M. No, and G. Servant, Energy budget of cosmological first-order phase transitions, J. Cosmol. Astropart. Phys. 06 (2010) 028.
- S. De Curtis, L. D. Rose, A. Guiggiani, Á. G. Muyor, and G. Panico, Bubble wall dynamics at the electroweak phase transition, J. High Energy Phys. 03 (2022) 163.
- S. De Curtis, L. Delle Rose, A. Guiggiani, Á. Gil Muyor, and G. Panico, Collision integrals for cosmological phase transitions, J. High Energy Phys. 05 (2023) 194.
- S. De Curtis, L. Delle Rose, A. Guiggiani, Á. Gil Muyor, and G. Panico, Non-linearities in cosmological bubble wall dynamics, J. High Energy Phys. 05 (2024) 009.
- C. Branchina, A. Conaci, L. Delle Rose, and S. De Curtis, Bubble wall velocity with out-of-equilibrium corrections, Phys. Rev. D 113, 035024 (2026).
- C. Branchina, A. Conaci, L. Delle Rose, and S. De Curtis, Electroweak phase transition and bubble wall velocity in local thermal equilibrium, Phys. Rev. D 112, 095008 (2025).
- G. C. Dorsch and D. A. Pinto, Bubble wall velocities with an extended fluid Ansatz, J. Cosmol. Astropart. Phys. 04 (2024) 027.
- G. C. Dorsch, T. Konstandin, E. Perboni, and D. A. Pinto, Non-singular solutions to the Boltzmann equation with a fluid Ansatz, J. Cosmol. Astropart. Phys. 04 (2025) 033.
- A. Ekstedt, O. Gould, J. Hirvonen, B. Laurent, L. Niemi, P. Schicho, and J. van de Vis, How fast does the wallgo? A package for computing wall velocities in first-order phase transitions, J. High Energy Phys. 04 (2025) 101.
- J. van de Vis, P. Schicho, L. Niemi, B. Laurent, J. Hirvonen, and O. Gould, WallGo investigates: Theoretical uncertainties in the bubble wall velocity, arXiv:2510.27691.
- W.-Y. Ai, B. Garbrecht, and C. Tamarit, Bubble wall velocities in local equilibrium, J. Cosmol. Astropart. Phys. 03 (2022) 015.
- W.-Y. Ai, B. Laurent, and J. van de Vis, Bounds on the bubble wall velocity, J. High Energy Phys. 02 (2025) 119.
- A. Dashko and A. Ekstedt, Bubble-wall speed with loop corrections, J. High Energy Phys. 03 (2025) 024.
- A. Ekstedt, T. Konstandin, and J. van de Vis, Scalar damping in cosmological phase transitions, arXiv:2512.16663.
- T. Krajewski, M. Lewicki, I. Nałęcz, and M. Zych, Steady-state bubbles beyond local thermal equilibrium, J. High Energy Phys. 06 (2025) 118.
- T. Krajewski, M. Lewicki, and M. Zych, Bubble-wall velocity in local thermal equilibrium: Hydrodynamical simulations vs analytical treatment, J. High Energy Phys. 05 (2024) 011.
- T. Krajewski, M. Lewicki, M. Vasar, V. Vaskonen, H. Veermäe, and M. Zych, Thermalization effects on the dynamics of growing vacuum bubbles, J. High Energy Phys. 03 (2025) 178.
- T. Krajewski, M. Lewicki, and M. Zych, Hydrodynamical constraints on the bubble wall velocity, Phys. Rev. D 108, 103523 (2023).
- W.-Y. Ai, M. Carosi, B. Garbrecht, C. Tamarit, and M. Vanvlasselaer, Bubble wall dynamics from nonequilibrium quantum field theory, J. High Energy Phys. 08 (2025) 077.
- J. Ellis, M. Lewicki, J. M. No, and V. Vaskonen, Gravitational wave energy budget in strongly supercooled phase transitions, J. Cosmol. Astropart. Phys. 06 (2019) 024.
- M. Lewicki and V. Vaskonen, On bubble collisions in strongly supercooled phase transitions, Phys. Dark Universe 30, 100672 (2020).
- M. Lewicki and V. Vaskonen, Gravitational wave spectra from strongly supercooled phase transitions, Eur. Phys. J. C 80, 1003 (2020).
- M. Lewicki and V. Vaskonen, Gravitational waves from colliding vacuum bubbles in gauge theories, Eur. Phys. J. C 81, 437 (2021).
- M. Lewicki and V. Vaskonen, Gravitational waves from bubble collisions and fluid motion in strongly supercooled phase transitions, Eur. Phys. J. C 83, 109 (2023).
- M. Lewicki and V. Vaskonen, Impact of cosmic expansion on gravitational wave spectra from strongly supercooled first-order phase transitions, arXiv:2511.15687.
- M. Hindmarsh, S. J. Huber, K. Rummukainen, and D. J. Weir, Gravitational waves from the sound of a first order phase transition, Phys. Rev. Lett. 112, 041301 (2014).
- M. Hindmarsh, S. J. Huber, K. Rummukainen, and D. J. Weir, Numerical simulations of acoustically generated gravitational waves at a first order phase transition, Phys. Rev. D 92, 123009 (2015).
- M. Hindmarsh, Sound shell model for acoustic gravitational wave production at a first-order phase transition in the early Universe, Phys. Rev. Lett. 120, 071301 (2018).
- M. Hindmarsh, S. J. Huber, K. Rummukainen, and D. J. Weir, Shape of the acoustic gravitational wave power spectrum from a first order phase transition, Phys. Rev. D 96, 103520 (2017).
- M. Hindmarsh and M. Hijazi, Gravitational waves from first order cosmological phase transitions in the sound shell model, J. Cosmol. Astropart. Phys. 12 (2019) 062.
- R. Jinno, T. Konstandin, and H. Rubira, A hybrid simulation of gravitational wave production in first-order phase transitions, J. Cosmol. Astropart. Phys. 04 (2021) 014.
- C. Gowling and M. Hindmarsh, Observational prospects for phase transitions at LISA: Fisher matrix analysis, J. Cosmol. Astropart. Phys. 10 (2021) 039.
- A. Roper Pol, S. Procacci, and C. Caprini, Characterization of the gravitational wave spectrum from sound waves within the sound shell model, Phys. Rev. D 109, 063531 (2024).
- R. Sharma, J. Dahl, A. Brandenburg, and M. Hindmarsh, Shallow relic gravitational wave spectrum with acoustic peak, J. Cosmol. Astropart. Phys. 12 (2023) 042.
- J. Correia, M. Hindmarsh, K. Rummukainen, and D. J. Weir, Gravitational waves from strong first-order phase transitions, Phys. Rev. D 112, 123546 (2025).
- R. Jinno, T. Konstandin, H. Rubira, and I. Stomberg, Higgsless simulations of cosmological phase transitions and gravitational waves, J. Cosmol. Astropart. Phys. 02 (2023) 011.
- C. Caprini, R. Jinno, T. Konstandin, A. Roper Pol, H. Rubira, and I. Stomberg, Gravitational waves from first-order phase transitions: From weak to strong, J. High Energy Phys. 07 (2025) 217.
- LISA Cosmology Working Group Collaboration, Gravitational waves from first-order phase transitions in LISA: Reconstruction pipeline and physics interpretation, J. Cosmol. Astropart. Phys. 10 (2024) 020.
- B. Laurent and J. M. Cline, Fluid equations for fast-moving electroweak bubble walls, Phys. Rev. D 102, 063516 (2020).
- J. M. Cline, A. Friedlander, D.-M. He, K. Kainulainen, B. Laurent, and D. Tucker-Smith, Baryogenesis and gravity waves from a UV-completed electroweak phase transition, Phys. Rev. D 103, 123529 (2021).
- B. Laurent and J. M. Cline, First principles determination of bubble wall velocity, Phys. Rev. D 106, 023501 (2022).
- R. Allahverdi et al., The first three seconds: A review of possible expansion histories of the early universe, Open J. Astrophys. 4, astro.2006.16182 (2021).
- Y. Gouttenoire, G. Servant, and P. Simakachorn, Kination cosmology from scalar fields and gravitational-wave signatures, arXiv:2111.01150.
- C. Caprini, R. Durrer, T. Konstandin, and G. Servant, General properties of the gravitational wave spectrum from phase transitions, Phys. Rev. D 79, 083519 (2009).
- R.-G. Cai, S. Pi, and M. Sasaki, Universal infrared scaling of gravitational wave background spectra, Phys. Rev. D 102, 083528 (2020).
- LISA Collaboration, Laser interferometer space antenna, arXiv:1702.00786.
- LISA Collaboration, LISA definition study report, arXiv:2402.07571.
- M. Punturo et al., The Einstein telescope: A third-generation gravitational wave observatory, Classical Quantum Gravity 27, 194002 (2010).
- S. Hild et al., Sensitivity studies for third-generation gravitational wave observatories, Classical Quantum Gravity 28, 094013 (2011).
- S. Dodelson and M. S. Turner, Nonequilibrium neutrino statistical mechanics in the expanding universe, Phys. Rev. D 46, 3372 (1992).
- S. Hannestad and J. Madsen, Neutrino decoupling in the early universe, Phys. Rev. D 52, 1764 (1995).
- A. D. Dolgov, S. H. Hansen, and D. V. Semikoz, Nonequilibrium corrections to the spectra of massless neutrinos in the early universe, Nucl. Phys. B503, 426 (1997).
- G. Mangano, G. Miele, S. Pastor, T. Pinto, O. Pisanti, and P. D. Serpico, Relic neutrino decoupling including flavor oscillations, Nucl. Phys. B729, 221 (2005).
- P. F. de Salas and S. Pastor, Relic neutrino decoupling with flavour oscillations revisited, J. Cosmol. Astropart. Phys. 07 (2016) 051.
- M. Escudero Abenza, Precision early universe thermodynamics made simple: and neutrino decoupling in the Standard Model and beyond, J. Cosmol. Astropart. Phys. 05 (2020) 048.
- K. Akita and M. Yamaguchi, A precision calculation of relic neutrino decoupling, J. Cosmol. Astropart. Phys. 08 (2020) 012.
- J. Froustey, C. Pitrou, and M. C. Volpe, Neutrino decoupling including flavour oscillations and primordial nucleosynthesis, J. Cosmol. Astropart. Phys. 12 (2020) 015.
- J. J. Bennett, G. Buldgen, P. F. De Salas, M. Drewes, S. Gariazzo, S. Pastor, and Y. Y. Y. Wong, Towards a precision calculation of in the standard model II: Neutrino decoupling in the presence of flavour oscillations and finite-temperature QED, J. Cosmol. Astropart. Phys. 04 (2021) 073.
- M. Maggiore, Gravitational wave experiments and early universe cosmology, Phys. Rep. 331, 283 (2000).
- L. A. Boyle and A. Buonanno, Relating gravitational wave constraints from primordial nucleosynthesis, pulsar timing, laser interferometers, and the CMB: Implications for the early Universe, Phys. Rev. D 78, 043531 (2008).
- S. Kuroyanagi, T. Takahashi, and S. Yokoyama, Blue-tilted tensor spectrum and thermal history of the Universe, J. Cosmol. Astropart. Phys. 02 (2015) 003.
- D. G. Figueroa and E. H. Tanin, Ability of LIGO and LISA to probe the equation of state of the early Universe, J. Cosmol. Astropart. Phys. 08 (2019) 011.
- Planck Collaboration, Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020).
- K. Abazajian et al., CMB-S4 science case, reference design, and project plan, arXiv:1907.04473.
- CMB-HD Collaboration, Snowmass2021 CMB-HD white paper, arXiv:2203.05728.
- COrE Collaboration, COrE (cosmic origins explorer) a white paper, arXiv:1102.2181.
- EUCLID Collaboration, Euclid definition study report, arXiv:1110.3193.
- LIGO Scientific Collaboration, Exploring the sensitivity of next generation gravitational wave detectors, Classical Quantum Gravity 34, 044001 (2017).
- D. Reitze et al., Cosmic explorer: The U.S. contribution to gravitational-wave astronomy beyond LIGO, Bull. Am. Astron. Soc. 51, 035 (2019).
- J. Crowder and N. J. Cornish, Beyond LISA: Exploring future gravitational wave missions, Phys. Rev. D 72, 083005 (2005).
- D. Brzeminski, A. Hook, and G. Marques-Tavares, Precision early universe cosmology from stochastic gravitational waves, J. High Energy Phys. 11 (2022) 061.
- G. Franciolini, D. Racco, and F. Rompineve, Footprints of the QCD crossover on cosmological gravitational waves at pulsar timing arrays, Phys. Rev. Lett. 132, 081001 (2024).
- A. Gonstal, M. Lewicki, and B. Swiezewska, Reconstructing early universe evolution with gravitational waves from supercooled phase transitions, J. High Energy Phys. 08 (2025) 039.
- S. W. Hawking, I. G. Moss, and J. M. Stewart, Bubble collisions in the very early universe, Phys. Rev. D 26, 2681 (1982).
- H. Kodama, M. Sasaki, and K. Sato, Abundance of primordial holes produced by cosmological first order phase transition, Prog. Theor. Phys. 68, 1979 (1982).
- M. Lewicki, P. Toczek, and V. Vaskonen, Primordial black holes from strong first-order phase transitions, J. High Energy Phys. 09 (2023) 092.
- J. Liu, L. Bian, R.-G. Cai, Z.-K. Guo, and S.-J. Wang, Primordial black hole production during first-order phase transitions, Phys. Rev. D 105, L021303 (2022).
- K. Kawana, T. Kim, and P. Lu, PBH formation from overdensities in delayed vacuum transitions, Phys. Rev. D 108, 103531 (2023).
- Y. Gouttenoire and T. Volansky, Primordial black holes from supercooled phase transitions, Phys. Rev. D 110, 043514 (2024).
- A. Salvio, Supercooling in radiative symmetry breaking: Theory extensions, gravitational wave detection and primordial black holes, J. Cosmol. Astropart. Phys. 12 (2023) 046.
- I. Baldes and M. O. Olea-Romacho, Primordial black holes as dark matter: Interferometric tests of phase transition origin, J. High Energy Phys. 01 (2024) 133.
- M. Lewicki, P. Toczek, and V. Vaskonen, Black holes and gravitational waves from slow first-order phase transitions, Phys. Rev. Lett. 133, 221003 (2024).
- S. Kanemura, M. Tanaka, and K.-P. Xie, Primordial black holes from slow phase transitions: A model-building perspective, J. High Energy Phys. 06 (2024) 036.
- S. Balaji, M. Fairbairn, and M. O. Olea-Romacho, Magnetogenesis with gravitational waves and primordial black hole dark matter, Phys. Rev. D 109, 075048 (2024).
- D. Gonçalves, A. Kaladharan, and Y. Wu, Primordial black holes from first-order phase transition in the singlet-extended SM, Phys. Rev. D 111, 035009 (2025).
- B. Carr, A. J. Iovino, G. Perna, V. Vaskonen, and H. Veermäe, Primordial black holes: Constraints, potential evidence and prospects, arXiv:2601.06024.
- G. Franciolini, Y. Gouttenoire, and R. Jinno, Curvature perturbations from first-order phase transitions: Implications to black holes and gravitational waves, arXiv:2503.01962.
- X. Wang, C. Balázs, R. Ding, and C. Tian, How large are curvature perturbations from slow first-order phase transitions? A gauge-invariant analysis, arXiv:2601.14412.
- J. Liu, L. Bian, R.-G. Cai, Z.-K. Guo, and S.-J. Wang, Constraining first-order phase transitions with curvature perturbations, Phys. Rev. Lett. 130, 051001 (2023).
- S. Koren, Y. Tsai, and R. Wang, Boiling after the dust settles: Constraining first-order phase transitions during dark energy domination, arXiv:2509.07076.
- K. Greene, D. W. R. Ho, S. Kumar, and Y. Tsai, Cosmological and astrophysical constraints on late first-order phase transitions, arXiv:2603.00272.
- Atacama Cosmology Telescope Collaboration, The atacama cosmology telescope: DR6 power spectra, likelihoods and parameters, J. Cosmol. Astropart. Phys. 11 (2025) 062.
- Planck Collaboration, Planck 2018 results. X. Constraints on inflation, Astron. Astrophys. 641, A10 (2020).