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Unexpected shape of the primordial black hole mass function

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Broad enhancements of the primordial power spectrum make heavy black holes, not light ones, dominate.

desk verdict Real new result on bimodal PBH mass functions, with a load-bearing but clearly flagged extrapolation of the collapse threshold; worth a serious referee. read the letter →

arxiv 2412.07709 v3 pith:KD2LXODQ submitted 2024-12-10 astro-ph.CO gr-qchep-th

Jacopo Fumagalli , Jaume Garriga , Cristiano Germani , Ravi K. Sheth This is my paper · ORCID
classification astro-ph.COgr-qchep-th PACS 98.80.-k04.70.-s
keywords primordialblackholesmassfunctioncompactioncriticalcollapsepeakstatisticsbroadpowerspectrumTypeIIfluctuationsdarkmatter
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 establishes a counterintuitive result for primordial black holes (PBHs) formed from nearly Gaussian curvature perturbations: when the curvature power spectrum is enhanced over a broad range of scales, between an infrared scale $k_{\rm IR}$ and an ultraviolet scale $k_{\rm UV}$, the PBH mass function is bimodal, and the dominant, heavier peak sits near the infrared scale rather than the ultraviolet one. Standard estimates for a broad, nearly scale-invariant spectrum predict that light black holes formed at the ultraviolet scale dominate, because they form earlier during radiation domination. The authors show analytically and numerically that the statistics of peaks select thin spherical shells with infrared radius and ultraviolet thickness, pushing the typical collapsing configurations toward large values of the curvature parameter $w$ and toward the Type I/Type II boundary. If this is right, predicted PBH abundances are higher at large masses, so overproduction bounds on the power spectrum amplitude tighten and the mass range in which PBHs could be all of the dark matter narrows.

What carries the argument

The engine is the joint peak statistics of the compaction function $C=g(1-3g/8)$, with $g=-(4/3)r\,\partial_r\zeta$, conditioned on a radial extremum $v=0$; the curvature parameter $w\equiv-r^2\partial_r^2g$ controls the collapse threshold $g_c(w)$. For a broad spectrum the correlators simplify to $\tilde\gamma\simeq\gamma\simeq1/\sqrt{\ln\alpha}\ll1$ and $\sigma_w^2\simeq(2/9)A_s(r\alpha)^4$, and the saddle point of the abundance integrand, Eq.~(17), sits at $w_{\max}/\sigma_w=(\gamma\nu_c+\sqrt{\gamma^2\nu_c^2+4\lambda})/2$ with $\lambda\simeq2.28$ set by the large-$w$ peak-theory shape factor $f\simeq(w/\sigma_w)^3$. Combined with $\sigma_g^2\simeq(8/9)A_s[\ln\alpha+{\rm CosIntegral}(2r)+(\sin^2r-r\sin2r)/r^2]$, whose maximum is at $r\simeq1.69/k_{\rm IR}$, this yields the infrared peak and the analytic approximation Eq.~(24). The condition $A_s(\ln\alpha)^2\gtrsim0.05$ separates the bimodal regime from a single-peak regime whose abundance is negligible.

What would settle it

A set of numerical relativity simulations that initialize thin spherical-shell compaction profiles with $w\gg1$ and scale separation $\alpha\gtrsim100$, measuring the critical threshold $g_c(w)$ and the resulting PBH mass, would settle the claim. If $g_c(w)$ does not keep rising toward $4/3$, or if such shells do not collapse, the predicted infrared peak at $r\simeq1.69/k_{\rm IR}$ and the scaling $M_{\rm heavy}/M_{\rm UV}\propto\alpha^{2-4\gamma_{\rm cr}}$ fail. A cheaper partial test is to evaluate the full mass-function integral for a smooth broad spectrum and check whether the bimodal shape and infrared dominance survive.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that the PBH mass function from a flat plateau $P_\zeta(k)=A_s\,\theta(k-k_{\rm IR})\theta(k_{\rm UV}-k)$ with $\alpha=k_{\rm UV}/k_{\rm IR}\gtrsim25$ develops two peaks: a subdominant ultraviolet peak near $r\simeq4/k_{\rm UV}$ and a dominant infrared peak near $r\simeq1.69/k_{\rm IR}$, with the heavy peak scaling as $M_{\rm heavy}/M_{\rm UV}\propto\alpha^{2-4\gamma_{\rm cr}}$ for $\gamma_{\rm cr}\simeq0.36$. The infrared peak arises because, conditioned on meeting the collapse threshold, the most probable compaction function is a thin spherical shell of radius $\sim1/k_{\rm IR}$ and thickness $\sim1/k_{\rm UV}$, corresponding to large $w$ and to $g$ close to $4/3$, i.e. near the Type I/Type II boundary. The paper derives this result with a saddle-point estimate of the peak-theory integral and verifies it by numerically evaluating the full mass-function integral, for power-spectrum amplitudes satisfying $A_s(\ln\alpha)^2\gtrsim0.05$.

Load-bearing premise

The heavy, infrared peak rests on extrapolating the analytic collapse threshold to very narrow, strongly curved profiles with $w\gg1$, where the paper notes dedicated simulations are still missing; if the threshold saturates at a different value there, the peak's height and location shift.

Editorial extensions

If this is right

  • If the central claim holds, standard estimates for broad spectra miss the dominant heavy-PBH population, so overproduction bounds on $A_s$ tighten because the infrared peak adds a large-mass contribution that was previously neglected.
  • The viable window for PBHs as all of the dark matter narrows, especially in the asteroid-mass range: the subdominant ultraviolet population must avoid Hawking-evaporation bounds, which pushes the dominant infrared population to larger masses.
  • Gravitational-wave and lensing constraints that assume a single-peaked mass function need to be re-evaluated for enhancements spanning $N\gtrsim3$ e-folds, where $\alpha\gtrsim25$ already produces bimodality.
  • Typical collapsing configurations are thin shells with large $w$, so Type II fluctuations, or the Type I/Type II boundary, become statistically relevant even though their per-profile probability is small.
  • The narrow-spectrum approximation is not valid for the overall mass function once $\alpha\gtrsim25$, and even the ultraviolet peak shifts to $r\simeq4/k_{\rm UV}$ once $\alpha\gtrsim1.5$, according to the paper's numerical checks.

Reading between the lines

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

  • A direct test would be numerical relativity simulations that evolve thin-shell compaction profiles with $w\gg1$ and scale separation $\alpha\gtrsim100$; the paper explicitly states that such simulations are needed, and their outcome would confirm or refute the infrared peak.
  • The same peak-statistics mechanism should operate for smooth broad spectra, not just the sharp plateau used for analytic control; the paper reports numerical robustness, so a lognormal or running-power-law enhancement is a clean place to look for the same bimodal signature.
  • If the infrared peak is real, mapping a fixed gravitational-wave stochastic background onto $A_s$ should be revised: a given background would be compatible with a lower amplitude than single-peak estimates suggest, because the heavy population is more abundant.
  • The result implies that replacing a broad spectrum by a Dirac-delta spectrum of the same logarithmic area, a common shortcut, systematically underestimates the heavy-mass tail; only the ultraviolet peak is captured by that shortcut.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This Letter computes the PBH mass function for a flat, enhanced curvature power spectrum bounded by infrared and ultraviolet scales. Using compaction-function peak theory together with the critical-collapse mass relation, the authors integrate the abundance over smoothing radius, peak curvature w, and amplitude g, and find that the mass function is bimodal: a UV feature at r≈4/k_UV and a dominant heavy feature at r≈1.69/k_IR, with an analytic saddle-point estimate reproducing the numerical result. They interpret the heavy peak as arising from thin spherical shells whose profiles have large curvature w, close to the Type I/Type II boundary, and they discuss consequences for PBH overproduction bounds and for Type II collapse.

Significance. If correct, the main result overturns the standard expectation that a broad enhancement is dominated by light PBHs associated with the UV cutoff, and it would strengthen overproduction constraints in the asteroid-mass window. The paper's strengths are its transparent analytic estimates (Eqs. 17-22 and 24), which agree with the direct numerical integration, and the explicit condition in Eq. (16) that delimits the bimodal regime. The bimodality is not put in by hand; it emerges from the statistics. The principal caveat is that the dominant IR peak depends on an extrapolated form of the collapse threshold at large w, Eq. (7), and this dependence is acknowledged but not quantified. On balance the result is significant and plausible, but it needs a robustness or sensitivity analysis before publication.

major comments (3)
  1. [Sec. III, Eq. (7) and Fig. 2] The dominant IR feature rests on the large-w behavior of the collapse threshold. For α=100 and r=1.69, the relevant values of w are of order γνcσw≃(2/3)(rα)^2/ln α, i.e. thousands, far beyond the regime in which Eq. (7) has been calibrated. The exact numerical curves in Fig. 2 use the same analytic fit gc(w) in the lower integration limit of Eq. (12), so they cannot independently confirm the extrapolation. As the authors note in the Outlook, dedicated simulations are required. If the true threshold saturates more slowly, or with a different coefficient, the amplitude and scale of the IR peak (Eqs. 20 and 24, with Mheavy/MUV∝α^{2−4γcr}) shift; if it saturates more quickly, the [gc(w),4/3] window narrows and the IR contribution may no longer dominate. I therefore regard the central prediction as conditional on Eq. (7) and ask for an explicit sensitivity analysis, for example using alternative asymptotic forms for gc(w), in the revised manuscript.
  2. [Sec. V and Eq. (12)] The claim that Type II fluctuations may contribute significantly is not backed by a computation. Equation (12) restricts the g integral to g≤4/3, so all Type II fluctuations are excluded by construction. The fact that the dominant Type I configurations are close to the boundary is suggestive, but the statement in the Abstract that the results imply a "higher-than-expected abundance of PBH originating from Type II initial fluctuations" goes beyond the quantitative content of the paper. Please either include an explicit estimate of a Type II contribution with suitable modeling, or clearly soften the claim to a qualitative conjecture.
  3. [Footnote 2 and Eq. (22)] The robustness of the result for power spectra without sharp cutoffs is asserted in footnote 2 but not demonstrated. Since Eq. (22) contains sharp-boundary terms such as CosIntegral(2r), and since the IR peak is located at r≈1.69, it is important to check that this feature is not a cutoff artifact. A figure or a short quantitative statement showing that smooth cutoffs produce the same bimodal structure would remove this uncertainty.
minor comments (5)
  1. [Sec. IV] There is a duplicated word in "As shown in in Figure 2"; please correct it.
  2. [Sec. II] There is a duplicated word in "a wide class of of profiles"; please correct it.
  3. [Sec. III, Eq. (16)] Equation (16) is quoted as the bimodality condition, but the derivation of the numerical coefficient 0.05 is not shown; a brief derivation or reference would help the reader assess its range of validity.
  4. [Sec. II, below Eq. (3)] The phrase "withv(r)=0" should read "with v(r)=0".
  5. [References] Reference [3] is printed with a DOI but without a visible publication venue; completing the bibliographic entry would be helpful.

Circularity Check

0 steps flagged · score 2.0 of 10

No construction-level circularity: the bimodal mass function is an emergent integral over an externally calibrated threshold; self-citations are not load-bearing.

full rationale

The claimed prediction—a bimodal mass function with an IR peak—is not an input to the calculation. The calculation starts from the top-hat power spectrum Eq. (15), the compaction-function peak statistics of Refs. [12,13], and the analytic threshold gc(w) of Eq. (7) from Ref. [14]. Although Refs. [12,13,14] are co-authored by the present authors, they are not circularly invoked: Ref. [14] is said to match the numerical simulations of Ref. [15] to within a few percent, and the critical-scaling mass formula is taken from Ref. [16]. The IR feature emerges from integrating Eq. (12): for rα≫1 the joint PDF in Eq. (10) peaks at large w (Eq. (19)), and the saddle-point estimate (20) combined with the α-independent maximum of σg(r) at r≃1.69/kIR (Eq. (22)) produces the heavy peak. No parameter of the calculation is fitted to the target mass function; the analytic IR formula (24) is checked against the same numerical integration, not against external data. The paper explicitly flags the one fragile step—the large-w extrapolation of gc(w) toward 4/3 (Outlook: 'dedicated simulations ... are needed to confirm the extrapolation ... of a threshold saturation in the Type I case'). That is an acknowledged assumption about input physics, not a reduction of the output to the input; if the extrapolation is wrong the IR peak would shift, but this is a correctness risk, not circularity. Because the formalism is substantially self-cited, the score is 2 rather than 0; the self-citations are present but not load-bearing in the circular sense.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The calculation is built on the standard peak-theory framework and on numerical fits from earlier PBH literature. No new entities or ad hoc constants are introduced to force the bimodal shape; the main external loads are the Gaussian random field assumption, the fitted collapse threshold, and the critical-scaling parameters K and gamma_cr. As and alpha are scenario inputs rather than fitted parameters.

free parameters (2)
  • K (critical collapse normalization) = approximately 6
    Taken from Musco et al. [16], weakly profile-dependent; assumed constant across w. It scales the absolute mass and abundance in Eqs. (14), (23), and (24).
  • gamma_cr (critical collapse exponent) = approximately 0.36
    Taken from Escriva et al. [14]; sets the mass scaling and the slope of the infrared mass tail in Eq. (24).
assumptions (6)
  • domain assumption Fourier modes zeta_k are independent and Gaussian-distributed.
    Invoked in Section II; required for the bivariate Gaussian peak statistics that produce the mass function.
  • domain assumption High peaks of a Gaussian random field are approximately spherically symmetric (BBKS).
    Used to reduce the collapse configuration to a radial profile g(r; x0); basis for the spherical peak conditions.
  • standard math The superhorizon compaction function is C = g(1 - 3g/8), with Type I/II boundary at g=4/3.
    Follows from the gradient expansion of the metric and is taken from Harada et al. [8]; used to translate the collapse threshold to a threshold on g.
  • domain assumption The threshold gc(w) is given by the numerical fit of Escriva, Germani and Sheth [14], Eq. (7), and saturates at 4/3 for large w.
    Central to the large-w saddle point that produces the infrared peak; the paper itself calls for simulations to confirm the large-w extrapolation.
  • domain assumption The collapsed mass follows M approximately K M_H(r) (C - C_c(w))^gamma_cr with K=6 and gamma_cr=0.36.
    Critical scaling from Musco et al. [16] and Escriva et al. [14]; used to convert the statistics of g and w into a mass function.
  • domain assumption A flat plateau power spectrum with sharp cutoffs (Eq. 15) represents a broad enhancement.
    Chosen for analytical control; the authors state they verified robustness for smooth cutoffs, but the sharp-cutoff form is the main explicit example.

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Cite this review

Pith. "Pith review of Unexpected shape of the primordial black hole mass function." pith.science (2026). https://pith.science/paper/KD2LXODQ

@misc{pith2026241207709,
  author       = {Pith},
  title        = {Pith review of: Unexpected shape of the primordial black hole mass function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KD2LXODQ}},
  note         = {Machine review of arXiv:2412.07709}
}
read the original abstract

In a Universe with nearly-Gaussian initial curvature perturbations, the abundance of primordial black holes can be derived from the curvature power spectrum. When the latter is enhanced within a narrow range around a characteristic scale, the resulting mass function has a single distinct peak, corresponding to Schwarzschild radii set by the horizon entry time of that scale. In contrast, we show (both numerically and by providing an analytic estimation) that a broad enhancement - such as a plateau bounded by infrared and ultraviolet scales - produces a bimodal mass function, with a primary peak close to the infrared scale. We find that the typical initial gravitational potential (compaction function), conditioned on meeting the threshold for critical collapse, is generated by a thin spherical shell with infrared radius and a thickness comparable to the ultraviolet scale. This suggests a higher-than-expected abundance of PBH originating from Type II initial fluctuations. Our results significantly impact overproduction bounds on the amplitude of the power spectrum, and tighten the viable mass range for primordial black holes as dark matter.

Figures

Figures reproduced from arXiv: 2412.07709 by the authors.

Figure 1
Figure 1. FIG. 1. PDF marginalized over [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Bimodal PBH mass functions for flat power spec [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Mass function for narrow power spectra with a fixed [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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Reference graph

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