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Primordial black hole dark matter: A quantitative parameter sensitivity comparison across formation mechanisms and particle candidates

Stefano Profumo*

  • *Contact author: profumo@ucsc.edu
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Phys. Rev. D 114, 063025 – Published 10 September, 2026

DOI: https://doi.org/10.1103/nk1q-5k51

Abstract

Primordial black holes (PBHs) in the asteroid-mass window (1017–1022  g) can account for all of the dark matter without violating any observational constraint, yet are routinely dismissed as fine-tuned. I put that dismissal to the test by applying three complementary sensitivity measures uniformly across a broad landscape: three noninflationary PBH production mechanisms, six classes of inflationary PBH models, and seven particle dark matter benchmarks, all evaluated against the same observable target. Three distinct naturalness universality classes emerge, determined entirely by the analytic structure of the abundance map rather than by the nature of the dark matter candidate. Biased-domain-wall PBHs, in their least model-dependent (free-Vb) form, have the same low sensitivity, Δ=4.5, as off-resonance weakly interacting massive particles and freeze-in particles (Δ=2), a sensitivity that, because it is constant over the entire parameter space of the construction, also coincides trivially with its own Wilson-normalized average within that parameter space (Section Definition and conventions), an equivalence that concerns only the space over which Δ is computed and is not a naturalness statement about the construction as a whole; a further reduction to Δ=2 is possible only under the additional, independently motivated but not required, assumption that the domain-wall bias is generated by Planck-suppressed operators; early matter-domination PBHs occupy an intermediate tier alongside coannihilating weakly interacting massive particles (WIMPs), unified by a structural identity in which the sensitivity measure equals the logarithm of the ratio of the formation scale to the matter–radiation equality scale; first-order phase transition PBHs, once the more accurate super-exponential collapse probability is used in place of the single-exponential approximation, instead belong to the same highly sensitive tier as resonant WIMP annihilation and single-field inflationary collapse, for a structurally distinct reason; single-field ultraslow-roll inflationary collapse is severely tuned for a distinct reason: a double exponential in which the power spectrum amplitude is itself exponentially sensitive to the inflaton potential coefficients, on top of the exponential collapse sensitivity of the abundance map. My main conclusion is that the claim that PBH dark matter is generically fine-tuned conflates the worst case with a landscape spanning every naturalness tier. The Barbieri-Giudice sensitivity computed here and the Wilson-normalized measure of Iovino and Riotto answer distinct and mutually consistent questions about the same construction, a distinction I clarify within the two-layer decomposition.

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References (113)

  1. G. Jungman, M. Kamionkowski, and K. Griest, Supersymmetric dark matter, Phys. Rep. 267, 195 (1996).
  2. G. Bertone, D. Hooper, and J. Silk, Particle dark matter: Evidence, candidates and constraints, Phys. Rep. 405, 279 (2005).
  3. R. D. Peccei and H. R. Quinn, CP conservation in the presence of pseudoparticles, Phys. Rev. Lett. 38, 1440 (1977).
  4. S. Weinberg, A new light boson?, Phys. Rev. Lett. 40, 223 (1978).
  5. F. Wilczek, Problem of strong P and T invariance in the presence of instantons, Phys. Rev. Lett. 40, 279 (1978).
  6. L. J. Hall, K. Jedamzik, J. March-Russell, and S. M. West, Freeze-in production of FIMP dark matter, J. High Energy Phys. 03 (2010) 080.
  7. J. McDonald, Thermally generated gauge singlet scalars as self-interacting dark matter, Phys. Rev. Lett. 88, 091304 (2002).
  8. S. Nussinov, Technocosmology: Could a technibaryon excess provide a “natural” missing mass candidate?, Phys. Lett. B 165, 55 (1985).
  9. D. E. Kaplan, M. A. Luty, and K. M. Zurek, Asymmetric dark matter, Phys. Rev. D 79, 115016 (2009).
  10. B. J. Carr and S. W. Hawking, Black holes in the early universe, Mon. Not. R. Astron. Soc. 168, 399 (1974).
  11. A. M. Green and A. R. Liddle, Constraints on the density perturbation spectrum from primordial black holes, Phys. Rev. D 56, 6166 (1997).
  12. B. Carr and F. Kühnel, Primordial black holes as dark matter: Recent developments, Annu. Rev. Nucl. Part. Sci. 70, 355 (2020).
  13. A. M. Green and B. J. Kavanagh, Primordial black holes as a dark matter candidate, J. Phys. G 48, 043001 (2021).
  14. A. Escrivà, F. Kuhnel, and Y. Tada, Primordial black holes, Black Holes in the Era of Gravitational-Wave Astronomy (Elsevier, New York, 2024).
  15. P. Grothaus, M. Lindner, and Y. Takanishi, Naturalness of neutralino dark matter, J. High Energy Phys. 07 (2012) 094.
  16. R. Barbieri and G. F. Giudice, Upper bounds on supersymmetric particle masses, Nucl. Phys. B306, 63 (1988).
  17. M. E. Cabrera, J. A. Casas, A. Delgado, S. Robles, and R. Ruiz de Austri, Naturalness of MSSM dark matter, J. High Energy Phys. 08 (2016) 058.
  18. S. F. King and J. P. Roberts, Natural implementation of neutralino dark matter, J. High Energy Phys. 09 (2006) 036.
  19. M. P. Hertzberg and M. Yamada, Primordial black holes from polynomial potentials in single field inflation, Phys. Rev. D 97, 083509 (2018).
  20. I. Stamou, Mechanisms for producing primordial black holes from inflationary models beyond fine-tuning, Universe 10, 241 (2024).
  21. A. Kalaja, N. Bellomo, N. Bartolo, D. Bertacca, S. Matarrese, I. Musco, A. Raccanelli, and L. Verde, From primordial black holes abundance to primordial curvature power spectrum (and back), J. Cosmol. Astropart. Phys. 10 (2019) 031.
  22. A. J. Iovino and A. Riotto, Are primordial black holes truly fine-tuned?, J. Cosmol. Astropart. Phys. 07 (2026) 047.
  23. J. R. Ellis, K. Enqvist, D. V. Nanopoulos, and F. Zwirner, Observables in low-energy superstring models, Mod. Phys. Lett. A 01, 57 (1986).
  24. G. W. Anderson and D. J. Castano, Measures of fine tuning, Phys. Lett. B 347, 300 (1995).
  25. N. Aghanim et al. (Planck Collaboration), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020); 652, C4(E) (2021).
  26. A. Strumia, Naturalness of supersymmetric models, in 34th Rencontres de Moriond: Electroweak Interactions and Unified Theories (1999), pp. 441–446, arXiv:hep-ph/9904247.
  27. M. E. Cabrera, J. A. Casas, and R. Ruiz de Austri, Bayesian approach and naturalness in MSSM analyses for the LHC, J. High Energy Phys. 03 (2009) 075.
  28. S. P. Martin, A supersymmetry primer, Adv. Ser. Dir. High Energy Phys. 18, 1 (1998).
  29. K. Griest and D. Seckel, Three exceptions in the calculation of relic abundances, Phys. Rev. D 43, 3191 (1991).
  30. J. Edsjo and P. Gondolo, Neutralino relic density including coannihilations, Phys. Rev. D 56, 1879 (1997).
  31. H. Baer, V. Barger, and A. Mustafayev, Implications of a 125 GeV Higgs scalar for LHC SUSY and neutralino dark matter searches, Phys. Rev. D 85, 075010 (2012).
  32. P. S. Cole, A. D. Gow, C. T. Byrnes, and S. P. Patil, Primordial black holes from single-field inflation: A fine-tuning audit, J. Cosmol. Astropart. Phys. 08 (2023) 031.
  33. K. Kohri, D. H. Lyth, and A. Melchiorri, Black hole formation and slow-roll inflation, J. Cosmol. Astropart. Phys. 04 (2007) 038.
  34. E. V. Bugaev and P. A. Klimai, Primordial black hole constraints for curvaton models with predicted large non-Gaussianity, Int. J. Mod. Phys. D 22, 1350034 (2013).
  35. V. Vennin, Stochastic inflation and primordial black holes, arXiv:2009.08715.
  36. S. Clesse and J. García-Bellido, Massive primordial black holes from hybrid inflation as dark matter and the seeds of galaxies, Phys. Rev. D 92, 023524 (2015).
  37. G. Ballesteros and M. Taoso, Primordial black hole dark matter from single field inflation, Phys. Rev. D 97, 023501 (2018).
  38. H. Motohashi and W. Hu, Primordial black holes and slow-roll violation, Phys. Rev. D 96, 063503 (2017).
  39. K. Kannike, L. Marzola, M. Raidal, and H. Veermäe, Single field double inflation and primordial black holes, J. Cosmol. Astropart. Phys. 09 (2017) 020.
  40. J. Aalbers et al. (LZ Collaboration), Dark matter search results from 4.2  tonne−years of exposure of the LUX-ZEPLIN (LZ) experiment, Phys. Rev. Lett. 135, 011802 (2025).
  41. G. Ballesteros, M. A. G. Garcia, and M. Pierre, How warm are non-thermal relics? Lyman-α bounds on out-of-equilibrium dark matter, J. Cosmol. Astropart. Phys. 03 (2021) 101.
  42. Q. Decant, J. Heisig, D. C. Hooper, and L. Lopez-Honorez, Lyman-α constraints on freeze-in and superWIMPs, J. Cosmol. Astropart. Phys. 03 (2022) 041.
  43. M. Buschmann, J. W. Foster, A. Hook, A. Peterson, D. E. Willcox, W. Zhang, and B. R. Safdi, Dark matter from axion strings with adaptive mesh refinement, Nat. Commun. 13, 1049 (2022).
  44. J. M. Alarcon, J. Martin Camalich, and J. A. Oller, The chiral representation of the πN scattering amplitude and the pion-nucleon sigma term, Phys. Rev. D 85, 051503 (2012).
  45. F. Ferrer, E. Massò, G. Panico, O. Pujolas, and F. Rompineve, Primordial black holes from the QCD axion, Phys. Rev. Lett. 122, 101301 (2019).
  46. Y. Gouttenoire and E. Vitagliano, Primordial black holes and wormholes from domain wall networks, Phys. Rev. D 109, 123507 (2024).
  47. R. Z. Ferreira, A. Notari, O. Pujolàs, and F. Rompineve, Collapsing domain wall networks: Impact on pulsar timing arrays and primordial black holes, J. Cosmol. Astropart. Phys. 06 (2024) 020.
  48. J. Georg and S. Watson, A preferred mass range for primordial black hole formation and black holes as dark matter revisited, J. High Energy Phys. 09 (2016) 138.
  49. J. Georg, B. Melcher, and S. Watson, Primordial black holes and co-decaying dark matter, J. Cosmol. Astropart. Phys. 11 (2019) 014.
  50. I. Dalianis and C. Kouvaris, Gravitational waves from density perturbations in an early matter domination era, J. Cosmol. Astropart. Phys. 07 (2021) 046.
  51. B. Carr, K. Kohri, Y. Sendouda, and J. Yokoyama, Constraints on primordial black holes, Rep. Prog. Phys. 84, 116902 (2021).
  52. 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).
  53. M. Lewicki, P. Toczek, and V. Vaskonen, Primordial black holes from strong first-order phase transitions, J. High Energy Phys. 09 (2023) 092.
  54. T. C. Gehrman, B. Shams Es Haghi, K. Sinha, and T. Xu, Baryogenesis, primordial black holes and MHz–GHz gravitational waves, J. Cosmol. Astropart. Phys. 02 (2023) 062.
  55. A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press, Cambridge, England, 2000).
  56. G. Franciolini and A. Urbano, Primordial black hole dark matter from inflation: The reverse engineering approach, Phys. Rev. D 106, 123519 (2022).
  57. O. Özsoy and G. Tasinato, Inflation and primordial black holes, Universe 9, 203 (2023).
  58. J. García-Bellido and E. Ruiz Morales, Primordial black holes from single field models of inflation, Phys. Dark Universe 18, 47 (2017).
  59. C. Germani and T. Prokopec, On primordial black holes from an inflection point, Phys. Dark Universe 18, 6 (2017).
  60. S. R. Geller, W. Qin, E. McDonough, and D. I. Kaiser, Primordial black holes from multifield inflation with nonminimal couplings, Phys. Rev. D 106, 063535 (2022).
  61. W. Qin, S. R. Geller, S. Balaji, E. McDonough, and D. I. Kaiser, Planck constraints and gravitational wave forecasts for primordial black hole dark matter seeded by multifield inflation, Phys. Rev. D 108, 043508 (2023).
  62. D. L. Lorenzoni, S. R. Geller, Z. Ireland, D. I. Kaiser, E. McDonough, and K. A. Wittmeier, Light scalar fields Foster production of primordial black holes, arXiv:2504.13251.
  63. D. L. Lorenzoni, S. R. Geller, D. I. Kaiser, and E. McDonough, Primordial black holes from inflation with a spectator field, arXiv:2512.04199.
  64. J. R. Espinosa, D. Racco, and A. Riotto, Cosmological signature of the standard model Higgs vacuum instability: Primordial black holes as dark matter, Phys. Rev. Lett. 120, 121301 (2018).
  65. T. W. B. Kibble, Topology of cosmic domains and strings, J. Phys. A 9, 1387 (1976).
  66. S. M. Barr and D. Seckel, Planck-scale corrections to axion models, Phys. Rev. D 46, 539 (1992).
  67. R. Holman, S. D. H. Hsu, T. W. Kephart, E. W. Kolb, R. Watkins, and L. M. Widrow, Solutions to the strong CP problem in a world with gravity, Phys. Lett. B 282, 132 (1992).
  68. M. Kawasaki, K. Saikawa, and T. Sekiguchi, Axion dark matter from topological defects, Phys. Rev. D 91, 065014 (2015).
  69. K. J. Bae, H. Baer, V. Barger, and R. W. Deal, The cosmological moduli problem and naturalness, J. High Energy Phys. 02 (2008) 138.
  70. R. Allahverdi et al., The first three seconds: A review of possible expansion histories of the early universe, Open J. Astrophys. 4, 1 (2021).
  71. M. Y. Khlopov and A. G. Polnarev, Primordial black holes as a cosmological test of grand unification, Phys. Lett. 97, 383 (1980).
  72. T. Harada, C.-M. Yoo, K. Kohri, and T. Nakama, Primordial black hole formation in the matter-dominated phase of the universe, Phys. Rev. D 96, 083517 (2017).
  73. W. H. Press and P. Schechter, Formation of galaxies and clusters of galaxies by self-similar gravitational condensation, Astrophys. J. 187, 425 (1974).
  74. Z. Zhou, J. Jiang, Y.-F. Cai, M. Sasaki, and S. Pi, Primordial black holes and gravitational waves from resonant amplification during inflation, Phys. Rev. D 102, 103527 (2020).
  75. K. N. Ananda, C. Clarkson, and D. Wands, The cosmological gravitational wave background from primordial density perturbations, Phys. Rev. D 75, 123518 (2007).
  76. D. Baumann, P. J. Steinhardt, K. Takahashi, and K. Ichiki, Gravitational wave spectrum induced by primordial scalar perturbations, Phys. Rev. D 76, 084019 (2007).
  77. C. Caprini et al., Detecting gravitational waves from cosmological phase transitions with LISA: An update, J. Cosmol. Astropart. Phys. 03 (2020) 024.
  78. A. D. Linde, Fate of the false vacuum at finite temperature: Theory and applications, Phys. Lett. 100B, 37 (1981).
  79. E. Witten, Cosmic separation of phases, Phys. Rev. D 30, 272 (1984).
  80. 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.
  81. S. W. Hawking, I. G. Moss, and J. M. Stewart, Bubble collisions in the very early universe, Phys. Rev. D 26, 2681 (1982).
  82. I. G. Moss, Singularity formation from colliding bubbles, Phys. Rev. D 50, 676 (1994).
  83. H. Kodama, M. Sasaki, and K. Sato, Abundance of primordial holes produced by cosmological first-order phase transition, Prog. Theor. Phys. 68, 1979 (1982).
  84. Y. Wu and S. Profumo, Superexponential primordial black hole production via delayed vacuum decay, Phys. Rev. D 111, 103524 (2025).
  85. 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).
  86. D. H. Lyth and D. Wands, Generating the curvature perturbation without an inflaton, Phys. Lett. B 524, 5 (2002).
  87. K. Enqvist and M. S. Sloth, Adiabatic CMB perturbations in pre-big bang string cosmology, Nucl. Phys. B626, 395 (2002).
  88. A. A. Starobinsky and J. Yokoyama, Equilibrium state and density fluctuations in de Sitter space with and without λϕ4 self-interaction, Phys. Rev. D 50, 6357 (1994).
  89. C. Pattison, V. Vennin, H. Assadullahi, and D. Wands, Quantum diffusion during inflation and primordial black holes, J. Cosmol. Astropart. Phys. 10 (2021) 046.
  90. J. García-Bellido, A. D. Linde, and D. Wands, Density perturbations and black hole formation in hybrid inflation, Phys. Rev. D 54, 6040 (1996).
  91. Y.-F. Cai, X. Chen, M. H. Namjoo, M. Sasaki, D.-G. Wang, and Z. Wang, Revisiting non-Gaussianity from non-attractor inflation models, J. Cosmol. Astropart. Phys. 05 (2018) 012.
  92. G. Ballesteros, J. Rey, M. Taoso, and A. Urbano, Primordial black holes as dark matter and gravitational waves from single-field polynomial inflation, J. Cosmol. Astropart. Phys. 07 (2020) 025.
  93. E. W. Kolb and M. S. Turner, The Early Universe (Addison-Wesley, Redwood City, CA, 1990).
  94. V. Silveira and A. Zee, Scalar phantoms, Phys. Lett. 161B, 136 (1985).
  95. J. McDonald, Gauge singlet scalars as cold dark matter, Phys. Rev. D 50, 3637 (1994).
  96. C. P. Burgess, M. Pospelov, and T. ter Veldhuis, The minimal model of nonbaryonic dark matter: A singlet scalar, Nucl. Phys. B619, 709 (2001).
  97. M. Ibe, H. Murayama, and T. T. Yanagida, Breit-Wigner enhancement of dark matter annihilation, Phys. Rev. D 79, 095009 (2009).
  98. L. Visinelli and P. Gondolo, Dark matter axions revisited, Phys. Rev. D 80, 035024 (2009).
  99. S. Borsanyi et al., Calculation of the axion mass based on high-temperature lattice quantum chromodynamics, Nature (London) 539, 69 (2016).
  100. R. L. Davis, Goldstone bosons in string models of galaxy formation, Phys. Rev. D 35, 3705 (1987).
  101. R. A. Battye and E. P. S. Shellard, Axion string constraints, Phys. Rev. Lett. 73, 2954 (1994).
  102. A. Ayala, I. Domínguez, M. Giannotti, A. Mirizzi, and O. Straniero, Revisiting the bound on axion-photon coupling from globular clusters, Phys. Rev. Lett. 113, 191302 (2014).
  103. N. Du et al. (ADMX Collaboration), A search for invisible axion dark matter with the axion dark matter experiment, Phys. Rev. Lett. 120, 151301 (2018).
  104. K. M. Backes et al. (HAYSTAC Collaboration), A quantum-enhanced search for dark matter axions, Nature (London) 590, 238 (2021).
  105. K. Abazajian et al. (CMB-S4 Collaboration), CMB-S4 science book, first edition, arXiv:1610.02743.
  106. Y. Kahn, B. R. Safdi, and J. Thaler, Broadband and resonant approaches to axion dark matter detection, Phys. Rev. Lett. 117, 141801 (2016).
  107. L. Brouwer et al. (DMRadio Collaboration), Proposal for a definitive search for GUT-scale QCD axions, Phys. Rev. D 106, 112003 (2022).
  108. D. M. Ghilencea and G. G. Ross, The fine-tuning cost of the likelihood in SUSY models, Nucl. Phys. B868, 65 (2013).
  109. I. Musco, Threshold for primordial black holes: Dependence on the shape of the cosmological perturbations, Phys. Rev. D 100, 123524 (2019).
  110. A. Escrivà, C. Germani, and R. K. Sheth, Universal threshold for primordial black hole formation, Phys. Rev. D 101, 044022 (2020).
  111. G. Franciolini, Y. Gouttenoire, and R. Jinno, Curvature perturbations from first-order phase transitions: Implications to black holes and gravitational waves, Phys. Rev. Lett. 136, 171404 (2026).
  112. X. Wang, C. Balázs, R. Ding, and C. Tian, How large are curvature perturbations from slow first-order phase transitions? A gauge-invariant analysis, Phys. Rev. D 113, 123026 (2026).
  113. W.-Y. Ai and K.-P. Xie, Reviving primordial black hole formation in slow first-order phase transitions, arXiv:2605.11332.

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Phys. Rev. D 114, 063025– Published 10 September, 2026
  • Received 13 June 2026
  • Accepted 6 August 2026
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