Primordial Black Hole mass growth from neutrinos during the radiation era

Ma\"el Gonin

REVIEW 4 major objections 5 minor 82 references

Primordial Black Hole mass growth from neutrinos during the radiation era

T0 review · 4 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Primordial black holes in the roughly 10^3 to 10^7 solar-mass range can grow substantially during the radiation era by absorbing neutrinos, overturning the usual assumption that PBH masses are frozen until matter domination.

desk verdict A novel neutrino-absorption growth mechanism for intermediate-mass PBHs, but the quantitative claims are undermined by an incorrect mass-function transformation and an internal ODE inconsistency. read the letter →

arxiv 2607.09285 v2 pith:6E67F2T5 submitted 2026-07-10 astro-ph.CO hep-th

Maël Gonin This is my paper · ORCID

classification astro-ph.COhep-th PACS 98.80.-k95.35.+d97.60.Lf

keywords primordialblackholesneutrinoabsorptionradiationeramassgrowththermalbathQCDtransitiondarkmatterspectrum

topics Dark Matter

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper challenges a long-standing assumption in primordial black hole physics: that black holes cannot gain mass during the radiation-dominated era because radiation is too tightly coupled to flow into them. It argues that neutrinos are different, because once the neutrino mean free path exceeds the black hole's horizon radius, the hole acts as a blackbody absorber of neutrinos and can grow substantially. For black holes formed near the QCD transition, with masses around 10^3 to 10^7 solar masses and a collapse fraction gamma = 0.55, the growth is large enough to shift the predicted electron-positron annihilation peak in the mass spectrum, create an additional intermediate-mass peak, and raise the dark matter fraction carried by PBHs. If the mechanism is real, heavy PBHs no longer require very large primordial density fluctuations, which relaxes constraints from spectral distortions and scalar-induced gravitational waves while changing how observational bounds must be applied.

What carries the argument

The load-bearing mechanism is the geometric-optics absorption cross section of a Schwarzschild black hole, σ = (27/64π) M^2/M_P^4, applied to neutrinos whose mean free path λ_ν = 1/(n_ν σ_weak) exceeds the horizon radius r_S. A weight function W = exp(-κ r_S/λ_ν), with κ = 0.5, 1, or 2, smoothly interpolates between absorbing and non-absorbing regimes, and the mass evolution is carried by the equation dR/dx = 12π γ_eff (δ_hf/α) x^3 R^3 W. The physics that decides how much growth occurs is the ratio δ_hf/α, the neutrino energy density relative to the total radiation density, which peaks after the QCD transition and creates the 'sweet spot' where growth is maximal.

What would settle it

Compute, for a Schwarzschild black hole immersed in a thermal neutrino bath, the exact absorption probability for neutrinos with momenta near the thermal peak and compare it with the geometric-optics value used in the mass-growth equation; if the probability is substantially below the geometric value when r_S is comparable to the mean free path, the predicted growth factors are correspondingly overestimated.

Watch

Extended reading notes

Core claim

The central claim is that the standard criterion for whether a PBH can absorb surrounding radiation, namely that the radiation mean free path exceeds the Schwarzschild radius, is satisfied for neutrinos at the relevant epochs even though it fails for photons. Starting from the weak-interaction neutrino mean free path, the paper defines a start temperature at which absorption becomes possible and solves a semi-classical mass-growth equation, dR/dx = 12π γ_eff (δ_hf/α) x^3 R^3 W, using the geometric-optics cross section σ = (27/64π) M^2/M_P^4 and a smooth weight function W = exp(-κ r_S/λ_ν) to model the transition from non-absorbing to absorbing regimes. With a collapse fraction γ = 0.55, PBHs

Load-bearing premise

The whole result rests on one premise: a black hole absorbs neutrinos as a perfect blackbody once the neutrino mean free path exceeds the hole's horizon radius, with a smoothly guessed transition in between; if any significant fraction of neutrinos scatter before reaching the horizon, the predicted growth largely disappears.

Editorial extensions

If this is right

PBHs in the 10^3 to 10^7 solar-mass range can grow substantially in the radiation era, with the largest growth for PBHs formed near the QCD transition when neutrinos carry a peak share of the radiation density.
The e+e- annihilation peak in the PBH mass spectrum shifts to larger masses, and an additional peak can appear in the intermediate-mass range depending on the collapse fraction γ.
The dark matter fraction in PBHs increases relative to its initial value; for γ = 0.55, the step-function transition raises f_PBH from 0.1 to 0.126.
To avoid runaway absorption, the collapse fraction must satisfy γ ≲ 0.55, otherwise the growth equation diverges within the model.
Observational constraints on PBHs must be remapped from formation mass to final mass; CMB and accretion bounds are not relieved and may be tightened for this mass window, while constraints tied to primordial fluctuations are partially evaded.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same absorption logic should apply to any weakly interacting relic that decouples early; if a feebly interacting species carries a non-negligible energy density at PBH formation, it would contribute to growth and could shift the optimal mass window.
The γ ≲ 0.55 bound is derived using the geometric cross section and an ad hoc transition weight; a first-principles calculation of neutrino absorption probabilities in the Schwarzschild metric near r_S ~ λ_ν would determine whether the growth factors are over- or under-estimated.
Because growth is fastest shortly after formation, PBHs formed with super-critical overdensities (δ > δ_c) would grow more than the δ = δ_c approximation used here, moving the additional peak to higher masses.
A clean observational discriminator is the intermediate-mass black hole mass spectrum: if the 10^3 to 10^7 solar-mass window remains empty in gravitational-wave and microlensing searches despite a large initial f_PBH, then either γ is lower or the absorption efficiency is suppressed.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What you should know first: this paper proposes something genuinely new. The author argues that in the radiation era, PBHs in the ~10^3–10^7 M_sun range can absorb neutrinos because the neutrino mean free path can exceed the Schwarzschild radius, overturning the standard Carr–Hawking no-growth picture. That is a real idea, and it deserves to be taken seriously. The paper is also refreshingly transparent: it uses CosEoS, explains the absorption regime, and explicitly lists many simplifications.

Now the soft spots. Two are load-bearing. First, the growth equation is internally inconsistent: Eq. (8) has dR/dx ∝ R^3, while Appendix B derives dR/dx = g(x) R^2 and solution R = 1/(1-G). Those are different equations with different runaway conditions. The paper never says which one was solved. Second, Eq. (11) mis-transforms the mass function under mass-dependent growth. Number conservation requires mapping M_i to M_f = R(M_i)M_i and including the Jacobian 1 + d ln R/d ln M_i. The paper instead multiplies the initial spectrum by R(M) at the same mass. That is wrong, and it undermines the claimed peak shifts and the new intermediate-mass peak in Fig. 4. The integrated f_PBH is unchanged by the Jacobian, which is probably why the error slipped through, but the differential spectrum is not.

Beyond that, the large growth only appears for gamma close to the divergence (0.55), and the transition factor W = exp(-kappa r_S / lambda_mfp) is an ad hoc interpolation. The author acknowledges these limitations. Still, as presented, the main quantitative claims are not robust.

Who is this for? People working on PBH mass evolution, PBH dark matter constraints, and early-universe plasma physics. They would get a useful prompt to think about neutrino absorption, but they should not take the spectra or tables at face value.

I would send this to peer review rather than desk-reject. The mechanism is novel, the topic is important, and the flaws are fixable: correct the ODE, fix the mass-function transformation, and scan gamma away from the runaway. But I would not cite the quantitative results until that happens.

Referee Report

4 major / 5 minor

Summary. The paper argues that primordial black holes (PBHs) in the mass range roughly 10^3–10^7 M_sun can grow substantially during the radiation era by absorbing neutrinos, contrary to the usual Carr–Hawking conclusion. The author models the onset of absorption by comparing the neutrino mean free path to the Schwarzschild radius, introduces an exponential weight function W for the transition, solves a mass-growth ODE, and applies the resulting growth factor R(M) to an extended PBH mass spectrum generated from the thermal history (QCD and e+e− features). The main reported consequences are a shift of the e+e− peak, a possible new intermediate-mass peak, and an increase in f_PBH from 0.1 to about 0.11–0.13 for γ=0.55.

Significance. If the mechanism is correct, it would overturn a long-standing assumption that PBHs do not grow during the radiation era and would affect interpretations of intermediate-mass BHs, JWST 'little red dots', and constraints based on the initial fluctuation scale. The paper is transparent about its toy-model nature, uses the CosEoS code for the thermal history, and provides an explicit ODE and analytic solution in Appendix B. However, the quantitative results as presented are not yet trustworthy because of an incorrect spectrum remapping and an inconsistency between the main-text ODE and the appendix derivation; the physical transition function is also ad hoc. The paper is therefore promising as a proposal, but it needs substantial revision before the central claims can be accepted.

major comments (4)

[Section IV, Eq. (11)] Eq. (11), d f_abs/dlnM = R(M) d f_init/dlnM, is not a valid remapping under a mass-dependent growth factor R(M_i)=M_f/M_i. Properly, each initial bin at M_i moves to M_f=R(M_i)M_i, so f_f(M_f)dlnM_f = R(M_i) f_i(M_i)dlnM_i, i.e. f_f(M_f)=R(M_i) f_i(M_i)/(1+dlnR/dlnM_i), with M_i determined implicitly. The paper evaluates both R and f_i at the same mass M and omits the Jacobian. For constant R this would give a pure shift f_f(M)=R f_i(M/R), whereas Eq. (11) gives R f_i(M) with no shift. Since R rises sharply across 10^3–10^7 M_sun, the claimed peak shifts and additional peak in Fig. 4 are quantitatively unreliable. The integrated f_PBH is unchanged by this error, but the spectral-shape claims in Section IV are not supported.
[Section III Eq. (8) vs Appendix B Eq. (B4)] The main text writes dR/dx = 12πγ_eff δ_hf/α x^3 R^3 W, but Appendix B derives dR/dx = g(x) R^2 and solves R=1/(1−G). The R^2 form follows from dM/dt=σ_hf ρ_R ∝ M^2 T^4 and dt/dx ∝ x. As printed, Eq. (8) is inconsistent with the equation actually solved and with the stated solution. The author must state which ODE was integrated; if R^3 was used, all growth factors, Tables I–II, and the γ bounds change; if R^2 was used, Eq. (8) must be corrected.
[Section III, Eqs. (6)–(7) and Fig. 1] The absorption criterion is λ_mfp ≥ r_S, with an ad hoc weight W = exp(−κ r_S/λ_mfp), κ in {0.5, 1, 2}. No derivation from kinetic theory or plasma physics is provided. The geometric-optics cross section Eq. (B1) assumes particles reach the horizon, whereas the mean-free-path condition concerns the ambient plasma; a particle with λ_mfp ≫ r_S may also pass through the hole's vicinity without being captured. Because the magnitude of R, the position of the extra peak, and the γ_div bound all depend on this transition (see Table I), the mechanism needs either a transport-equation calculation for neutrinos in a Schwarzschild background or a quantitative comparison with the Bondi/Carr–Hawking regime. As it stands, the central predictions are conditional on an unvalidated parametrization.
[Section IV, Fig. 2 and Table II] The large-growth branch is obtained with γ=0.55, which is close to the divergence value from Eq. (10). Since γ is treated as a free parameter and is not independently constrained, the statement that PBHs can grow significantly is a demonstration of the vicinity of the pole rather than a robust prediction. A physical prior on γ from critical-collapse/numerical-relativity fits would be needed to turn this into a falsifiable claim; otherwise the conclusions for IMBHs and LRDs rest on a tuned input.

minor comments (5)

[Abstract, Section V, and conclusion] Typos and formatting: 'predictede+e−peak' and the truncated 's' in the conclusion (which should read 'little red dots') should be fixed throughout.
[Fig. 2 caption] The caption refers to a red step function but the colors of the curves are not described; please add a legend or explicit color labels.
[Appendix A, Eq. (A2)] The amplitude A is introduced as a free normalization, but its relation to the primordial power spectrum amplitude is not stated; please clarify dimensions and typical values.
[Appendix B, Eq. (B2)–(B3)] The notation ρ_R and g_ρ in δ_hf is confusing because only neutrinos are absorbed. Please write g_ρ ≡ g_ν^ρ explicitly in the derivation to avoid the factor ambiguity.
[References] Several bibliographic entries have inconsistent or missing year fields (e.g. Refs. [18], [22], [23], [38], [40]); please check against the journal style.

Circularity Check

1 steps flagged · score 6.0 of 10

Spectral predictions are defined into Eq. (11): the updated mass function is R(M) × initial spectrum at the same M, so the 'additional peak' is R(M)'s own structure and the claimed e+e- peak shift is not derived; the growth ODE itself is independent.

self definitional [Section IV, Eq. (11) and Fig. 4; cf. Section III Eq. (8), Fig. 2]
"Comparing Fig. 2 and Fig. 4, we see the same structure from R(M_PBH) appearing in the PBH mass spectra. Since R depends on γ, the shifts of existing structures, f_abs_PBH (see Table II), the additional peak amplitude, and position also depend on γ."

The updated spectrum is defined by Eq. (11) as d f_abs/dlnM = R(M) d f_init/dlnM, evaluated at the same mass M. Thus any peak or feature in R(M) is multiplied directly into the output; the 'additional peak' at the absorption sweet spot is exactly R(M)'s peak imprinted by construction. The paper's own sentence confirms that the structure of R appears in the spectra. No number-conserving remapping M_final=R(M_init)M_init with Jacobian is performed, so the claimed e+e- peak shift is not even a consequence of Eq. (11). Since R(M) is controlled by the free parameters γ, κ and the ad hoc weight W, the headline spectral imprints reduce to the input model rather than being independent predictions.

full rationale

The mass-growth calculation in Section III is not circular in its core: Eq. (8) is an ODE for R(x) using microphysical inputs (λ_mfp from Eq. (6), σ_hf from Eq. (B1), γ_eff, and the explicitly ad hoc weight W), and the paper openly treats γ, κ, and A as free or uncertain parameters rather than fitting them to the claimed outputs. The growth curves in Figs. 2–3 are therefore a model-dependent computation, and the use of Ref. [28] for the evolution equation is an external input, not a self-citation. The self-citations [19,22] concern CosEoS and EoS tables, which are not the load-bearing part of the neutrino-absorption claim. The circularity is located in the spectrum-update step, Eq. (11). The updated mass function is defined as R(M) times the initial mass function at the same M, so any structure in R(M)—including the absorption-sweet-spot peak and the γ-dependent features—automatically appears in the output; the paper even states that the same structure from R appears in the spectra. The claimed e+e- peak shift is not derivable from Eq. (11) without a mass remapping and Jacobian, and the increase in f_PBH follows trivially from f_abs=∫R f_init dlnM with R>1. These spectral imprints are thus predictions in name only: they are the input R(M) (controlled by γ, κ, W) written into the spectrum by definition. This is a partial circularity (score 6), not a complete one, because the growth factor itself is obtained from an independent ODE and the paper is transparent about its toy-model status.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

No new particles or forces are invented. The free parameters (γ, A, κ, f_init) are all phenomenological: γ controls the amplitude of growth, A normalizes the abundance, κ sets the absorption onset, and f_init anchors the normalization. The axioms are standard cosmological/PBH modeling assumptions plus the ad hoc transition criterion, which is the weakest point.

free parameters (4)

collapse fraction γ = 0.1, 0.3, 0.4, 0.55 (tables and figures)
The ratio M_PBH = γ M_H at formation; treated as a free parameter. The claimed significant mass growth requires γ=0.55, close to the runaway boundary γ_div≈0.55.
fluctuation amplitude A = normalized to f_PBH^init = 0.1
A sets the amplitude of the Gaussian fluctuation spectrum (Eq. A2) and is used to normalize f_PBH; it is not constrained by observations in this study.
transition weight parameter κ = 0.5, 1, 2
Controls the smoothness of the non-absorbing/absorbing transition via W = exp(-κ r_S/λ_mfp) in Eq. (7); chosen by hand with no calibration.
initial dark-matter fraction f_PBH^init = 0.1
Used to normalize the spectra and compute f_PBH^abs; representative value, not derived from data.

assumptions (6)

domain assumption Geometric-optics cross section σ = (27/64π) M_BH^2/M_P^4 holds for all PBHs considered (r_S T >> 1 at all times).
Used in Eq. (B1) to compute absorption; convergence to geometric optics for spin 1/2 is cited from Refs. [48,50,51] but not re-derived.
domain assumption Only neutrinos contribute to absorption; cross-section is spin-blind and δ_hf is proportional to g_ν^ρ.
States in Section III/B.1; neglects absorption of photons, e+e−, and other species, which the author argues are blocked by short mean free paths.
ad hoc to paper The Carr–Hawking hydrodynamic/Bondi argument is bypassed whenever λ_mfp > r_S; the transition can be modeled by the weight W.
This is the load-bearing physical switch introduced in Section III, Eq. (7); no microscopic derivation is given for why a free-streaming neutrino is fully absorbed at r_S.
ad hoc to paper All PBH-forming fluctuations are exactly at threshold: δ = δ_c, so γ_eff = γ(1+δ_c).
Stated after Eq. (9); ignores super-critical fluctuations, which the author says are outside scope.
domain assumption Gaussian fluctuations and Press–Schechter statistics determine the PBH abundance.
Used in Appendix A, Eq. (A1)-(A3); standard in the PBH literature but still a modeling choice.
domain assumption The standard-model plasma EoS is correctly captured by CosEoS, and lepton/baryon asymmetries are negligible.
The thermal history and g_ρ(T) are taken from CosEoS references; the author explicitly defers asymmetric cases to future work (Section V).

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Pith. "Pith review of Primordial Black Hole mass growth from neutrinos during the radiation era." pith.science (2026). https://pith.science/paper/6E67F2T5

@misc{pith2026260709285,
  author       = {Pith},
  title        = {Pith review of: Primordial Black Hole mass growth from neutrinos during the radiation era},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6E67F2T5}},
  note         = {Machine review of arXiv:2607.09285}
}

abstract

We present a new picture of primordial black holes mass evolution through neutrino absorption. Using semi-classical approach and a closer look at the kinetic of the early plasma we revisit the thermal absorption of radiation by a population of primordial black holes ranging from $10^{-3}-10^9 M_\odot$ embedded in a thermal bath. We find significant mass growth of intermediate mass and supermassive PBHs, the effect shift the predicted $e^+e^-$ peak from thermal history. Depending on the value of the collapse fraction an additional peak around the intermediate mass range, might become significant. Moreover, because PBH grow from the thermal bath the fraction of DM in PBH $f_{\rm PBH}$ also change. These results revise the previous view on mass evolution of PBH and have implications for dark matter PBH observations.

Figures

Figures reproduced from arXiv: 2607.09285 by the authors.

**Figure 1.** FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** FIG. 2. Mass growth [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

Figures from the paper (2 more)

**Figure 3.** Figure 3: FIG. 3. Mass growth [PITH_FULL_IMAGE:figures/full_fig_p004_3.png]

**Figure 4.** Figure 4: FIG. 4. PBH mass spectra initially normalized at [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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Works this paper leans on

82 extracted references · 55 linked inside Pith

[1]

The high frequencyω≫1/r S and low frequency regimeω≪1/r S

Geometrical cross section A BH in a radiation bath has an associated cross sec- tion, the formula giving it depends on the absorption regime. The high frequencyω≫1/r S and low frequency regimeω≪1/r S. In our mass range, the high frequency regime, also written withr ST≫1 is fulfilled at all time. The PBH geometrical absorption cross section follows [25–27,...
[2]

During phase transitionδ c value dip henceγ eff follows

Onγ eff In the main text we defined an effective collapse frac- tionγ eff ∝(1 +δ c) defined by the overdensity, buta init is defined on the background.γ eff is now a function of the temperature as well. During phase transitionδ c value dip henceγ eff follows. Because the universe gets softer, the energy overdensity threshold is smaller and the PBH, if for...
[3]

B. J. Carr and S. W. Hawking, Mon. Not. Roy. Astron. Soc.168, 399 (1974)

1974
[4]

B. J. Carr, Astrophys. J.201, 1 (1975)

1975
[5]

B. Carr, K. Kohri, Y. Sendouda, and J. Yokoyama, Rept. Prog. Phys.84, 116902 (2021), arXiv:2002.12778 [astro- ph.CO]

arXiv 2021
[6]

Byrnes, G

C. Byrnes, G. Franciolini, T. Harada, P. Pani, and M. Sasaki, eds.,Primordial Black Holes, Springer Series in Astrophysics and Cosmology (Springer, 2025)

2025
[7]

B. Carr, A. J. Iovino, G. Perna, V. Vaskonen, and H. Veerm¨ ae, Riv. Nuovo Cim.49, 225 (2026), arXiv:2601.06024 [astro-ph.CO]

arXiv 2026
[8]

Baguiet al.(LISA Cosmology Working Group), Living Rev

E. Baguiet al.(LISA Cosmology Working Group), Living Rev. Rel.28, 1 (2025), arXiv:2310.19857 [astro-ph.CO]

arXiv 2025

Show all 82 references

[9]

A. G. Abacet al.(LIGO Scientific, VIRGO, KAGRA), Astrophys. J. Lett.1004, L22 (2026), arXiv:2508.18082 [gr-qc]

2026 arXiv
[10]

Abacet al.(LIGO Scientific, VIRGO, KAGRA), arXiv e-prints (2026), arXiv:2605.27225 [gr-qc]

N. Abacet al.(LIGO Scientific, VIRGO, KAGRA), arXiv e-prints (2026), arXiv:2605.27225 [gr-qc]

2026 arXiv
[11]

B. Carr, S. Clesse, J. Garc ´ ıa-Bellido, and F. K¨ uhnel, Phys. Dark Univ.31, 100755 (2021), arXiv:1906.08217 [astro-ph.CO]

2021 arXiv
[12]

S. Bird, I. Cholis, J. B. Mu˜ noz, Y. Ali-Ha ¨ ımoud, M. Kamionkowski, E. D. Kovetz, A. Raccanelli, and A. G. Riess, Phys. Rev. Lett.116, 201301 (2016), arXiv:1603.00464 [astro-ph.CO]

2016 arXiv
[13]

Sasaki, T

M. Sasaki, T. Suyama, T. Tanaka, and S. Yokoyama, Phys. Rev. Lett.117, 061101 (2016), [Erratum: Phys.Rev.Lett. 121, 059901 (2018)], arXiv:1603.08338 [astro-ph.CO]

2016 arXiv
[14]

Clesse and J

S. Clesse and J. Garc ´ ıa-Bellido, Phys. Dark Univ.15, 142 (2017), arXiv:1603.05234 [astro-ph.CO]

2017 arXiv
[15]

M. W. Choptuik, Phys. Rev. Lett.70, 9 (1993)

1993
[16]

C. R. Evans and J. S. Coleman, Phys. Rev. Lett.72, 1782 (1994), arXiv:gr-qc/9402041

1994 arXiv
[17]

J. C. Niemeyer and K. Jedamzik, Phys. Rev. D59, 124013 (1999), arXiv:astro-ph/9901292

1999 arXiv
[18]

Laine and M

M. Laine and M. Meyer, JCAP07(7), 035, arXiv:1503.04935 [hep-ph]

arXiv
[19]

Borsanyiet al., Nature539, 69 (2016), arXiv:1606.07494 [hep-lat]

S. Borsanyiet al., Nature539, 69 (2016), arXiv:1606.07494 [hep-lat]

2016 arXiv
[20]

C. T. Byrnes, M. Hindmarsh, S. Young, and M. R. S. Hawkins, JCAP2018(8), 041, arXiv:1801.06138 [astro- ph.CO]

arXiv
[21]

Gonin, G

M. Gonin, G. Hasinger, D. Blaschke, O. Ivanytskyi, and G. R¨ opke, Eur. Phys. J. A61, 170 (2025), arXiv:2505.05463 [hep-ph]

2025 arXiv
[22]

Ferreira, E

O. Ferreira, E. S. Fraga, M. Hippert, and J. Schaffner-Bielich, Phys. Rev. D112, 094009 (2025), arXiv:2507.06518 [hep-ph]

2025
[23]

Formaggio, F

L. Formaggio, F. Di Clemente, G. Yadav, A. Drago, and C. Ratti, Phys. Rev. D113, 023522 (2026), arXiv:2508.00094 [astro-ph.CO]

2026 arXiv
[24]

Gonin, O

M. Gonin, O. Ivanytskyi, D. Blaschke, and G. Hasinger, arXiv e-prints (2026), arXiv:2604.12581 [astro-ph.CO]

2026 arXiv
[25]

B¨ odeker, F

D. B¨ odeker, F. K¨ uhnel, I. M. Oldengott, and D. J. Schwarz, Phys. Rev. D103, 063506 (2021), arXiv:2011.07283 [astro-ph.CO]

2021 arXiv
[26]

Khlopov, Symmetry16, 1487 (2024)

M. Khlopov, Symmetry16, 1487 (2024)

2024
[27]

Y. B. Zel’dovich and I. D. Novikov, Sov. Astron.10, 602 (1967)

1967
[28]

P. S. Custodio and J. E. Horvath, Phys. Rev. D58, 023504 (1998), arXiv:astro-ph/9802362

1998 arXiv
[29]

P. S. Custodio and J. E. Horvath, Gen. Rel. Grav.34, 1895 (2002), arXiv:gr-qc/0203031

2002 arXiv
[30]

M. R. Haque, R. Karmakar, and Y. Mambrini, arXiv e- prints (2026), arXiv:2601.16717 [astro-ph.CO]

2026
[31]

Husdal, Galaxies4, 78 (2016), arXiv:1609.04979 [astro-ph.CO]

L. Husdal, Galaxies4, 78 (2016), arXiv:1609.04979 [astro-ph.CO]

2016 arXiv
[32]

Franciolini, I

G. Franciolini, I. Musco, P. Pani, and A. Urbano, Phys. Rev. D106, 123526 (2022), arXiv:2209.05959 [astro- ph.CO]

2022 arXiv
[33]

Escriv` a, E

A. Escriv` a, E. Bagui, and S. Clesse, JCAP05(05), 004, arXiv:2209.06196 [astro-ph.CO]

arXiv
[34]

Musco, K

I. Musco, K. Jedamzik, and S. Young, Phys. Rev. D109, 083506 (2024), arXiv:2303.07980 [astro-ph.CO]

2024 arXiv
[35]

Musco and J

I. Musco and J. C. Miller, Class. Quant. Grav.30, 145009 (2013), arXiv:1201.2379 [gr-qc]

2013 arXiv
[36]

K. H. Choi, J. Creswell, F. Kuhnel, and D. J. Schwarz, Phys. Rev. D113, 063528 (2026), arXiv:2501.17936 [astro-ph.CO]

2026 arXiv
[37]

[28, 42, 43]

We refer to a plasma fulfilling this set of conditions as a ‘thermal bath’, following Refs. [28, 42, 43]
[38]

Y. B. Zel’dovich and I. D. Novikov, Soviet Physics Us- pekhi8, 522 (1966)

1966
[39]

Bondi, Mon

H. Bondi, Mon. Not. Roy. Astron. Soc.112, 195 (1952)

1952
[40]

S. Das, M. R. Haque, J. Kalita, R. Karmakar, and D. Maity, Phys. Rev. D112, 123540 (2025), arXiv:2505.15419 [astro-ph.CO]

2025
[41]

D. S. Kallifatides, T. Papanikolaou, and E. N. Saridakis, arXiv e-prints (2026), arXiv:2601.18708 [astro-ph.CO]

2026
[42]

Chatterjee, J

A. Chatterjee, J. Kalita, and D. Maity, JHEP04(04), 026, arXiv:2512.07284 [hep-th]

arXiv
[43]

Dai and D

D.-C. Dai and D. Stojkovic, Phys. Rev. D108, 084024 (2023), arXiv:2309.13511 [gr-qc]

2023 arXiv
[44]

Barrau, K

A. Barrau, K. Martineau, and C. Renevey, Phys. Rev. D 106, 023509 (2022), arXiv:2203.13297 [gr-qc]

2022 arXiv
[45]

Barrau, K

A. Barrau, K. Martineau, and H. Zelgoum, Mod. Phys. Lett. A41, 2550227 (2026), arXiv:2511.01326 [gr-qc]

2026
[46]

S. W. Hawking, Commun. Math. Phys.43, 199 (1975), 8 [Erratum: Commun.Math.Phys. 46, 206 (1976)]

1975
[47]

D. N. Page, Phys. Rev. D13, 198 (1976)

1976
[48]

A. D. Dolgov, Phys. Rept.370, 333 (2002), arXiv:hep- ph/0202122

2002
[49]

E. W. Kolb and M. S. Turner,The Early Universe, Vol. 69 (Taylor and Francis, 2019)

2019
[50]

W. G. Unruh, Phys. Rev. D14, 3251 (1976)

1976
[51]

N. G. Sanchez, Phys. Rev. D18, 1030 (1978)

1978
[52]

Doran, A

C. Doran, A. Lasenby, S. Dolan, and I. Hinder, Phys. Rev. D71, 124020 (2005), arXiv:gr-qc/0503019

2005 arXiv
[53]

Dolan, C

S. Dolan, C. Doran, and A. Lasenby, Phys. Rev. D74, 064005 (2006), arXiv:gr-qc/0605031

2006 arXiv
[54]

L. C. B. Crispino, E. S. Oliveira, A. Higuchi, and G. E. A. Matsas, Phys. Rev. D75, 104012 (2007)

2007
[55]

[25] [Eqs

Interestingly, the critical value 32/81 already appears in Ref. [25] [Eqs. (2)–(3) and surrounding discussion], pre- dating its modern derivation [28]
[56]

De Luca, G

V. De Luca, G. Franciolini, P. Pani, and A. Riotto, JCAP 04(04), 052, arXiv:2003.02778 [astro-ph.CO]

2003 arXiv
[57]

De Luca, G

V. De Luca, G. Franciolini, P. Pani, and A. Riotto, Phys. Rev. D102, 043505 (2020), arXiv:2003.12589 [astro- ph.CO]

2020 arXiv
[58]

P. D. Serpico, V. Poulin, D. Inman, and K. Kohri, Phys. Rev. Res.2, 023204 (2020), arXiv:2002.10771 [astro- ph.CO]

2020 arXiv
[59]

Facchinetti, M

G. Facchinetti, M. Lucca, and S. Clesse, Phys. Rev. D 107, 043537 (2023), arXiv:2212.07969 [astro-ph.CO]

2023 arXiv
[60]

Agius, R

D. Agius, R. Essig, D. Gaggero, F. Scarcella, G. Suczewski, and M. Valli, JCAP07(07), 003, arXiv:2403.18895 [hep-ph]

arXiv
[61]

B. Carr, S. Clesse, J. Garcia-Bellido, M. Hawkins, and F. Kuhnel, Phys. Rept.1054, 1 (2024), arXiv:2306.03903 [astro-ph.CO]

2024 arXiv
[62]

Carr and J

B. Carr and J. Silk, Mon. Not. Roy. Astron. Soc.478, 3756 (2018), arXiv:1801.00672 [astro-ph.CO]

2018 arXiv
[63]

H¨ aberle, N

M. H¨ aberle, N. Neumayer, A. Seth, A. Bellini, M. Li- bralato, H. Baumgardt, M. Whitaker, A. Dumont, M. Alfaro-Cuello, J. Anderson, C. Clontz, N. Kacharov, S. Kamann, A. Feldmeier-Krause, A. Milone, M. S. Nitschai, R. Pechetti, and G. van de Ven, Nature (Lon- don)631, 285 (20...

2024 arXiv
[64]

Huang, Q

Y. Huang, Q. Li, J. Liu, X. Dong, H. Zhang, Y. Lu, and C. Du, Natl. Sci. Rev.12, nwae347 (2025), arXiv:2406.00923 [astro-ph.GA]

2025 arXiv
[65]

Bogdanet al., Nature Astron.8, 126 (2024), arXiv:2305.15458 [astro-ph.GA]

A. Bogdanet al., Nature Astron.8, 126 (2024), arXiv:2305.15458 [astro-ph.GA]

2024 arXiv
[66]

Maiolino, J

R. Maiolino, J. Scholtz, J. Witstok, S. Carniani, F. D’Eugenio, A. de Graaff, H. ¨Ubler, S. Tacchella, E. Curtis-Lake, S. Arribas, A. Bunker, S. Charlot, J. Chevallard, M. Curti, T. J. Looser, M. V. Maseda, T. D. Rawle, B. Rodr ´ ıguez del Pino, C. J. Willott, E. Egami, D. J. ...

2024 arXiv
[67]

Mattheeet al., Astrophys

J. Mattheeet al., Astrophys. J.963, 129 (2024), arXiv:2306.05448 [astro-ph.GA]

2024 arXiv
[68]

Pacucci, B

F. Pacucci, B. Nguyen, S. Carniani, R. Maiolino, and X. Fan, Astrophys. J. Lett.957, L3 (2023), arXiv:2308.12331 [astro-ph.GA]

2023 arXiv
[69]

Liu and V

B. Liu and V. Bromm, Astrophys. J. Lett.937, L30 (2022), arXiv:2208.13178 [astro-ph.CO]

2022 arXiv
[70]

Dayal, Astron

P. Dayal, Astron. Astrophys.690, A182 (2024), arXiv:2407.07162 [astro-ph.GA]

2024 arXiv
[71]

Zhang, B

S. Zhang, B. Liu, V. Bromm, and F. K¨ uhnel, Astro- phys. J. Lett.1000, L19 (2026), arXiv:2512.14066 [astro- ph.GA]

2026
[72]

De Luca, L

V. De Luca, L. Del Grosso, G. Franciolini, K. Kritos, E. Berti, D. J. D’Orazio, and J. Silk, Phys. Rev. Lett. 136, 231402 (2026), arXiv:2512.19666 [astro-ph.CO]

2026
[73]

Chluba, A

J. Chluba, A. L. Erickcek, and I. Ben-Dayan, Astrophys. J.758, 76 (2012), arXiv:1203.2681 [astro-ph.CO]

2012 arXiv
[74]

Nakama, B

T. Nakama, B. Carr, and J. Silk, Phys. Rev. D97, 043525 (2018), arXiv:1710.06945 [astro-ph.CO]

2018 arXiv
[75]

C. T. Byrnes, J. Lesgourgues, and D. Sharma, JCAP09 (09), 012, arXiv:2404.18475 [astro-ph.CO]

arXiv
[76]

Dom` enech, Universe7, 398 (2021), arXiv:2109.01398 [gr-qc]

G. Dom` enech, Universe7, 398 (2021), arXiv:2109.01398 [gr-qc]

2021 arXiv
[77]

B. Cyr, T. Kite, J. Chluba, J. C. Hill, D. Jeong, S. K. Acharya, B. Bolliet, and S. P. Patil, Mon. Not. Roy. Astron. Soc.528, 883 (2024), arXiv:2309.02366 [astro- ph.CO]

2024 arXiv
[78]

Cecchini, G

C. Cecchini, G. Franciolini, and M. Pieroni, Phys. Rev. D111, 123536 (2025), arXiv:2503.10805 [astro-ph.CO]

2025 arXiv
[79]

Aghanimet al.(Planck), Astron

N. Aghanimet al.(Planck), Astron. Astrophys.641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]

2020 arXiv
[80]

Braglia, J

M. Braglia, J. Garcia-Bellido, and S. Kuroyanagi, JCAP 12(12), 012, arXiv:2110.07488 [astro-ph.CO]

arXiv
[81]

B. J. Carr, K. Kohri, Y. Sendouda, and J. Yokoyama, Phys. Rev. D81, 104019 (2010), arXiv:0912.5297 [astro- ph.CO]

2010 arXiv
[82]

B. Carr, F. Kuhnel, and L. Visinelli, Mon. Not. Roy. Astron. Soc.501, 2029 (2021), arXiv:2008.08077 [astro- ph.CO]

arXiv

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

The high frequencyω≫1/r S and low frequency regimeω≪1/r S

Geometrical cross section A BH in a radiation bath has an associated cross sec- tion, the formula giving it depends on the absorption regime. The high frequencyω≫1/r S and low frequency regimeω≪1/r S. In our mass range, the high frequency regime, also written withr ST≫1 is fulfilled at all time. The PBH geometrical absorption cross section follows [25–27,...

[2] [2]

During phase transitionδ c value dip henceγ eff follows

Onγ eff In the main text we defined an effective collapse frac- tionγ eff ∝(1 +δ c) defined by the overdensity, buta init is defined on the background.γ eff is now a function of the temperature as well. During phase transitionδ c value dip henceγ eff follows. Because the universe gets softer, the energy overdensity threshold is smaller and the PBH, if for...

[3] [3]

B. J. Carr and S. W. Hawking, Mon. Not. Roy. Astron. Soc.168, 399 (1974)

1974

[4] [4]

B. J. Carr, Astrophys. J.201, 1 (1975)

1975

[5] [5]

B. Carr, K. Kohri, Y. Sendouda, and J. Yokoyama, Rept. Prog. Phys.84, 116902 (2021), arXiv:2002.12778 [astro- ph.CO]

arXiv 2021

[6] [6]

Byrnes, G

C. Byrnes, G. Franciolini, T. Harada, P. Pani, and M. Sasaki, eds.,Primordial Black Holes, Springer Series in Astrophysics and Cosmology (Springer, 2025)

2025

[7] [7]

B. Carr, A. J. Iovino, G. Perna, V. Vaskonen, and H. Veerm¨ ae, Riv. Nuovo Cim.49, 225 (2026), arXiv:2601.06024 [astro-ph.CO]

arXiv 2026

[8] [8]

Baguiet al.(LISA Cosmology Working Group), Living Rev

E. Baguiet al.(LISA Cosmology Working Group), Living Rev. Rel.28, 1 (2025), arXiv:2310.19857 [astro-ph.CO]

arXiv 2025