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Thermodynamical uncertainties for primordial black holes from cosmological phase transitions

Maciej Kierkla1,*, Nicklas Ramberg2,3,4,†, Philipp Schicho5,‡, and Daniel Schmitt6,§

  • *Contact author: maciej.kierkla@fuw.edu.pl
  • †Contact author: nramberg@sissa.it
  • ‡Contact author: philipp.schicho@unige.ch
  • §Contact author: daniel.schmitt@kit.edu
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Phys. Rev. D 113, 095024 – Published 18 May, 2026

DOI: https://doi.org/10.1103/nj27-ltyg

Abstract

Strongly supercooled first-order phase transitions have been proposed as a primordial black hole (PBH) production mechanism. While previous works rely on simplified models with limited thermodynamic precision, we stress that reliable theoretical PBH predictions require precise nucleation dynamics within realistic extensions of the Standard Model. By employing high-temperature dimensional reduction and computing the one-loop fluctuation determinants, we provide a state-of-the-art thermodynamic analysis and obtain a universal lower bound on the transition timescale, β/H*≃5. Then, we estimate the corresponding PBH abundance for classically conformal gauge-Higgs theories. Accounting for constraints from successful percolation and QCD chiral symmetry breaking, the parameter space where PBHs are viable dark matter candidates is severely limited.

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References (144)

  1. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. Lett. 116, 061102 (2016).
  2. G. Agazie et al. (NANOGrav Collaboration), Astrophys. J. Lett. 951, L10 (2023).
  3. G. Agazie et al. (NANOGrav Collaboration), Astrophys. J. Lett. 951, L8 (2023).
  4. G. Agazie et al. (NANOGrav Collaboration), Astrophys. J. Lett. 951, L9 (2023).
  5. G. Agazie et al. (NANOGrav Collaboration), Astrophys. J. Lett. 952, L37 (2023).
  6. A. Afzal et al. (NANOGrav Collaboration), Astrophys. J. Lett. 951, L11 (2023).
  7. A. D. Johnson et al. (NANOGrav Collaboration), Phys. Rev. D 109, 103012 (2024).
  8. G. Agazie et al. (NANOGrav Collaboration), Astrophys. J. Lett. 956, L3 (2023).
  9. G. Agazie et al. (NANOGrav Collaboration), Astrophys. J. Lett. 951, L50 (2023).
  10. J. Antoniadis et al., Astron. Astrophys. 690, A118 (2024).
  11. J. Antoniadis et al., Astron. Astrophys. 678, A48 (2023).
  12. J. Antoniadis et al., Astron. Astrophys. 678, A50 (2023).
  13. J. Antoniadis et al., Astron. Astrophys. 678, A49 (2023).
  14. J. Antoniadis et al., Astron. Astrophys. 685, A94 (2024).
  15. C. Smarra et al., Phys. Rev. Lett. 131, 171001 (2023).
  16. R. N. Manchester, Classical Quantum Gravity 30, 224010 (2013).
  17. B. B. P. Perera et al., Mon. Not. R. Astron. Soc. 490, 4666 (2019).
  18. D. J. Reardon et al., Mon. Not. R. Astron. Soc. 455, 1751 (2016).
  19. R. N. Manchester et al., Pub. Astron. Soc. Aust. 30, 17 (2013).
  20. M. Kerr et al., Pub. Astron. Soc. Aust. 37, e020 (2020).
  21. K. Kajantie, M. Laine, K. Rummukainen, and M. E. Shaposhnikov, Phys. Rev. Lett. 77, 2887 (1996).
  22. Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Szabo, Nature (London) 443, 675 (2006).
  23. LISA Collaboration, arXiv:1702.00786.
  24. P. Auclair et al. (LISA Cosmology Working Group Collaboration), Living Rev. Relativity 26, 5 (2023).
  25. K. A. Meissner and H. Nicolai, Phys. Lett. B 648, 312 (2007).
  26. R. Foot, A. Kobakhidze, K. L. McDonald, and R. R. Volkas, Phys. Rev. D 77, 035006 (2008).
  27. J. R. Espinosa, T. Konstandin, J. M. No, and M. Quiros, Phys. Rev. D 78, 123528 (2008).
  28. S. Iso, N. Okada, and Y. Orikasa, Phys. Lett. B 676, 81 (2009).
  29. S. Iso, N. Okada, and Y. Orikasa, Phys. Rev. D 80, 115007 (2009).
  30. S. Iso and Y. Orikasa, Prog. Theor. Exp. Phys. 2013, 023B08 (2013).
  31. A. Farzinnia, H.-J. He, and J. Ren, Phys. Lett. B 727, 141 (2013).
  32. C. Englert, J. Jaeckel, V. V. Khoze, and M. Spannowsky, J. High Energy Phys. 04 (2013) 060.
  33. M. Hashimoto, S. Iso, and Y. Orikasa, Phys. Rev. D 89, 016019 (2014).
  34. V. V. Khoze, C. McCabe, and G. Ro, J. High Energy Phys. 08 (2014) 026.
  35. T. Hur and P. Ko, Phys. Rev. Lett. 106, 141802 (2011).
  36. M. Heikinheimo, A. Racioppi, M. Raidal, C. Spethmann, and K. Tuominen, Mod. Phys. Lett. A 29, 1450077 (2014).
  37. M. Holthausen, J. Kubo, K. S. Lim, and M. Lindner, J. High Energy Phys. 12 (2013) 076.
  38. J. Kubo, K. S. Lim, and M. Lindner, Phys. Rev. Lett. 113, 091604 (2014).
  39. Y. Ametani, M. Aoki, H. Goto, and J. Kubo, Phys. Rev. D 91, 115007 (2015).
  40. J. Kubo and M. Yamada, Prog. Theor. Exp. Phys. 2015, 093B01 (2015).
  41. H. Hatanaka, D.-W. Jung, and P. Ko, J. High Energy Phys. 08 (2016) 094.
  42. P. Baratella, A. Pomarol, and F. Rompineve, J. High Energy Phys. 03 (2019) 100.
  43. J. Ellis, M. Lewicki, and J. M. No, J. Cosmol. Astropart. Phys. 04 (2019) 003.
  44. C. Marzo, L. Marzola, and V. Vaskonen, Eur. Phys. J. C 79, 601 (2019).
  45. J. Ellis, M. Lewicki, and V. Vaskonen, J. Cosmol. Astropart. Phys. 11 (2020) 020.
  46. T. de Boer, M. Lindner, and A. Trautner, Phys. Lett. B 861, 139241 (2025).
  47. M. Kierkla, A. Karam, and B. Swiezewska, J. High Energy Phys. 03 (2023) 007.
  48. L. Sagunski, P. Schicho, and D. Schmitt, Phys. Rev. D 107, 123512 (2023).
  49. M. Kierkla, B. Swiezewska, T. V. I. Tenkanen, and J. van de Vis, J. High Energy Phys. 02 (2024) 234.
  50. D. Schmitt and L. Sagunski, J. Cosmol. Astropart. Phys. 02 (2025) 075.
  51. M. Kierkla, P. Schicho, B. Swiezewska, T. V. I. Tenkanen, and J. van de Vis, J. High Energy Phys. 07 (2025) 153.
  52. J. a. Gonçalves, D. Marfatia, A. P. Morais, and R. Pasechnik, J. High Energy Phys. 02 (2025) 110.
  53. J. a. Gonçalves, D. Marfatia, A. P. Morais, and R. Pasechnik, Phys. Lett. B 869, 139829 (2025).
  54. S. Balan, T. Bringmann, F. Kahlhoefer, J. Matuszak, and C. Tasillo, J. Cosmol. Astropart. Phys. 08 (2025) 062.
  55. H. Kodama, M. Sasaki, and K. Sato, Prog. Theor. Phys. 68, 1979 (1982).
  56. J. Liu, L. Bian, R.-G. Cai, Z.-K. Guo, and S.-J. Wang, Phys. Rev. D 105, L021303 (2022).
  57. K. Hashino, S. Kanemura, and T. Takahashi, Phys. Lett. B 833, 137261 (2022).
  58. T. H. Jung and T. Okui, Phys. Rev. D 110, 115014 (2024).
  59. M. J. Baker, M. Breitbach, J. Kopp, and L. Mittnacht, Phys. Lett. B 868, 139625 (2025).
  60. M. J. Baker, M. Breitbach, J. Kopp, and L. Mittnacht, Phys. Rev. D 111, 063544 (2025).
  61. I. Baldes and M. O. Olea-Romacho, J. High Energy Phys. 01 (2024) 133.
  62. Y. Gouttenoire and T. Volansky, Phys. Rev. D 110, 043514 (2024).
  63. Y. Gouttenoire, Phys. Lett. B 855, 138800 (2024).
  64. Y. Gouttenoire, Phys. Rev. Lett. 131, 171404 (2023).
  65. A. Salvio, Phys. Lett. B 852, 138639 (2024).
  66. A. Salvio, J. Cosmol. Astropart. Phys. 12 (2023) 046.
  67. R. Jinno, J. Kume, and M. Yamada, Phys. Lett. B 849, 138465 (2024).
  68. I. K. Banerjee and U. K. Dey, J. High Energy Phys. 07 (2024) 006; 08 (2024) 54.
  69. I. K. Banerjee, U. K. Dey, and S. Khalil, J. High Energy Phys. 12 (2024) 009.
  70. M. M. Flores, A. Kusenko, and M. Sasaki, Phys. Rev. D 110, 015005 (2024).
  71. W.-Y. Ai, L. Heurtier, and T. H. Jung, Phys. Rev. D 113, 023542 (2026).
  72. A. Conaci, L. Delle Rose, P. S. B. Dev, and A. Ghoshal, J. High Energy Phys. 12 (2024) 196.
  73. S. Kanemura, M. Tanaka, and K.-P. Xie, J. High Energy Phys. 06 (2024) 036.
  74. K. Hashino, S. Kanemura, T. Takahashi, M. Tanaka, and C.-M. Yoo, J. Cosmol. Astropart. Phys. 09 (2025) 006.
  75. S. Balaji, J. a. Gonçalves, D. Marfatia, A. P. Morais, and R. Pasechnik, J. Cosmol. Astropart. Phys. 10 (2025) 064.
  76. M. Lewicki, P. Toczek, and V. Vaskonen, J. High Energy Phys. 09 (2023) 092.
  77. M. Lewicki, P. Toczek, and V. Vaskonen, Phys. Rev. Lett. 133, 221003 (2024).
  78. M. Lewicki, P. Toczek, and V. Vaskonen, Phys. Dark Universe 50, 102075 (2025).
  79. G. Franciolini, Y. Gouttenoire, and R. Jinno, arXiv:2503.01962.
  80. J. Li and P. Nath, Phys. Rev. D 111, 123007 (2025).
  81. I. K. Banerjee, F. Rescigno, and A. Salvio, J. Cosmol. Astropart. Phys. 07 (2025) 007.
  82. M. Arteaga, A. Ghoshal, and A. Strumia, J. Cosmol. Astropart. Phys. 05 (2025) 029.
  83. D. Gonçalves, A. Kaladharan, and Y. Wu, Phys. Rev. D 111, 035009 (2025).
  84. R.-G. Cai, Y.-S. Hao, and S.-J. Wang, Sci. China Phys. Mech. Astron. 67, 290411 (2024).
  85. D. Borah, S. J. Das, and I. Saha, Phys. Rev. D 110, 035014 (2024).
  86. K. Kawana, T. H. Kim, and P. Lu, Phys. Rev. D 108, 103531 (2023).
  87. P. H. Ginsparg, Nucl. Phys. B170, 388 (1980).
  88. T. Appelquist and R. D. Pisarski, Phys. Rev. D 23, 2305 (1981).
  89. K. Kajantie, M. Laine, K. Rummukainen, and M. E. Shaposhnikov, Nucl. Phys. B458, 90 (1996).
  90. J. Hirvonen, J. Löfgren, M. J. Ramsey-Musolf, P. Schicho, and T. V. I. Tenkanen, J. High Energy Phys. 07 (2022) 135.
  91. O. Gould and T. V. I. Tenkanen, J. High Energy Phys. 01 (2024) 048.
  92. A. Ekstedt, Eur. Phys. J. C 82, 173 (2022).
  93. A. Ekstedt, O. Gould, and J. Hirvonen, J. High Energy Phys. 12 (2023) 056.
  94. M. Lewicki, P. Toczek, and V. Vaskonen, https://github.com/vianvask/deltaPT (2024).
  95. S. R. Coleman and E. J. Weinberg, Phys. Rev. D 7, 1888 (1973).
  96. F. Loebbert, J. Miczajka, and J. Plefka, Phys. Rev. D 99, 015026 (2019).
  97. M. Aaboud et al. (ATLAS Collaboration), J. High Energy Phys. 10 (2017) 182.
  98. M. Escudero, S. J. Witte, and N. Rius, J. High Energy Phys. 08 (2018) 190.
  99. T. Hambye and A. Strumia, Phys. Rev. D 88, 055022 (2013).
  100. T. Prokopec, J. Rezacek, and B. Świeżewska, J. Cosmol. Astropart. Phys. 02 (2019) 009.
  101. F. Bernardo, P. Klose, P. Schicho, and T. V. I. Tenkanen, J. High Energy Phys. 08 (2025) 109.
  102. M. Dine, R. G. Leigh, P. Huet, A. D. Linde, and D. A. Linde, Phys. Lett. B 283, 319 (1992).
  103. M. Dine, R. G. Leigh, P. Y. Huet, A. D. Linde, and D. A. Linde, Phys. Rev. D 46, 550 (1992).
  104. J. Baacke and V. G. Kiselev, Phys. Rev. D 48, 5648 (1993).
  105. J. Baacke and G. Lavrelashvili, Phys. Rev. D 69, 025009 (2004).
  106. G. V. Dunne and H. Min, Phys. Rev. D 72, 125004 (2005).
  107. M. Matteini, M. Nemevšek, Y. Shoji, and L. Ubaldi, J. High Energy Phys. 04 (2025) 120.
  108. V. Brdar, M. Finetti, M. Matteini, A. P. Morais, and M. Nemevšek, Comput. Phys. Commun. 323, 110119 (2026).
  109. A. Ekstedt, Phys. Rev. D 106, 095026 (2022).
  110. A. Ekstedt, J. High Energy Phys. 08 (2022) 115.
  111. O. Gould and J. Hirvonen, Phys. Rev. D 104, 096015 (2021).
  112. O. Gould, A. Kormu, and D. J. Weir, Phys. Rev. D 111, L051901 (2025).
  113. J. Hirvonen, Phys. Rev. D 111, 116020 (2025).
  114. J. Hirvonen and O. Gould, Phys. Rev. Lett. 136, 081601 (2026).
  115. J. S. Langer, Ann. Phys. (N.Y.) 54, 258 (1969).
  116. P. Hanggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251 (1990).
  117. A. Berera, J. Mabillard, B. W. Mintz, and R. O. Ramos, Phys. Rev. D 100, 076005 (2019).
  118. O. Gould, S. Güyer, and K. Rummukainen, Phys. Rev. D 106, 114507 (2022); 110, 119903(E) (2024).
  119. V. K. S. Shante and S. Kirkpatrick, Adv. Phys. 20, 325 (1971).
  120. A. H. Guth and S. H. H. Tye, Phys. Rev. Lett. 44, 631 (1980).
  121. A. H. Guth and E. J. Weinberg, Phys. Rev. D 23, 876 (1981).
  122. K. Enqvist, J. Ignatius, K. Kajantie, and K. Rummukainen, Phys. Rev. D 45, 3415 (1992).
  123. A. Ekstedt, O. Gould, J. Hirvonen, B. Laurent, L. Niemi, P. Schicho, and J. van de Vis, J. High Energy Phys. 04 (2025) 101.
  124. A. H. Guth, J. Phys. A 40, 6811 (2007).
  125. A. Borde and A. Vilenkin, Phys. Rev. Lett. 72, 3305 (1994).
  126. A. H. Guth, Phys. Rev. D 23, 347 (1981).
  127. M. S. Turner, E. J. Weinberg, and L. M. Widrow, Phys. Rev. D 46, 2384 (1992).
  128. P. Athron, C. Balázs, and L. Morris, J. Cosmol. Astropart. Phys. 03 (2023) 006.
  129. P. Athron, C. Balázs, A. Fowlie, L. Morris, and L. Wu, Prog. Part. Nucl. Phys. 135, 104094 (2024).
  130. J. Braun and H. Gies, J. High Energy Phys. 06 (2006) 024.
  131. B. von Harling and G. Servant, J. High Energy Phys. 01 (2018) 159.
  132. S. Iso, P. D. Serpico, and K. Shimada, Phys. Rev. Lett. 119, 141301 (2017).
  133. A. Escrivà, C. Germani, and R. K. Sheth, Phys. Rev. D 101, 044022 (2020).
  134. G. Franciolini, I. Musco, P. Pani, and A. Urbano, Phys. Rev. D 106, 123526 (2022).
  135. Y. Wu and S. Profumo, Phys. Rev. D 111, 103524 (2025).
  136. D. Croon, O. Gould, P. Schicho, T. V. I. Tenkanen, and G. White, J. High Energy Phys. 04 (2021) 055.
  137. S. Coleman and F. De Luccia, Phys. Rev. D 21, 3305 (1980).
  138. L. Giombi and M. Hindmarsh, J. Cosmol. Astropart. Phys. 03 (2024) 059.
  139. M. Kierkla, Theoretical uncertainties for primordial black holes from cosmological phase transitions (2025), 10.58132/IMHIWV.
  140. B. Ruijl, T. Ueda, and J. Vermaseren, arXiv:1707.06453.
  141. A. Ekstedt, P. Schicho, and T. V. I. Tenkanen, Comput. Phys. Commun. 288, 108725 (2023).
  142. L. Niemi, P. Schicho, and Tuomas V. I. Tenkanen, Phys. Rev. D 103, 115035 (2021); 109, 039902(E) (2024).
  143. M. Lewicki, M. Merchand, L. Sagunski, P. Schicho, and D. Schmitt, Phys. Rev. D 110, 023538 (2024).
  144. A. Bhatnagar, D. Croon, and P. Schicho, J. High Energy Phys. 03 (2026) 014.

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Phys. Rev. D 113, 095024– Published 18 May, 2026
  • Received 10 July 2025
  • Revised 19 September 2025
  • Accepted 13 April 2026
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