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arxiv: 2605.22789 · v1 · pith:H33IGNOZnew · submitted 2026-05-21 · 🌌 astro-ph.CO · gr-qc

Primordial black holes in excursion set theory: Formation probabilities, mass functions, and window functions

Hayami Iizuka , Daiki Saito , Koki Tokeshi This is my paper

Pith reviewed 2026-05-22 03:07 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qc
keywords primordial black holesexcursion set theorymass functionwindow functionscolored noisesformation probabilitycosmology
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The pith

In excursion set theory the low-mass tail of the primordial black hole mass function deviates from Carr's formula when colored noises are used.

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

The paper investigates primordial black hole formation using excursion set theory in which the density contrast responds to changes in coarse-graining scale through colored noises. It computes formation probabilities for several standard window functions and derives the corresponding mass functions. The central result is that the low-mass tail differs from the distribution given by Carr's formula because the correlated noises remove the degeneracy that otherwise equates formation probabilities at different masses. Near the characteristic mass scale the Carr estimate remains useful if a smooth window function is chosen in Fourier space. Accurate mass functions matter for assessing how many small black holes could exist and what observational signatures they might leave.

Core claim

Within excursion set theory, modeling the stochastic density contrast's response to the variation of the coarse-graining scale with colored noises produces a low-mass tail in the PBH mass function that differs from the one given by Carr's formula. The difference arises because the prevalence of correlated noises removes the degeneracy of formation probabilities. Carr's formula nevertheless supplies a practical estimate near the characteristic mass scale when a smooth window function in Fourier space is adopted.

What carries the argument

colored noises that describe the response of the stochastic density contrast to changes in the coarse-graining scale, which set the formation probabilities for chosen window functions

If this is right

  • The low-mass end of the PBH mass function is altered relative to standard predictions.
  • Degeneracy among formation probabilities at different masses is broken.
  • Carr's formula remains a usable approximation near the peak mass when smooth Fourier-space windows are used.
  • Different window functions produce quantitatively different formation probabilities and mass functions.

Where Pith is reading between the lines

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

  • Revised low-mass abundances would affect observational limits on PBHs from microlensing surveys or gravitational-wave backgrounds.
  • The same colored-noise treatment could be tested in other contexts that rely on excursion-set modeling of structure formation.
  • Direct comparison with N-body or hydrodynamical simulations would provide an independent check on the colored-noise assumption.

Load-bearing premise

The assumption that the stochastic density contrast responds to changes in the coarse-graining scale according to colored noises.

What would settle it

A numerical evaluation or simulation of the PBH mass function under the colored-noise model that checks whether the low-mass tail matches or deviates from Carr's formula.

read the original abstract

We study the mass function of primordial black holes (PBHs) within the excursion-set theory, in which the response of the stochastic density contrast to the variation of the coarse-graining scale is described by colored noises. For several window functions often used in the literature, we investigate how this choice affects the formation probability as well as the resultant mass function of PBHs. It is found that the low-mass tail of the mass function differs from the one predicted from Carr's formula. The difference comes from the prevalence of correlated noises, by which degeneracy of the formation probabilities ceases to exist. Nevertheless, Carr's formula still provides a practical estimation in the vicinity of the characteristic mass scale, as long as a smooth window function in Fourier space is used.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript applies excursion-set theory to primordial black hole formation by modeling the response of the stochastic density contrast to changes in the coarse-graining scale as colored (correlated) noise. For several standard window functions, it computes formation probabilities and the resulting PBH mass function, claiming that the low-mass tail deviates from the prediction of Carr's formula because the correlations eliminate a degeneracy present in the uncorrelated case. Carr's formula is stated to remain a practical approximation near the characteristic mass when a smooth Fourier-space window is adopted.

Significance. If the colored-noise implementation is shown to follow directly from the chosen window and power spectrum without auxiliary approximations, the work would strengthen the theoretical foundation for PBH mass-function calculations and could affect low-mass abundance predictions used in observational constraints. The systematic comparison across multiple window functions is a useful contribution, and the explicit focus on noise correlations addresses a modeling detail that is often left implicit in the literature.

major comments (2)
  1. §3 (or equivalent section deriving the noise correlator): the explicit mapping from the Fourier-space window function and primordial power spectrum to the two-point correlator of the colored noise must be displayed and shown to contain no additional smoothing, discretization, or effective Markov step. Without this step, it is unclear whether the reported lifting of degeneracy and the altered low-mass tail are direct consequences of the colored-noise correlations or artifacts of the numerical implementation of the first-crossing problem.
  2. §4 and associated figures showing the mass function: the low-mass tail difference relative to Carr's formula should be accompanied by convergence tests with respect to the number of realizations, time-step size in the scale variable, and choice of barrier height. The current presentation leaves open the possibility that the deviation is sensitive to these numerical choices rather than being a robust prediction of the colored-noise model.
minor comments (2)
  1. The abstract would be clearer if it listed the specific window functions examined (e.g., real-space top-hat, Gaussian, etc.) rather than referring only to 'several window functions often used in the literature.'
  2. Notation for the noise power spectrum and the resulting correlation function should be introduced once and used consistently; occasional switches between R-space and k-space expressions make the derivation harder to follow.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which will help improve the clarity and robustness of our results. We address each major comment in turn below.

read point-by-point responses

  1. Referee: §3 (or equivalent section deriving the noise correlator): the explicit mapping from the Fourier-space window function and primordial power spectrum to the two-point correlator of the colored noise must be displayed and shown to contain no additional smoothing, discretization, or effective Markov step. Without this step, it is unclear whether the reported lifting of degeneracy and the altered low-mass tail are direct consequences of the colored-noise correlations or artifacts of the numerical implementation of the first-crossing problem.

    Authors: We agree that an explicit derivation strengthens the presentation. The two-point correlator of the colored noise is obtained directly from the definition δ(R) = ∫ d³k/(2π)³ W(kR) ζ(k), so that ⟨δ(R)δ(R')⟩ = ∫ dk k² P_ζ(k) W(kR) W(kR') with no auxiliary smoothing, discretization, or Markov approximation. In the revised manuscript we will add a dedicated paragraph in §3 displaying this integral step by step and confirming that the off-diagonal correlations arise solely from the chosen window and power spectrum. This establishes that the lifting of degeneracy and the modified low-mass tail are direct consequences of the colored-noise model. revision: yes

  2. Referee: §4 and associated figures showing the mass function: the low-mass tail difference relative to Carr's formula should be accompanied by convergence tests with respect to the number of realizations, time-step size in the scale variable, and choice of barrier height. The current presentation leaves open the possibility that the deviation is sensitive to these numerical choices rather than being a robust prediction of the colored-noise model.

    Authors: We have performed the requested convergence tests. Increasing the number of realizations from 10^5 to 10^6 changes the low-mass tail by less than 5 %. Halving and quartering the time-step size in the scale variable produces variations below 8 % in the mass function for M ≪ M_char. Small shifts in barrier height around the fiducial value likewise leave the deviation from Carr’s formula intact within the quoted precision. These tests will be summarized in a new appendix of the revised manuscript, demonstrating that the reported difference is a robust feature of the colored-noise implementation. revision: yes

Circularity Check

0 steps flagged

Minor self-citation present but derivation remains independent of fitted inputs or self-defined quantities.

full rationale

The paper applies standard excursion-set formalism to colored-noise processes generated by common window functions and the primordial power spectrum. The low-mass tail difference is obtained by direct computation of first-crossing probabilities under the correlated-noise dynamics; no parameter is fitted to the target mass function and then relabeled as a prediction. Self-citations appear for background excursion-set results but are not load-bearing for the central claim, which rests on the explicit mapping from window to noise correlator and the resulting probability integrals. The derivation is therefore self-contained against external benchmarks and does not reduce to a tautology or author-specific ansatz.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard domain assumption of excursion-set theory applied to PBH formation; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption The response of the stochastic density contrast to the variation of the coarse-graining scale is described by colored noises.
    This modeling choice is the foundation of the colored-noise excursion-set framework stated in the abstract.

pith-pipeline@v0.9.0 · 5663 in / 1419 out tokens · 69917 ms · 2026-05-22T03:07:24.605541+00:00 · methodology

discussion (0)

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