Clustering of Primordial Black Holes in Excursion Set Theory

Encieh Erfani; Hamed Kameli

REVIEW 3 major objections 4 minor 1 cited by

Clustering of Primordial Black Holes in Excursion Set Theory

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Primordial black hole clustering is set by the spectral tilt of the primordial power spectrum, with a one-to-one correspondence between tilt and clustered mass range.

desk verdict A useful idea for PBH pair clustering via two-trajectory excursion set theory, but the central cross-correlation is unvalidated and the full text is unreadable in the copy I saw. read the letter →

arxiv 2508.01896 v1 pith:4CDCTT6H submitted 2025-08-03 astro-ph.CO hep-ph

Hamed Kameli , Encieh Erfani This is my paper · ORCID

classification astro-ph.COhep-ph

keywords primordialblackholesexcursionsettheoryclusteringpowerspectrumbluetiltcriticaldensitythresholdmergerratestochastictrajectories

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 aims to show that the clustering of primordial black holes is not an extra coincidence but is determined by the same primordial power spectrum that decides whether the black holes form at all, through an extension of excursion set theory. The authors compute the joint probability that two black holes form within a chosen separation by treating the density fluctuations at the two sites as a pair of random walks with a shared history. They report that an enhanced, blue-tilted spectrum increases both the formation rate in specific mass ranges and the probability of close pair formation, with a one-to-one correspondence between the spectral tilt and the mass range that clusters. The point of the claim is that if it holds, PBH clustering and the resulting merger rates become a direct prediction of the primordial spectrum rather than a separate assumption.

What carries the argument

The central object is a pair of excursion-set trajectories: two random walks that represent the density contrast at two points smoothed from large to small scales. While the walks are independent at small scales, they share the same long-wavelength history, so their cross-correlation is fixed by the matter power spectrum. A PBH pair is counted when both walks first cross the critical density barrier within the clustering distance; the single-walk first-crossing rate gives the abundance, and the joint first-crossing rate gives the clustering probability.

What would settle it

An N-body simulation of PBH formation from a blue-tilted power spectrum would settle the matter: if the simulated pair correlation as a function of separation and mass disagrees with the joint first-crossing probability predicted by the two-walk model, the shared-history correlation is not capturing the true density-field statistics.

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Extended reading notes

Core claim

The central claim is that, in excursion set theory, the joint probability of forming two primordial black holes at a given separation is governed by the cross-correlation of two stochastic trajectories that share a smoothed history. For an enhanced, blue-tilted power spectrum (extra power at small scales), the first-crossing probability of the collapse barrier rises in certain mass windows, and the two trajectories become more likely to cross the barrier together, so PBH formation and PBH clustering are amplified in the same windows. The paper states this as a one-to-one correspondence between the blue-tilted spectral index and the mass ranges in which PBHs form and cluster. It further claims that the pair-formation probability decreases asymptotically with the clustering distance and that a higher critical density threshold suppresses the clustered abundance.

Load-bearing premise

The calculation assumes that the density field at two separated points is faithfully represented by two random walks whose shared history and cross-correlation come directly from the matter power spectrum, rather than by the exact joint statistics of peaks in the density field at fixed separation.

Editorial extensions

If this is right

If the one-to-one tilt–mass correspondence holds, measuring the PBH abundance in a mass window constrains the spectral tilt in that window, and vice versa.
Because the clustering probability falls with separation, close pairs dominate the initial binary population, which shifts predicted merger-time distributions toward shorter times.
A higher critical collapse threshold suppresses clustered pairs more than isolated formation, so environment-dependent threshold models would predict less clustering.
The formalism replaces heuristic clustering parameters with a direct computation from the primordial power spectrum, making PBH merger-rate estimates more directly tied to inflationary models.

Reading between the lines

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

The same shared-history machinery could be applied to three trajectories to predict the initial geometry of PBH clusters and the rate of close triple interactions, which may be observable through gravitational-wave eccentricity.
The predicted one-to-one mapping implies a distinctive mass-dependent clustering signature that could be searched for in stochastic gravitational-wave background anisotropies.
The calculation predicts initial comoving clustering; later accretion and dynamical heating will modify it, so comparison with present-day binaries requires a separate evolution step the paper does not attempt.

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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

I can only judge this paper from its abstract: the full text in the version I have is garbled beyond recovery. Given that caveat, here is my honest read.

The actual idea is decent. Rather than treating PBH formation at one point, Kameli and Erfani write down two excursion-set trajectories with a shared history, so the joint probability of two nearby points both crossing the collapse barrier carries clustering information. That is a natural extension of EST, and the abstract's target—connecting the primordial power spectrum to PBH pair formation and merger rates—is a live LIGO-era question. The qualitative results they report (blue tilt boosts both abundance and clustering; clustering decays with separation; higher barrier suppresses pairs) are plausible and in line with physical intuition.

The soft spot is not the idea but the quantitative core. Everything hinges on the functional form of the correlation between the two smoothed density contrasts, introduced as a "shared history." The stress-test concern is exactly right: any valid two-trajectory EST calculation must recover the peak-background split clustering amplitude b(M)^2 ξ_m(r) in the large-separation limit, with b ~ δ_c/σ^2 for rare peaks. If the shared-history correlation is merely the variance in modes with wavelength greater than r, that limit is not automatic, especially for a blue-tilted spectrum. The abstract gives no equations, no error estimates, and no comparison to simulations or peak theory. So the one-to-one spectral-index–mass correspondence may be an artefact of the approximation. I cannot check from the abstract alone, and the corrupted text means I cannot check at all from this copy.

I want to be fair: there is no sign of circular fitting. The free parameters listed (barrier, tilt, amplitude) are model inputs, not post-hoc adjustments. The citation pattern cannot be assessed from the abstract. This is a paper whose worth depends entirely on whether the two-trajectory correlation passes consistency checks. That is checkable, and the authors should be asked to show it.

Who is this for? PBH theorists and anyone using EST for clustering. It deserves a serious referee, but the referee needs a readable manuscript and should demand (1) the peak-background split limit, (2) a comparison with an N-body or standard peak-bias result, and (3) the actual equations defining the shared history. If those hold up, this becomes a useful tool. I would not cite it myself until that happens.

Recommendation: send to peer review, with a request for revisions that address the consistency condition. Not a desk reject.

Referee Report

3 major / 4 minor

Summary. The paper proposes an extension of excursion set theory (EST) to compute the joint probability that two primordial black holes (PBHs) form within a clustering distance, based on two stochastic trajectories with a shared history. The abstract claims that an enhanced, blue-tilted power spectrum increases both the formation rate of PBHs in specific mass ranges and the probability that they form in close pairs, with a one-to-one correspondence between the spectral tilt and the mass ranges that cluster. It also claims that the clustering probability decreases asymptotically with clustering distance and is suppressed by a higher critical density threshold. The submitted full text is corrupted and unreadable, so the derivation and numerical results cannot be independently assessed from the manuscript as provided.

Significance. If the central claims are correct, the paper would provide a calculable link between the primordial power spectrum and PBH clustering, with direct implications for predicted merger rates and interpretations of gravitational-wave events. The two-trajectory EST approach is a natural extension of a well-established formalism, and the predicted tilt-mass-clustering correspondence is falsifiable. However, the paper as submitted does not supply the needed derivation or validation: the full text is unreadable, and the abstract alone provides no equations, no comparison to the standard peak-background split limit, and no numerical or simulation tests. The significance therefore rests on a derivation that cannot currently be checked from the submission.

major comments (3)

[Full text] The full text of the submitted file is unreadable (mojibake) and appears to contain an unrelated identifier from another arXiv record, so the central derivation of the joint crossing probability cannot be checked. This blocks technical evaluation in the present form and must be corrected before the paper can be refereed.
[Abstract] The central claim that two stochastic trajectories with a shared history yield a pair-crossing probability is not validated against the standard peak-background split clustering amplitude. In the large-separation limit, any valid two-trajectory EST calculation should reproduce b(M)^2 xi_m(r) with b approximately delta_c/sigma^2(M) for rare Gaussian peaks; the abstract provides no indication that this consistency condition is imposed or satisfied. If the derived pair probability does not reduce to this limit, the claimed enhancement of clustering for blue-tilted spectra is an artifact of the approximation rather than a prediction of the density field.
[Abstract] The claimed one-to-one correspondence between the blue-tilted spectral index and the mass ranges in which PBHs form and cluster is not defined. For a continuous power spectrum, a bijective mapping between a continuous parameter and a set of mass ranges requires a precise statement of the mass window and the sense in which the correspondence holds; without such a definition, the claim is ambiguous and cannot be tested against the derived formulas.

minor comments (4)

[Abstract] The term 'clustering distance' should be explicitly defined, for example as the comoving separation r at which the pair probability is evaluated.
[Abstract] The abstract would benefit from stating the smoothing filter (sharp-k or Gaussian) used in the excursion-set walks, since the covariance structure of the two trajectories depends sensitively on this choice.
[Full text] The corrupted full text contains what appears to be a header from arXiv:2508.01897v1 [cs.SD], which is unrelated to the stated subject; a clean resubmission should remove all extraneous content.
[Abstract] The asymptotic decrease of clustering probability with distance is stated only qualitatively; a scaling relation or a figure would make the result more informative and easier to compare with future simulations.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the available abstract derives clustering probabilities from stated model inputs, and no specific reduction to a fit or self-citation can be exhibited.

full rationale

The only readable portion of the manuscript is the abstract; the full text is corrupted in the supplied file. The abstract describes an excursion set theory calculation that extends the formalism to compute the joint probability of forming PBH pairs using two stochastic trajectories with a shared history. The inputs are the power spectrum, the clustering distance, and the critical density threshold (barrier); the outputs are formation rates, clustering probabilities, and a claimed one-to-one correspondence between blue-tilted spectral index and mass ranges. There is no indication in the abstract of fitted parameters being relabeled as predictions, no self-citation chain carrying a load-bearing premise, and no uniqueness theorem imported from the authors' prior work. The claimed one-to-one correspondence is presented as a derived result rather than as a definition or an ansatz. Under the hard rule that circularity may only be claimed when a specific reduction can be quoted and exhibited, no such reduction is available from the evidence provided. Accordingly, the appropriate finding is no significant circularity, with score 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central calculation rests on standard excursion-set assumptions (Gaussian random fields, first-passage above a threshold) plus the paper-specific modeling of two trajectories sharing a history. The power spectrum amplitude, tilt, and barrier height enter as inputs; none appear to be fitted to data within the abstract.

free parameters (3)

critical density threshold barrier = unspecified
The barrier height determines when a density fluctuation collapses to a PBH. The abstract notes that a higher barrier suppresses clustering abundance, indicating results depend on its chosen value.
spectral index (blue tilt) = unspecified
The spectral index or its blue tilt is a model input controlling the shape of the power spectrum. The claimed one-to-one mapping between tilt and mass ranges relies on this parameter.
power spectrum amplitude = unspecified
An enhanced power spectrum is the key driver of both formation and clustering; its amplitude is an input, not derived.

assumptions (4)

domain assumption The primordial density field is a Gaussian random field with a specified power spectrum.
Standard in EST and assumed in the abstract's description of stochastic trajectories.
domain assumption PBH formation occurs wherever the smoothed overdensity exceeds a critical threshold (barrier), and the fraction depends on the first crossing of this barrier.
Abstract describes the barrier as controlling collapse and clustering; this is the standard excursion set ansatz.
ad hoc to paper Two trajectories separated by a distance have a joint distribution determined by their shared history and cross-correlation.
The abstract's extension to two trajectories with shared history is the paper's modeling choice for clustering; it is not derived in the abstract.
standard math The trajectories are Markovian random walks or can be treated as such.
The classic EST approximation; assumed when computing first-passage probabilities.

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Cite this review

Pith. "Pith review of Clustering of Primordial Black Holes in Excursion Set Theory." pith.science (2026). https://pith.science/paper/4CDCTT6H

@misc{pith2026250801896,
  author       = {Pith},
  title        = {Pith review of: Clustering of Primordial Black Holes in Excursion Set Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4CDCTT6H}},
  note         = {Machine review of arXiv:2508.01896}
}

read the original abstract

We investigate the clustering of Primordial Black Holes (PBHs) within the framework of Excursion Set Theory (EST). The EST formalism is extended to compute the joint probability of forming PBH pairs within a clustering distance, based on two stochastic trajectories with a shared history. Our results show that an enhanced power spectrum not only increases the formation of PBHs in specific mass ranges but also enhances their clustering probability. We find a one-to-one correspondence between the blue-tilted spectral index and the mass ranges in which PBHs form and cluster. Additionally, we demonstrate that the clustering probability decreases asymptotically with increasing clustering distance, while a higher critical density threshold (barrier) leads to a suppression of clustering abundance.

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