From Primordial Black Holes Abundance to Primordial Curvature Power Spectrum (and back)

Alba Kalaja; Alvise Raccanelli; Daniele Bertacca; Ilia Musco; Licia Verde; Nicola Bartolo; Nicola Bellomo; Sabino Matarrese

REVIEW 2 major objections 5 minor 5 cited by

From Primordial Black Holes Abundance to Primordial Curvature Power Spectrum (and back)

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Primordial black hole abundance, translated through non-linear collapse physics and peak theory, caps the small-scale curvature power spectrum about an order of magnitude more tightly than previously thought, and a single such black hole…

desk verdict The most careful PBH-to-spectrum pipeline to date, with a robust qualitative exclusion of scale-invariant spectra, but the headline order-of-magnitude tightening depends on a crude transfer-function estimate that needs an error bar before it is definitive. read the letter →

arxiv 1908.03596 v2 pith:QPOE7B5T submitted 2019-08-09 astro-ph.CO

Alba Kalaja , Nicola Bellomo , Nicola Bartolo , Daniele Bertacca , Sabino Matarrese , Ilia Musco , Alvise Raccanelli , Licia Verde This is my paper · ORCID

classification astro-ph.CO

keywords primordialblackholescurvaturepowerspectrumpeaktheorynumericalrelativitygravitationalcollapsethresholdsmall-scaleperturbationsdarkmatternon-Gaussianity

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

Primordial black holes (PBHs) forming from large curvature perturbations are one of the few probes of the primordial curvature power spectrum on scales far smaller than the cosmic microwave background can reach. This paper builds a sharper bridge between PBH abundance and that power spectrum, replacing simplified collapse criteria with numerically simulated thresholds, an exact non-linear density–curvature relation, and peak theory for the smoothed overdensity field. The main quantitative claim is that the maximum allowed amplitude of the curvature power spectrum on scales $k \sim 10^5$–$10^{15}\,\mathrm{Mpc}^{-1}$ is about an order of magnitude lower than earlier estimates. The paper also concludes that even a single PBH formed this way would be incompatible with a scale-invariant spectrum extrapolated from CMB scales, so detecting just one would signal a distinctly non-slow-roll inflationary epoch.

What carries the argument

The load-bearing object is the compaction function, twice the mass excess inside a sphere divided by its areal radius, $C = 2\delta M/R$; the paper adopts the criterion that a PBH forms when the value of $C$ at its maximum radius and horizon-crossing time exceeds the shape-dependent threshold $\delta_{I,c}(\alpha)$. The second piece is the exact non-linear map from the curvature perturbation $\zeta$ to the density contrast, $\delta \propto e^{2\zeta}[\nabla^2\zeta - \tfrac{1}{2}(\nabla\zeta)^2]$, which replaces the usual linearized Laplacian relation and changes the inferred peak scale, horizon-crossing mass, and threshold. The third piece is standard peak theory for Gaussian random fields, whose spectral moments $\sigma_0,\sigma_1,\sigma_2$ are fed through the abundance equation so that the variance $\sigma_0$ is fixed by the observed PBH fraction, and the mean peak profile is converted via equation (6.1) into the curvature power spectrum $\mathcal{P}_\zeta(k)$ on the modes relevant to collapse.

What would settle it

Re-derive the PBH abundance at fixed $\mathcal{P}_\zeta$ using the same non-linear $\zeta\to\delta$ map but with a non-Gaussian peak distribution, or with numerical simulations that include the full mode coupling, and compare the resulting $f_{\rm PBH}$ to the Gaussian peak-theory prediction; a difference of more than about an order of magnitude at fixed amplitude would shift the central constraint by more than the claimed tightening. A direct check is also possible observationally: a future measurement of the stochastic gravitational-wave background from the same scalar perturbations on the scales of Figure 6 would confirm or rule out the amplitude ceiling.

Watch

Extended reading notes

Core claim

The paper's central claim is that, with the collapse physics treated properly, PBH abundance constraints become an order of magnitude stronger: on scales $k \simeq 10^5$–$10^{15}\,\mathrm{Mpc}^{-1}$ the allowed amplitude of $\mathcal{P}_\zeta(k)$ drops by about a factor of ten relative to earlier analyses. The improvement comes from three corrections that all push in the same direction: the collapse threshold $\delta_{I,c}(\alpha)$ ranges from $0.4135$ to $2/3$ across the profile family instead of being a single number; the full non-linear relation between density and curvature damps and shrinks the overdensity profile, with linear theory underestimating the perturbation size and horizon mass by up to a factor of about six; and standard peak theory gives a variance $\sigma_0$ that is $10$–$30\%$ smaller than the usual simplified abundance estimate, so previous work overestimated the power-spectrum amplitude by $20$–$70\%$. Combining these elements, the reconstruction formula yields both the peak amplitude and the shape of $\mathcal{P}_\zeta(k)$ on modes $k_t$ to $5k_t$, and the same machinery shows that generating even one PBH in the observable Universe would require $\mathcal{P}_\zeta \sim 10^{-3}$–$10^{-2}$, orders of magnitude above the CMB-scale value.

Load-bearing premise

The calculation assumes the smoothed overdensity field is Gaussian so that peak theory can be applied, an assumption the authors themselves call too strict; the non-linearities they quantify make the true field non-Gaussian, and because PBH abundance is set by the tail of the peak-height distribution, relaxing this assumption could move the reconstructed power-spectrum amplitude substantially.

Editorial extensions

If this is right

The maximum allowed amplitude of the primordial curvature power spectrum on scales $k \sim 10^5$–$10^{15}\,\mathrm{Mpc}^{-1}$ is roughly an order of magnitude lower than previous PBH-based estimates.
A single curvature-origin PBH in the observable Universe would force $\mathcal{P}_\zeta$ to rise to $\sim 10^{-3}$–$10^{-2}$, so a scale-invariant spectrum at the CMB amplitude is incompatible with even one such object.
The reconstructed power-spectrum shape on modes $k_t \lesssim k \lesssim 5k_t$ steepens as the profile parameter $\alpha$ grows, so measuring the high-$k$ slope of a spike constrains the typical perturbation profile.
Because $\sigma_0$ depends on the abundance only through $(-\log f_{\rm PBH})^{-1/2}$, the resulting amplitude limits are nearly insensitive to which observational $f_{\rm PBH}$ bound is used; the collapse modelling, not the abundance data, dominates the uncertainty.
If PBHs are detected but future gravitational-wave searches see no scalar-induced stochastic background at the corresponding scales, the curvature-perturbation formation mechanism would be disfavoured in favour of alternatives such as topological defects.

Reading between the lines

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

A non-Gaussian extension of peak theory could move the constraints either way; because the abundance is set by the tail of the peak-height distribution, even mild small-scale non-Gaussianity could shift $\mathcal{P}_\zeta$ by as much as the order-of-magnitude tightening claimed here.
The filtering prescription — smooth on scales far below the perturbation rather than at the horizon scale — transfers directly to other rare-object statistics, such as heavy halos or cosmic-string loops, where window-function choice is a known source of systematic uncertainty.
Running the same pipeline backwards with a full critical-collapse mass function would predict a specific PBH mass distribution for each spike shape, letting a measured mass function break the degeneracy between $\mathcal{P}_\zeta$ peak amplitude and profile parameter $\alpha$.
For broad or plateau-like power spectra, the black-hole-in-black-hole problem becomes relevant; implementing an excursion-set peak theory would likely change the derived constraints relative to the monochromatic-spike case considered here.

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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

Let me give you the short version first. This is the most complete PBH-abundance-to-curvature-spectrum pipeline I have seen: numerical relativity thresholds that depend on profile shape, a physically motivated smoothing prescription, a non-linear density-curvature relation, and peak theory instead of Press-Schechter. The qualitative headline—one single CPBH would be incompatible with a scale-invariant spectrum extrapolated from CMB scales—is robust, because the gap is orders of magnitude. The quantitative claim, that the updated upper limits on P_zeta are about an order of magnitude tighter than previous estimates, is plausible but not yet nailed down.

The genuinely new pieces are the combined pipeline and the shape reconstruction for k > k_t. Section 4.3's exact expansion of the non-linear overdensity field is a useful contribution in its own right, and the paper is transparent about where shortcuts remain. It explicitly calls the Gaussian peak-theory assumption "too strict" and gives the T_NL/T_LIN estimate as crude.

The soft spots. The stress-test note is right: T_NL/T_LIN enters squared in Eq. (6.1), and the estimate from Eq. (5.6) is an average profile comparison with no error bar. A 30% uncertainty there shifts the headline amplitude by about a factor of two, which is a large fraction of the claimed order-of-magnitude improvement. It is not fatal for the qualitative single-PBH statement, but it means the tight quantitative bounds in Fig. 6 are conditional on that number. Second, peak theory is applied to a field the authors themselves show is non-Gaussian; no quantification of the bias on the abundance tail is given. Third, the monochromatic, single-shape population is an admitted simplification; it is standard in the field, but again unquantified. These are not reasons to dismiss the paper—they are reasons to ask for error bands before calling the tightening definitive.

Who is this for? Anyone working on PBH constraints, inflationary small-scale power spectrum, or forecasting LISA/SKA synergies will read it. I would cite it for the methodology and for the qualitative exclusion. It deserves a serious referee, not a desk rejection; I'd send it out with a request for a sensitivity analysis on T_NL/T_LIN and at least a parametric estimate of non-Gaussian corrections to the peak height distribution. That is a major revision, not a fundamental flaw.

Referee Report

2 major / 5 minor

Summary. Using numerical-relativity collapse thresholds, a physically motivated smoothing prescription, and Gaussian peak theory, this paper constructs an end-to-end pipeline that maps observational upper limits on the primordial black hole abundance f_PBH into upper limits on the primordial curvature power spectrum P_zeta(k) over k ~ 10^5 - 10^15 Mpc^-1, and demonstrates how to invert the mapping from a spectrum to the abundance. The central quantitative claims are (i) that the maximal allowed amplitude of a spike-type P_zeta is about one order of magnitude tighter than previous estimates (Fig. 6 and Section 7, point 5), and (ii) that even a single PBH in the observable universe is incompatible with a scale-invariant spectrum (Fig. 7). The paper also reconstructs the shape of the spike on scales k_t to 5 k_t (Fig. 8), provides a step-by-step recipe for using the reconstructed shape to determine the profile parameter alpha and the abundance (Section 6, items 1-6), and derives carefully the relativistic degree-of-freedom conversion between beta and f_PBH (Appendix B).

Significance. If the central quantitative claim survives scrutiny, this is a genuinely useful methodological step forward: it is the first PBH-to-spectrum mapping that combines simulation-calibrated shape-dependent thresholds, a physical filtering prescription, and peak theory in one self-consistent pipeline. The paper is transparent and unusually complete in its exposition: the reconstruction steps are enumerated (Section 6, items 1-6), the g* factors are derived rather than approximated (Appendix B), and the Section 4.3 treatment of how non-linearities mix all n-point functions of zeta into the density power spectrum is a valuable formal result. The qualitative conclusions are robust and falsifiable: the incompatibility of one single CPBH with a scale-invariant spectrum, the sensitivity of the bounds to the modelling of collapse, and the predicted SKA/LISA gravitational-wave signatures.

major comments (2)

[Sec. 5.1, Eq. (5.6); Sec. 6, Eq. (6.1) and Fig. 6] The headline claim of Section 7 (point 5) and Figure 6 — that the new pipeline gives constraints about one order of magnitude tighter than previous estimates — is controlled by the numerical factors T_NL/T_LIN ~ 1.2-1.8 quoted in Section 5.1, because the reconstructed amplitude scales as P_zeta proportional to 1/T_NL^2 in Eq. (6.1). The paper explicitly calls this estimate 'crude' and defers the full calculation to Ref. [188], but no uncertainty or robustness test is attached to the quoted factors, and the estimator in Eq. (5.6) is not uniquely defined (the radial range over which the profiles are averaged, the weighting, and the normalization convention are not specified). Since T_NL enters squared, a 30% uncertainty in T_NL/T_LIN changes the reconstructed P_zeta by roughly a factor of two, which is a substantial fraction of the claimed order-of-magnitude improvement; the red band in Fig. 6 therefore does not yet carry the precision implied by the abstract and the conclusions. The companion assumption nu' ~ nu in Section 5.1 is an uncontrolled approximation of similar size. I stress that this concern is specific to the quantitative bound in Fig. 6: the 'one single PBH' conclusion of Fig. 7 rests on a gap of several orders of magnitude and is robust. A concrete remedy is to propagate an explicit uncertainty band around T_NL (or to show the conservative case T_NL = T_LIN) in Fig. 6 and to rescale the 'order of magnitude' claim accordingly.
[Sec. 4.3, Eqs. (4.13)-(4.15); Sec. 5 (first paragraphs); Sec. 6, Eq. (6.4)] The abundance mapping that fixes sigma_0 in Eq. (6.4) assumes that the smoothed overdensity field is Gaussian for the purposes of peak theory (Section 5, first paragraphs), and Section 6 states that the reconstruction is performed 'assuming Gaussian initial conditions'. However, Section 4.3 shows that the non-linear relation (3.7) makes the smoothed density field non-Gaussian even when zeta is Gaussian: Eqs. (4.13)-(4.14) give a non-zero one-point function and two-point contributions from all higher n-point functions of zeta, including the <zeta zeta zeta zeta> terms that survive at Gaussian zeta. The Gaussian assumption is admitted in the text as 'too strict', and the filtering of the density field in the presence of curvature is likewise acknowledged to be approximate (Section 4.2, Eqs. 4.8-4.10), but no bounds are placed on how these approximations shift the reconstructed sigma_0 and hence P_zeta. This is consequential because the inference of sigma_0 from f_PBH runs through the exponential tail e^{-nu^2/2} of the peak-height distribution, which is precisely the part of the distribution modified by non-Gaussianity. The manuscript cites contemporaneous work on the non-linear density-curvature relation and PBH abundance (Refs. [160-162, 179]) but does not import any of these results into the quantitative constraints. I recommend either a quantitative evaluation of the leading non-Gaussian correction to the peak-height distribution (e.g., the one-loop terms in Eq. 4.14 at Gaussian zeta), or an explicit caveat in the abstract and conclusions that the tightened bound of Fig. 6 applies only under the Gaussian smoothed-field assumption.

minor comments (5)

[Sec. 4.3, text after Eq. (4.14)] The sentence introducing the filter functions says equation (4.14) is multiplied by W'_s(k1)W'_s(k1); the second factor should presumably read W'_s(k2), since the two-point function depends on two wavevectors.
[Sec. 4.3, discussion after Eq. (4.15)] The claim that the Gaussian and mixed higher-order terms are 'highly suppressed' because they contain four transfer functions is not compelling in the regime used later, where the transfer function is evaluated at horizon crossing with T_LIN(k tau ~ 1) ~ 0.9 (Section 5.1); the transfer-function counting does not by itself bound the loop integrals. A sentence with a dimensional estimate, or an explicit statement that these terms are dropped in the Gaussian limit, would avoid overstating the case.
[Sec. 6, paragraph after Eq. (6.4)] The assertion that different choices of (delta_I - delta_I,c) 'do not have any impact on the constraint itself' is stronger than what the preceding discussion shows: Eq. (6.5) fixes the mass-scale calibration, and a factor-3 shift in M_PBH for a given k_t changes which f_PBH upper limit is applied at that scale, which can shift the vertical position of the bound in regions where f_max varies steeply. Please state the resulting calibration uncertainty explicitly.
[Sec. 2, Executive Summary; Sec. 3.3] 'We prove that none of these quantities is accurately computed using linear theory' overstates the demonstration, which is performed for the one-parameter family of profiles of Eq. (3.5); 'show' would be more accurate than 'prove'.
[References [188], [211]] References [188] and [211] are listed as 'arXiv:ToAppear' without identifiers; since the argument of Section 5.1 explicitly defers the non-linear transfer function to [188], these placeholders should be completed or removed before publication.

Circularity Check

1 steps flagged · score 2.0 of 10

Secondary shape-inference claim is circular; central amplitude constraints are self-contained.

self definitional [Section 6, paragraph after Figure 8 (shape reconstruction discussion)]
"Moreover, by using equation (6.1), we can also compute the shape of the spike for modes k & kt comparable to or slightly larger than the typical mode kt. As we show in figure 8, the power spectrum to the right of the spike becomes increasingly steeper when α increases, i.e., when the profile becomes flatter. Determining the shape of the power spectrum allow us to determine the shape parameter α, which together with the peak amplitude Pζ(kt) fix σ0, thus the abundance fPBH of CPBHs produced by the spike in the primordial curvature power spectrum."

Equation (6.1) is assembled from Eq. (5.14), xi_s(r) = sigma0^2 delta_peak(r)/delta_peak(0), and Eq. (4.15). In Section 6, delta_peak is the assumed family of Eq. (3.5) (mapped to overdensity via Eq. 3.7), labelled by alpha. Thus, for k_t <= k <= 5 k_t, P_zeta(k) is, up to the k^4 W'^2 T_NL^2 prefactor, the Fourier transform of that assumed profile: the high-k shape is generated by alpha by construction. The statement that 'determining the shape of the power spectrum allow[s] us to determine the shape parameter alpha' therefore inverts an input into an output; the shape 'determination' is self-definitional. This does not affect the central amplitude constraints (Figs. 6-7), where sigma0 is solved from the external f_PBH via Eq. (6.4).

full rationale

The main pipeline maps the external abundance f_PBH to the amplitude of P_zeta through simulation-calibrated thresholds delta_I,c(alpha)/delta_peak,0,c(alpha), peak theory (Eqs. 5.8 and 6.4), and the transfer-function ratio T_NL/T_LIN estimated from the same numerical simulations. The load-bearing constants are not fitted to the target P_zeta; the resulting constraints are compared against external bounds (CMB, spectral distortions, gravitational-wave experiments) and previous PBH estimates, so the central amplitude claim has independent content. The Gaussianity assumption is explicitly flagged as an approximation rather than disguised as a derivation. The one genuinely circular statement is the shape-inference sentence in Section 6: because Eq. (6.1) constructs the spike shape from the assumed profile family of Eq. (3.5), 'determining alpha from the shape' is determining an input from an output. This is a secondary, non-load-bearing overstatement; it does not affect the headline amplitude bounds or the single-PBH incompatibility conclusion. The crude T_NL/T_LIN estimate is an uncertainty and robustness concern, not a circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on several modeling inputs: the choice of peak-shape family, simulation-calibrated thresholds, the Gaussian peak-theory assumption, and an approximate non-linear transfer function. None of these are fitted to the target power spectrum, so the pipeline is not circular, but the inputs carry systematic uncertainties that are not fully propagated into the final error budget.

free parameters (2)

shape parameter alpha = 0.15, 1, 30
Profile family (3.5) parameter; the paper chooses three representative values to produce a band of constraints. Not fitted to abundance data.
non-linear transfer function correction T_NL/T_LIN = 1.8 (alpha=0.15), 1.5 (alpha=1), 1.2 (alpha=30)
Estimated from equation (5.6) by comparing simulation profiles to linear theory; enters equation (6.1) as T_NL^2, directly scaling the reconstructed P_zeta.

assumptions (5)

domain assumption The overdensity field used in peak theory is Gaussian, although non-linearities induce non-Gaussianity.
Stated in Section 5 intro: 'the Gaussian assumption might be too strict... we will consider only the Gaussian case.' The final constraints in Section 6 assume Gaussian initial conditions.
domain assumption The one-parameter profile family of equation (3.5) spans the shapes of peaks that form PBHs.
Both the numerical simulations and the statistical reconstruction use this family; different real peak profiles would shift thresholds and reconstructed spectra.
domain assumption The critical collapse scaling law (3.12) with gamma_crit ~ 0.36 and K(alpha) from simulations applies.
Used in equation (6.4) to connect PBH mass and abundance to the peak-height distribution.
domain assumption All PBHs have a monochromatic mass and share the same formation epoch, with (delta_I - delta_I,c) = 0.01, linking mass to k_t via equation (6.5).
Section 6: 'We assume that all the CPBHs share the same formation time... and that all the CPBHs exceeded the critical threshold by the same amount (0.01).'
standard math The compaction function is conserved on super-horizon scales and the gradient expansion is valid.
Used in Section 3.2 to define the threshold criterion C(t_m, r_m) = delta_I(t_m, r_m).

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Cite this review

Pith. "Pith review of From Primordial Black Holes Abundance to Primordial Curvature Power Spectrum (and back)." pith.science (2026). https://pith.science/paper/QPOE7B5T

@misc{pith2026190803596,
  author       = {Pith},
  title        = {Pith review of: From Primordial Black Holes Abundance to Primordial Curvature Power Spectrum (and back)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QPOE7B5T}},
  note         = {Machine review of arXiv:1908.03596}
}

read the original abstract

In the model where Primordial Black Holes (PBHs) form from large primordial curvature (C) perturbations, i.e., CPBHs, constraints on PBH abundance provide in principle constraints on the primordial curvature power spectrum. This connection however depends necessarily on the details of PBH formation mechanism. In this paper we provide, for the first time, constraints on the primordial curvature power spectrum from the latest limits on PBH abundance, taking into account all the steps from gravitational collapse in real space to PBH formation. In particular, we use results from numerical relativity simulations and peak theory to study the conditions for PBH formation for a range of perturbation shapes, including non-linearities, perturbation profile and a careful treatment of smoothing and filtering scales. We then obtain updated PBH formation conditions and translate that into primordial spectrum constraints for a wide range of shapes and abundances. These updated constraints cover a range of scales not probed by other cosmological observables. Our results show that the correct and accurate modelling of non-linearities, filtering and typical perturbation profile, is crucial for deriving meaningful cosmological implications.

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Forward citations

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Reviewed August 14, 2026 · model on record in the stance chip above.

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