pith. sign in

arxiv: 2606.09482 · v1 · pith:77I7Z5CFnew · submitted 2026-06-08 · ✦ hep-ph · astro-ph.CO· gr-qc

Primordial Black Holes from Slow Phase Transitions with Delayed Reheating: A Peak-Theory Approach

Indra Kumar Banerjee This is my paper

Pith reviewed 2026-06-27 16:08 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.COgr-qc
keywords primordial black holesfirst-order phase transitiondelayed reheatingpeak theorydark matterearly matter dominationhoop conjecture
0
0 comments X

The pith

A slow first-order phase transition with delayed reheating can generate enough primordial black holes to comprise all dark matter, with the abundance varying by many orders of magnitude for small changes in reheating efficiency.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines PBH production during an early matter-dominated phase that arises when reheating is delayed after a slow first-order phase transition. It develops a peak-theory treatment of the resulting non-Gaussian overdensity field and uses large Monte Carlo simulations together with the hoop-conjecture criterion to compute the collapse probability. The calculation incorporates tidal torques to track initial PBH spin. The resulting collapse fraction proves extremely sensitive to the reheating efficiency, so that order-unity shifts in that efficiency produce many orders of magnitude change in the PBH abundance. Certain regions of parameter space allow the mechanism to supply the entire dark-matter density, and the same framework yields observational constraints on the transition and reheating parameters.

Core claim

During the early matter-dominated era created by delayed reheating after a slow first-order phase transition, the non-Gaussian overdensity distribution permits significant PBH formation; Monte Carlo evaluation of the hoop-conjecture collapse threshold shows that the collapse fraction is extremely sensitive to reheating efficiency, with order-unity efficiency changes producing many orders of magnitude variation in abundance, and that viable parameter regions exist in which the resulting PBHs account for all dark matter while also satisfying current observational bounds.

What carries the argument

The peak-theoretic approach to collapse during the early matter-dominated phase, which models the non-Gaussian overdensity statistics from the phase transition and applies the hoop-conjecture criterion inside Monte Carlo simulations to obtain the collapse probability.

If this is right

  • The PBH abundance changes by many orders of magnitude when the reheating efficiency varies by order unity.
  • Regions of parameter space exist in which PBHs produced by this channel constitute the entire dark matter density.
  • The average initial spin parameter of the resulting PBHs is of order 10 to the minus 3.
  • Current data on the presence or absence of PBHs can be used to constrain the phase transition strength and the duration of delayed reheating.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same sensitivity to reheating efficiency may translate into distinct gravitational-wave signatures from the phase transition or from PBH binaries that could be tested by future detectors.
  • Precision measurements of the dark matter density or of small-scale PBH clustering could provide an independent test of the predicted collapse threshold dependence on transition parameters.
  • The peak-theory treatment developed here could be applied to other early-universe sources of non-Gaussian overdensities, such as those arising from inflationary features.

Load-bearing premise

The overdensity field generated by the slow phase transition follows a non-Gaussian distribution whose statistics are fully captured by peak theory, and the hoop-conjecture criterion supplies a reliable collapse threshold throughout the early matter-dominated phase.

What would settle it

A measured primordial black hole abundance that fails to exhibit the predicted extreme sensitivity to order-unity changes in reheating efficiency would falsify the central claim.

read the original abstract

We study the possibility of significant PBH production from a slow first-order phase transition with delayed reheating. Since delayed reheating results in an early matter-dominated phase between percolation and reheating, we developed a peak-theoretic approach to PBH formation during this phase based on the non-Gaussian distribution of overdensity arising from the transition. To obtain the collapse probability, we performed large-scale Monte Carlo simulations and employed the hoop-conjecture criterion. We include tidal-torque terms to investigate the initial spin of the PBHs and find that the average spin parameter is $\mathcal{O}(10^{-3})$. Furthermore, we obtain an emergent overdensity threshold for collapse that depends on the phase transition properties and reheating efficiency. We find that the resulting PBH abundance is extremely sensitive to the reheating efficiency, with order-unity changes in efficiency leading to variations of many orders of magnitude in the collapse fraction. We identify regions of parameter space where the resulting PBHs can account for the entirety of the dark matter abundance. Finally, we also constrain the phase transition and reheating properties from current data on (non-)observations of PBHs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates primordial black hole (PBH) production during slow first-order phase transitions with delayed reheating, which induces an early matter-dominated phase. It develops a peak-theory framework for the non-Gaussian overdensity field, uses Monte Carlo simulations with the hoop-conjecture to compute collapse probabilities, accounts for tidal torques yielding average PBH spin of O(10^{-3}), identifies an emergent collapse threshold depending on transition parameters and reheating efficiency, shows that PBH abundance is highly sensitive to reheating efficiency with order-unity changes causing orders-of-magnitude variations in collapse fraction, finds parameter regions where PBHs can comprise all dark matter, and places constraints from PBH (non-)observations.

Significance. Should the central results hold, the work provides a new mechanism linking phase transition dynamics and reheating to PBH dark matter, with the extreme sensitivity offering testable implications for cosmology. The Monte Carlo approach and inclusion of spin are positive aspects. The reported ability to account for all dark matter in some regions would be a notable result if the collapse modeling is robust.

major comments (2)
  1. [Monte Carlo simulations and hoop-conjecture criterion] Monte Carlo simulations section (hoop-conjecture application): the collapse fraction is obtained by applying the hoop-conjecture criterion to the non-Gaussian overdensity field in the early matter-dominated phase. The paper extracts an emergent threshold depending on transition parameters and reheating efficiency, but provides no independent cross-check (e.g., compaction-function analysis or numerical-relativity validation) that the hoop criterion remains quantitatively reliable when pressure support is absent. Because the abundance varies by many orders of magnitude with O(1) changes in efficiency, even a modest systematic shift in the effective threshold would move the viable DM parameter space.
  2. [Peak-theory approach] Peak-theory approach (non-Gaussian overdensity): the claim that the overdensity statistics are fully captured by the peak-theory formalism rests on the assumption that the distribution generated by the slow phase transition can be modeled without additional corrections from the prolonged matter-dominated era. The manuscript should explicitly show how the peak-theory parameters are derived from the transition dynamics and reheating efficiency, as this directly controls the Monte Carlo inputs and the reported sensitivity.
minor comments (2)
  1. [Spin calculation] The abstract states that the average spin parameter is O(10^{-3}); the main text should define the precise spin parameter used (e.g., dimensionless Kerr parameter) and report its distribution or variance.
  2. Figure captions for the Monte Carlo results should include the number of realizations performed and any convergence tests performed on the collapse fraction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major comments point by point below. Where appropriate we have revised the text to improve clarity and acknowledge limitations.

read point-by-point responses

  1. Referee: [Monte Carlo simulations and hoop-conjecture criterion] Monte Carlo simulations section (hoop-conjecture application): the collapse fraction is obtained by applying the hoop-conjecture criterion to the non-Gaussian overdensity field in the early matter-dominated phase. The paper extracts an emergent threshold depending on transition parameters and reheating efficiency, but provides no independent cross-check (e.g., compaction-function analysis or numerical-relativity validation) that the hoop criterion remains quantitatively reliable when pressure support is absent. Because the abundance varies by many orders of magnitude with O(1) changes in efficiency, even a modest systematic shift in the effective threshold would move the viable DM parameter space.

    Authors: We appreciate the referee's concern about the quantitative reliability of the hoop-conjecture criterion in a pressureless environment. The hoop conjecture is a standard geometric criterion employed in the PBH literature for matter-dominated collapse (see e.g. references in our Sec. 4). Our Monte Carlo implementation extracts an emergent, parameter-dependent threshold directly from the simulated non-Gaussian field, which already incorporates the effects of the prolonged matter era. Nevertheless, we acknowledge that an independent compaction-function or numerical-relativity cross-check would strengthen the result. Performing such simulations lies outside the scope of the present work; we will therefore add an explicit discussion of this modeling limitation and the associated systematic uncertainty in the revised manuscript. revision: partial

  2. Referee: [Peak-theory approach] Peak-theory approach (non-Gaussian overdensity): the claim that the overdensity statistics are fully captured by the peak-theory formalism rests on the assumption that the distribution generated by the slow phase transition can be modeled without additional corrections from the prolonged matter-dominated era. The manuscript should explicitly show how the peak-theory parameters are derived from the transition dynamics and reheating efficiency, as this directly controls the Monte Carlo inputs and the reported sensitivity.

    Authors: The peak-theory parameters are derived in Sec. 3 from the bubble nucleation rate and the duration of the matter-dominated phase set by the reheating efficiency. The overdensity field is constructed as a superposition of contributions from randomly nucleated bubbles, yielding a non-Gaussian distribution whose peak statistics are then computed analytically before being fed into the Monte Carlo. We agree that the derivation can be presented more transparently; we will add a dedicated subsection (or short appendix) that walks through the mapping from transition parameters to the peak-theory inputs, making the connection to the reported sensitivity explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds by applying peak theory to a non-Gaussian overdensity field generated by the slow phase transition, then using Monte Carlo simulations with the hoop-conjecture criterion to extract collapse probabilities and an emergent threshold that depends on transition parameters and reheating efficiency. The resulting PBH abundance is computed from these quantities and shown to be sensitive to the efficiency parameter. No equation or step quoted in the provided text reduces the abundance, threshold, or collapse fraction to a definitional identity with the inputs; the numerical extraction supplies independent content, and the sensitivity is an output of the calculation rather than a self-referential fit. The chain remains self-contained against the stated assumptions and external criterion.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the non-Gaussian overdensity statistics generated by the phase transition and on the applicability of the hoop-conjecture collapse criterion during the matter-dominated phase; these are treated as given rather than derived from first principles in the abstract.

free parameters (1)
  • reheating efficiency
    Varied by order unity to produce orders-of-magnitude changes in collapse fraction; appears as a tunable input controlling the duration of the matter era.
axioms (2)
  • domain assumption Hoop-conjecture criterion determines collapse of overdense regions
    Invoked to convert simulated overdensity peaks into black-hole formation probability.
  • domain assumption Overdensity field from slow phase transition is non-Gaussian
    Stated as the basis for the peak-theory approach.

pith-pipeline@v0.9.1-grok · 5731 in / 1473 out tokens · 21839 ms · 2026-06-27T16:08:52.023284+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

64 extracted references · 19 linked inside Pith

  1. [1]

    Zel’dovich and I.D

    Y.B. Zel’dovich and I.D. Novikov,The Hypothesis of Cores Retarded during Expansion and the Hot Cosmological Model,Sov. Astron.10(1967) 602

    1967

  2. [2]

    Carr and S.W

    B.J. Carr and S.W. Hawking,Black holes in the early Universe,Mon. Not. Roy. Astron. Soc. 168(1974) 399

    1974

  3. [3]

    B.J. Carr, K. Kohri, Y. Sendouda and J. Yokoyama,New cosmological constraints on primordial black holes,Phys. Rev. D81(2010) 104019 [0912.5297]

    Pith/arXiv arXiv 2010

  4. [4]

    B. Carr, F. Kuhnel and M. Sandstad,Primordial Black Holes as Dark Matter,Phys. Rev. D94 (2016) 083504 [1607.06077]

    Pith/arXiv arXiv 2016

  5. [5]

    B. Carr, M. Raidal, T. Tenkanen, V. Vaskonen and H. Veerm¨ ae,Primordial black hole constraints for extended mass functions,Phys. Rev. D96(2017) 023514 [1705.05567]

    Pith/arXiv arXiv 2017

  6. [6]

    B. Carr, K. Kohri, Y. Sendouda and J. Yokoyama,Constraints on primordial black holes,Rept. Prog. Phys.84(2021) 116902 [2002.12778]

    Pith/arXiv arXiv 2021

  7. [7]

    Ivanov, P

    P. Ivanov, P. Naselsky and I. Novikov,Inflation and primordial black holes as dark matter, Phys. Rev. D50(1994) 7173

    1994

  8. [8]

    Kodama, M

    H. Kodama, M. Sasaki and K. Sato,Abundance of Primordial Holes Produced by Cosmological First Order Phase Transition,Prog. Theor. Phys.68(1982) 1979

    1982

  9. [9]

    Hsu,Black Holes From Extended Inflation,Phys

    S.D.H. Hsu,Black Holes From Extended Inflation,Phys. Lett. B251(1990) 343

    1990

  10. [10]

    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. D105(2022) L021303 [2106.05637]

    arXiv 2022

  11. [11]

    Hashino, S

    K. Hashino, S. Kanemura and T. Takahashi,Primordial black holes as a probe of strongly first-order electroweak phase transition,Phys. Lett. B833(2022) 137261 [2111.13099]

    arXiv 2022

  12. [12]

    Jung and T

    T.H. Jung and T. Okui,Primordial black holes from bubble collisions during a first-order phase transition,Phys. Rev. D110(2024) 115014 [2110.04271]

    arXiv 2024

  13. [13]

    Kawana, T

    K. Kawana, T. Kim and P. Lu,PBH formation from overdensities in delayed vacuum transitions,Phys. Rev. D108(2023) 103531 [2212.14037]

    arXiv 2023

  14. [14]

    Lewicki, P

    M. Lewicki, P. Toczek and V. Vaskonen,Primordial black holes from strong first-order phase transitions,JHEP09(2023) 092 [2305.04924]

    arXiv 2023

  15. [15]

    Gouttenoire and T

    Y. Gouttenoire and T. Volansky,Primordial black holes from supercooled phase transitions, Phys. Rev. D110(2024) 043514 [2305.04942]. – 17 –

    arXiv 2024

  16. [16]

    Baldes and M.O

    I. Baldes and M.O. Olea-Romacho,Primordial black holes as dark matter: interferometric tests of phase transition origin,JHEP01(2024) 133 [2307.11639]

    arXiv 2024

  17. [17]

    Gouttenoire,First-Order Phase Transition Interpretation of Pulsar Timing Array Signal Is Consistent with Solar-Mass Black Holes,Phys

    Y. Gouttenoire,First-Order Phase Transition Interpretation of Pulsar Timing Array Signal Is Consistent with Solar-Mass Black Holes,Phys. Rev. Lett.131(2023) 171404 [2307.04239]

    arXiv 2023

  18. [18]

    Salvio,Supercooling in Radiative Symmetry Breaking: Theory Extensions, Gravitational Wave Detection and Primordial Black Holes,JCAP12(2023) 046 [2307.04694]

    A. Salvio,Supercooling in Radiative Symmetry Breaking: Theory Extensions, Gravitational Wave Detection and Primordial Black Holes,JCAP12(2023) 046 [2307.04694]

    arXiv 2023

  19. [19]

    Gouttenoire,Primordial black holes from conformal Higgs,Phys

    Y. Gouttenoire,Primordial black holes from conformal Higgs,Phys. Lett. B855(2024) 138800 [2311.13640]

    arXiv 2024

  20. [20]

    Jinno, J

    R. Jinno, J. Kume and M. Yamada,Super-slow phase transition catalyzed by BHs and the birth of baby BHs,Phys. Lett. B849(2024) 138465 [2310.06901]

    arXiv 2024

  21. [21]

    Banerjee and U.K

    I.K. Banerjee and U.K. Dey,Spinning primordial black holes from first order phase transition, JHEP07(2024) 006 [2311.03406]

    arXiv 2024

  22. [22]

    Flores, A

    M.M. Flores, A. Kusenko and M. Sasaki,Revisiting formation of primordial black holes in a supercooled first-order phase transition,Phys. Rev. D110(2024) 015005 [2402.13341]

    arXiv 2024

  23. [23]

    Lewicki, P

    M. Lewicki, P. Toczek and V. Vaskonen,Black Holes and Gravitational Waves from Slow First-Order Phase Transitions,Phys. Rev. Lett.133(2024) 221003 [2402.04158]

    arXiv 2024

  24. [24]

    Lewicki, P

    M. Lewicki, P. Toczek and V. Vaskonen,Black holes and gravitational waves from phase transitions in realistic models,Phys. Dark Univ.50(2025) 102075 [2412.10366]

    arXiv 2025

  25. [25]

    W.-Y. Ai, L. Heurtier and T.H. Jung,Primordial black holes from an interrupted phase transition,Phys. Rev. D113(2026) 023542 [2409.02175]

    arXiv 2026

  26. [26]

    Conaci, L

    A. Conaci, L. Delle Rose, P.S.B. Dev and A. Ghoshal,Slaying axion-like particles via gravitational waves and primordial black holes from supercooled phase transition,JHEP12 (2024) 196 [2401.09411]

    arXiv 2024

  27. [27]

    Kanemura, M

    S. Kanemura, M. Tanaka and K.-P. Xie,Primordial black holes from slow phase transitions: a model-building perspective,JHEP06(2024) 036 [2404.00646]

    arXiv 2024

  28. [28]

    Banerjee, F

    I.K. Banerjee, F. Rescigno and A. Salvio,Primordial black holes (as dark matter) from the supercooled phase transitions with radiative symmetry breaking,JCAP07(2025) 007 [2412.06889]

    arXiv 2025

  29. [29]

    Hashino, S

    K. Hashino, S. Kanemura, T. Takahashi, M. Tanaka and C.-M. Yoo,Super-critical primordial black hole formation via delayed first-order electroweak phase transition,JCAP09(2025) 006 [2501.11040]

    arXiv 2025

  30. [30]

    Murai, K

    K. Murai, K. Sakurai and F. Takahashi,Primordial black hole formation via inverted bubble collapse,JHEP07(2025) 065 [2502.02291]

    arXiv 2025

  31. [31]

    Banerjee, U.K

    I.K. Banerjee, U.K. Dey and S. Khalil,Primordial Black Holes and Gravitational Waves in the U(1) B−L extended inert doublet model: a first-order phase transition perspective,JHEP12 (2024) 009 [2406.12518]

    arXiv 2024

  32. [32]

    Kawana and K.-P

    K. Kawana and K.-P. Xie,Primordial black holes from a cosmic phase transition: The collapse of Fermi-balls,Phys. Lett. B824(2022) 136791 [2106.00111]

    arXiv 2022

  33. [33]

    J.B. Dent, B. Dutta, J. Kumar and D. Marfatia,Primordial black holes from Q-balls produced in a first-order phase transition,2505.21830

    arXiv

  34. [34]

    H. Deng, J. Garriga and A. Vilenkin,Primordial black hole and wormhole formation by domain walls,JCAP04(2017) 050 [1612.03753]

    Pith/arXiv arXiv 2017

  35. [35]

    Deng and A

    H. Deng and A. Vilenkin,Primordial black hole formation by vacuum bubbles,JCAP12(2017) 044 [1710.02865]. – 18 –

    Pith/arXiv arXiv 2017

  36. [36]

    Deng,Primordial black hole formation by vacuum bubbles

    H. Deng,Primordial black hole formation by vacuum bubbles. Part II,JCAP09(2020) 023 [2006.11907]

    arXiv 2020

  37. [37]

    Gouttenoire and E

    Y. Gouttenoire and E. Vitagliano,Primordial black holes and wormholes from domain wall networks,Phys. Rev. D109(2024) 123507 [2311.07670]

    arXiv 2024

  38. [38]

    Hawking,Black Holes From Cosmic Strings,Phys

    S.W. Hawking,Black Holes From Cosmic Strings,Phys. Lett. B231(1989) 237

    1989

  39. [39]

    Polnarev and R

    A. Polnarev and R. Zembowicz,Formation of Primordial Black Holes by Cosmic Strings,Phys. Rev. D43(1991) 1106

    1991

  40. [40]

    Caprini et al.,Science with the space-based interferometer eLISA

    C. Caprini et al.,Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions,JCAP04(2016) 001 [1512.06239]

    Pith/arXiv arXiv 2016

  41. [41]

    Caprini et al.,Detecting gravitational waves from cosmological phase transitions with LISA: an update,JCAP03(2020) 024 [1910.13125]

    C. Caprini et al.,Detecting gravitational waves from cosmological phase transitions with LISA: an update,JCAP03(2020) 024 [1910.13125]

    Pith/arXiv arXiv 2020

  42. [42]

    Ellis, M

    J. Ellis, M. Lewicki, J.M. No and V. Vaskonen,Gravitational wave energy budget in strongly supercooled phase transitions,JCAP06(2019) 024 [1903.09642]

    Pith/arXiv arXiv 2019

  43. [43]

    Banerjee and U.K

    I.K. Banerjee and U.K. Dey,Probing the origin of primordial black holes through novel gravitational wave spectrum,JCAP07(2023) 024 [2305.07569]

    arXiv 2023

  44. [44]

    Banerjee and U.K

    I.K. Banerjee and U.K. Dey,Gravitational wave probe of primordial black hole origin via superradiance,JCAP04(2024) 049 [2311.02876]

    arXiv 2024

  45. [45]

    Franciolini, Y

    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(2026) 171404 [2503.01962]

    Pith/arXiv arXiv 2026

  46. [46]

    X. Wang, C. Bal´ azs, R. Ding and C. Tian,How large are curvature perturbations from slow first-order phase transitions? A gauge-invariant analysis,2601.14412

    arXiv

  47. [47]

    Ai and K.-P

    W.-Y. Ai and K.-P. Xie,Reviving primordial black hole formation in slow first-order phase transitions,2605.11332

    Pith/arXiv arXiv

  48. [48]

    Escriv` a,PBH Formation from Spherically Symmetric Hydrodynamical Perturbations: A Review,Universe8(2022) 66 [2111.12693]

    A. Escriv` a,PBH Formation from Spherically Symmetric Hydrodynamical Perturbations: A Review,Universe8(2022) 66 [2111.12693]

    arXiv 2022

  49. [49]

    Musco,Threshold for primordial black holes: Dependence on the shape of the cosmological perturbations,Phys

    I. Musco,Threshold for primordial black holes: Dependence on the shape of the cosmological perturbations,Phys. Rev. D100(2019) 123524 [1809.02127]

    arXiv 2019

  50. [50]

    Escriv` a, C

    A. Escriv` a, C. Germani and R.K. Sheth,Universal threshold for primordial black hole formation,Phys. Rev. D101(2020) 044022 [1907.13311]

    arXiv 2020

  51. [51]

    Hambye, A

    T. Hambye, A. Strumia and D. Teresi,Super-cool Dark Matter,JHEP08(2018) 188 [1805.01473]

    Pith/arXiv arXiv 2018

  52. [52]

    W. Ye, Y. Gong, T. Harada, Z. Kang, K. Kohri, D. Saito et al.,Primordial black hole formation and spin in matter domination revisited,Phys. Rev. D112(2025) 103524 [2508.10070]

    arXiv 2025

  53. [53]

    Zeldovich,Gravitational instability: An Approximate theory for large density perturbations,Astron

    Y.B. Zeldovich,Gravitational instability: An Approximate theory for large density perturbations,Astron. Astrophys.5(1970) 84

    1970

  54. [54]

    Misner, K.S

    C.W. Misner, K.S. Thorne and J.A. Wheeler,Gravitation, W. H. Freeman, San Francisco (1973)

    1973

  55. [55]

    Bardeen, J.R

    J.M. Bardeen, J.R. Bond, N. Kaiser and A.S. Szalay,The Statistics of Peaks of Gaussian Random Fields,Astrophys. J.304(1986) 15

    1986

  56. [56]

    B. Carr, K. Dimopoulos, C. Owen and T. Tenkanen,Primordial Black Hole Formation During Slow Reheating After Inflation,Phys. Rev. D97(2018) 123535 [1804.08639]

    Pith/arXiv arXiv 2018

  57. [57]

    B.J. Carr, K. Kohri, Y. Sendouda and J. Yokoyama,Constraints on primordial black holes from the Galactic gamma-ray background,Phys. Rev. D94(2016) 044029 [1604.05349]. – 19 – [58]Planckcollaboration,Planck 2018 results. VI. Cosmological parameters,Astron. Astrophys. 641(2020) A6 [1807.06209]

    Pith/arXiv arXiv 2016

  58. [58]

    Dodelson and F

    S. Dodelson and F. Schmidt,Modern Cosmology, Academic Press (2020)

    2020

  59. [59]

    Zhang, X

    L. Zhang, X. Chen, M. Kamionkowski, Z.-g. Si and Z. Zheng,Constraints on radiative dark-matter decay from the cosmic microwave background,Phys. Rev. D76(2007) 061301 [0704.2444]

    Pith/arXiv arXiv 2007

  60. [60]

    Adams, S

    J.A. Adams, S. Sarkar and D.W. Sciama,CMB anisotropy in the decaying neutrino cosmology, Mon. Not. Roy. Astron. Soc.301(1998) 210 [astro-ph/9805108]. [62]EGRETcollaboration,EGRET observations of the extragalactic gamma-ray emission, Astrophys. J.494(1998) 523 [astro-ph/9709257]

    Pith/arXiv arXiv 1998

  61. [61]

    Wright,On the density of pbh’s in the galactic halo,Astrophys

    E.L. Wright,On the density of pbh’s in the galactic halo,Astrophys. J.459(1996) 487 [astro-ph/9509074]

    Pith/arXiv arXiv 1996

  62. [62]

    MacGibbon and B.J

    J.H. MacGibbon and B.J. Carr,Cosmic rays from primordial black holes,Astrophys. J.371 (1991) 447

    1991

  63. [63]

    Boudaud and M

    M. Boudaud and M. Cirelli,Voyager 1e ± Further Constrain Primordial Black Holes as Dark Matter,Phys. Rev. Lett.122(2019) 041104 [1807.03075]

    Pith/arXiv arXiv 2019

  64. [64]

    Niikura et al.,Microlensing constraints on primordial black holes with Subaru/HSC Andromeda observations,Nature Astron.3(2019) 524 [1701.02151]

    H. Niikura et al.,Microlensing constraints on primordial black holes with Subaru/HSC Andromeda observations,Nature Astron.3(2019) 524 [1701.02151]. – 20 –

    Pith/arXiv arXiv 2019