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The Irrelevance of Primordial Black Hole Clustering in the LVK mass range

T0 review · 2 major / 3 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper argues that initial clustering of primordial black holes does not affect the binary merger rate in the LIGO-Virgo-KAGRA mass range, because CMB spectral-distortion bounds cap clustering scales below those relevant for the mergers.

desk verdict Useful bias formalism and a clear no-go claim, but the FIRAS constraint transfer from log-normal to top-hat spectra is not conservative and may undo the central conclusion. read the letter →

arxiv 2502.01617 v2 pith:TAJMSYFK submitted 2025-02-03 astro-ph.CO gr-qc

F. Crescimbeni , V. Desjacques , G. Franciolini , A. Ianniccari , A. J. Iovino , G. Perna , D. Perrone , A. Riotto
show 1 more author
H. Veermäe
This is my paper · ORCID
classification astro-ph.COgr-qc
keywords primordialblackholesPBHclusteringgravitationalwavemergersLVKmassrangeCMBspectraldistortionsFIRASconstraintsnon-Gaussianitysubsolarholebinaries
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 sets out to determine whether the spatial clustering of primordial black holes beyond a Poisson distribution changes the rate at which they merge into binaries detectable by the LIGO-Virgo-KAGRA gravitational-wave network. For masses above roughly $0.1\,M_\odot$, and for the single-field and curvaton formation scenarios that are commonly studied, it argues the answer is no. The crucial input is the COBE/FIRAS bound on CMB spectral distortions, which keeps the clustering coherence length below about $10^{-2}\,\mathrm{kpc}$, while binary separations that merge today lie in a window around $10^{-2}$ to $0.3\,\mathrm{kpc}$ for subsolar masses. The result is that the standard Poisson-based estimate of the PBH binary merger rate remains valid for current detectors, including for subsolar-mass events.

What carries the argument

The mechanism is a scale comparison carried by the volume-averaged PBH correlation function $\bar\xi_{\mathrm{PBH}}(R)$. The correlation is built from a linear bias factor $b_1$ obtained by a peak-background split on the compaction-function collapse threshold $C>C_c$, with local non-Gaussianity encoded in a general relation $\zeta=F(\zeta_g)$ (specialized to quadratic and curvaton forms). Clustering matters only where $\bar\xi_{\mathrm{PBH}}(R)\gg1$ inside the window $R_{\min}\lesssim R\lesssim R_{\max}$; the FIRAS bound on CMB spectral distortions fixes the outer scale $k_{\min}^{-1}\lesssim R_{\mathrm{FIRAS}}\simeq10^{-2}\,\mathrm{kpc}$, which removes that window for LVK masses.

What would settle it

Re-compute the spectral-distortion integrals of Eq. (4.5) for the top-hat power spectrum (3.7) instead of approximating it by a log-normal spectrum; if the resulting bound allows $k_{\min}^{-1}$ above about $0.1\,\mathrm{kpc}$ for $M=0.1\,M_\odot$, the no-go conclusion for subsolar masses fails. A second falsifier would be an observed excess of subsolar-mass PBH mergers over the Poisson rate at current detectors, which clustering was predicted to produce.

Watch

Extended reading notes

Core claim

The paper's central claim is that the initial PBH two-point correlation, although never exactly zero in these formation models, is too small on merger-relevant separations to alter the binary formation rate assumed in the Poisson case. The proof works by comparing three comoving scales: the minimum separation $R_{\min}\sim 9.5\times10^{-3}(M/M_\odot)^{7/16}\,\mathrm{kpc}$ below which binaries would already have merged, the maximum separation $R_{\max}\sim 0.31(M/M_\odot)^{1/3}\,\mathrm{kpc}$ beyond which a pair decouples too slowly from the Hubble flow, and the FIRAS-induced clustering scale $R_{\mathrm{FIRAS}}\simeq10^{-2}\,\mathrm{kpc}$. For masses $M\gtrsim0.1\,M_\odot$ the clustering scale is smaller than the merger window, and the volume-averaged correlation function $\bar\xi_{\mathrm{PBH}}(R)$ computed for maximally broad power spectra with quadratic or curvaton non-Gaussianity stays below the Poisson floor throughout the window. The conclusion therefore applies to common single-field models and, for $M\gtrsim1\,M_\odot$, is even stronger because the clustering scale falls below $R_{\min}$.

Load-bearing premise

The argument assumes that the FIRAS bound computed for a log-normal power spectrum with width $\sigma=1$ carries over to the broad top-hat spectra used in the benchmarks; if the true bound for those spectra were weaker, the clustering length could extend into the merger window for subsolar masses.

Editorial extensions

If this is right

  • The standard Poisson approximation for PBH binary merger rates in the LVK mass range survives, so existing rate estimates for subsolar and solar-mass PBH mergers do not need a clustering correction.
  • Because the no-go argument uses a maximally broad power spectrum, it covers the best-case clustering from broad single-field spectra; narrower spectra can only make clustering less relevant.
  • For PBHs heavier than about one solar mass, the clustering scale lies below $R_{\min}$, so binary formation proceeds as in the Poisson picture for all merger-relevant separations.
  • The conclusion does not extend to masses around $10^{-6}M_\odot$ relevant for future detectors such as the Einstein Telescope, where clustering may enhance merger rates.

Reading between the lines

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

  • A direct computation of the FIRAS $\mu$- and $y$-distortion integrals for the top-hat spectrum of Eq. (3.7), rather than the log-normal surrogate used in Fig. 3, would test the main transfer assumption; a weaker bound would reopen the subsolar window.
  • If future detectors observe a merger-rate excess at masses below about $10^{-6}M_\odot$ over the Poisson prediction, that would be a natural signature of the clustering channel the paper leaves open.
  • Because the paper's bias factor differs from the curvature-threshold prescription, simulations of PBH formation from broad spectra with strong local non-Gaussianity could settle which bias prescription is correct.
  • For multi-field models with a spectator-field modulation, the FIRAS bound does not apply in the same way, so those models are the place to look for revival of clustering effects in the LVK band.
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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

2 major / 3 minor

Summary. The paper argues that for primordial black holes (PBHs) in the LIGO-Virgo-KAGRA (LVK) mass range, including subsolar masses, the initial PBH clustering beyond the Poisson distribution is irrelevant for binary merger rates. The argument combines a peak-background split computation of the PBH bias from a compaction-function threshold, an analytic evaluation of the two-point correlation function for a broad top-hat curvature power spectrum, and a comparison of the clustering length kmin^-1 with the binary merger window Rmin < R < Rmax. The central step is the claim, anchored in Eq. (4.7), that COBE/FIRAS constraints force kmin^-1 <~ 10^-2 kpc, so that the volume-averaged correlation function is negligible in the merger window. The paper also shows that for lighter PBHs (M <~ 10^-6 Msun) clustering may become relevant, and it explicitly excludes multi-field scenarios.

Significance. If correct, the conclusion would justify the common assumption of an initial Poisson distribution for PBH binaries in the LVK mass range, simplifying merger-rate estimates in single-field and curvaton production models. The paper has several genuine strengths: the bias formula in Eq. (3.20) is derived from a nonlinear compaction threshold rather than a truncated quadratic expansion, the correlation functions in Eqs. (3.8) and (3.10) are explicit and analytic, and the authors clearly state the scope of their conclusion. However, the central no-go step relies on transferring a FIRAS bound computed for a log-normal spectrum to the top-hat spectra used in the benchmarks, and this transfer is not conservative in the direction the argument needs. The main claim is therefore not established as stated, although the framework and methods are valuable.

major comments (2)
  1. [§4, Eq. (4.7) and Fig. 3] The bound kmin^-1 <~ R_FIRAS ~ 10^-2 kpc is derived from Fig. 3 for a log-normal power spectrum with width sigma=1 that is said to mimic the broad spectra assumed in this work, but the benchmark models in Table 1 use the top-hat spectrum of Eq. (3.7). For a top-hat with low-k cutoff kmin, the mu-distortion is mu = As * integral_{kmin}^{infinity} W_mu(k) dk/k. Because W_mu is concentrated near k ~ 10^3-10^4 Mpc^-1 and is exponentially suppressed for k >~ 10^4, a top-hat with kmin = 10^5 Mpc^-1 produces essentially zero mu-distortion and FIRAS does not constrain kmin at all for such a spectrum. Conversely, the QL2 benchmark (As ~ 3*10^-4, b1 = 63) satisfies the FIRAS limit mu <= 4.7*10^-5 even with kmin = 10^3 Mpc^-1, since 3*10^-4 * 0.086 ~ 2.6*10^-5. Then kmin^-1 ~ 1 kpc, which exceeds Rmax for subsolar masses, and Eq. (3.10) gives xi_bar_PBH ~ 1.3 at R = 10^-2 kpc. The log-normal proxy is therefore not conservative for the no-go: it can understate the clustering scale that a FIRAS-compatible top-hat spectrum can have. Eq. (4.7) does not follow for the top-hat benchmarks, and the central claim is not established.
  2. [Table 1 and §4] The value kmin = 10^5 Mpc^-1 is imposed in every benchmark case, and this is exactly the value that makes the clustering length (10^-2 kpc) smaller than the merger window. The text does not derive this value from FIRAS constraints for the actual spectrum shape; it appeals instead to the log-normal calculation of Fig. 3. Since the mu-distortion vanishes for a top-hat with kmin >~ 10^4 Mpc^-1, the choice kmin = 10^5 Mpc^-1 should be justified either as a property of the specific inflationary models or as the result of a maximization over all FIRAS-compatible top-hat spectra. As written, the anti-clustering conclusion is an artifact of this imposed cutoff.
minor comments (3)
  1. [Eq. (4.5) and Eq. (3.7)] The symbol P_zeta is used both for the dimensional spectrum in Eq. (3.7) and as the amplitude-like quantity in the FIRAS integral of Eq. (4.5). Clarify whether P_zeta in Eq. (4.5) is the dimensionless power spectrum, and if so, state the explicit relation to the top-hat definition of Eq. (3.7).
  2. [§4, Eq. (4.6)] The expression for W_mu(k) is typeset ambiguously: the exponential ratio is written as e^{-(k_hat/1360)^2 / 1+...} rather than exp[- (k_hat/1360)^2 / (1+...)]. Please add parentheses to unambiguously indicate the intended function.
  3. [§3.2 and Fig. 2 caption] The sentence "The only non-linear (NL) broad case ... has been obtained fixing As = 10^-2.07" is unclear because all cases use the non-linear forms of Eq. (2.6) or Eq. (A1); specify which model this NL curve refers to and why it is singled out.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction found: the no-go argument is an externally anchored upper-limit exercise with benchmark parameters as inputs.

full rationale

The central derivation chain is: a broad top-hat curvature power spectrum P_zeta(k) -> compaction-function threshold statistics for the PBH abundance -> peak-background-split bias b1 -> the PBH correlation function xi_PBH and its volume average -> comparison with the binary merger scales Rmin and Rmax, with the clustering scale kmin^-1 bounded by COBE/FIRAS spectral-distortion limits. Every link is computed from stated definitions and measured inputs; no equation redefines the conclusion as an input. The benchmark amplitudes and non-Gaussian parameters in Table 1 are taken from published formation scenarios to reproduce stated f_PBH values, and the subsequent evaluation of xi_bar is a genuine calculation, not a fit to the desired answer. The criterion is also discriminating: the paper explicitly finds that clustering may reach O(10) for M about 10^-6 M_sun and would matter if kmin^{-1} were larger, which shows the conclusion is not true by construction. The FIRAS bound in Eq. (4.7) is cited to Ref. [70], a paper with overlapping authors, and is displayed in Fig. 3; however it is anchored to measured COBE/FIRAS limits and standard spectral-distortion window functions, so it is independent evidence rather than an imported conclusion. The main caveat, namely the transfer of the log-normal sigma=1 FIRAS bound to the top-hat benchmarks, is a load-bearing approximation and a potential correctness risk, not a circularity. Similarly, the restriction to common single-field and curvaton models is explicitly stated as a scope limitation, and no self-citation chain is used to forbid alternatives or to import a uniqueness theorem. The derivation is therefore self-contained given its stated inputs, and I find no significant circularity.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central no-go argument is an upper-limit exercise built on four anchor choices: threshold-statistic PBH formation, a top-hat spectrum chosen to maximize clustering, a FIRAS-bound transfer from log-normal to top-hat spectra, and a literature-based merger-rate dependence. These are assumptions about the model space, not invented entities; the paper contains no new particles or forces.

free parameters (6)
  • Amplitude As of curvature power spectrum = 10^-2.2 to 10^-3.5 (Table 1 cases)
    Chosen per benchmark so that PBH abundance f_PBH matches observed DM limits; these amplitudes enter the clustering amplitude directly.
  • Quadratic non-Gaussianity f_NL = 0.42 and 10.75
    Hand-picked to span common single-field models; controls bias b1 through F''_s/F'_s.
  • Curvaton decay fraction r_dec = 0.1 and 0.5
    Hand-picked choices controlling the NG strength in curvaton models.
  • Minimum wavenumber k_min = 10^5 Mpc^-1 (k_min^-1 = R_FIRAS)
    Imposed at the FIRAS upper limit to maximize clustering; the no-go argument assumes this is the largest allowed clustering scale.
  • Shape factor kappa = 4.5
    Fixed from the shape of the power spectrum following Ref. [103]; converts horizon mass to PBH mass.
  • Compaction threshold C_c = 0.56
    Taken from Ref. [103] for the assumed spectrum shape; threshold for PBH collapse.
assumptions (5)
  • domain assumption PBH formation is described by threshold statistics on the compaction function C with C_c=0.56 (Sec. 2).
    The entire abundance and bias calculation relies on this criterion; the paper notes peak theory disagrees (footnote 1) and uses threshold statistics only.
  • domain assumption A broad, sharply cut top-hat power spectrum maximizes clustering at fixed FIRAS constraints (Sec. 3.2, Eq. 3.7).
    Used to make the no-go argument robust; the paper does not prove maximality among all spectra.
  • domain assumption FIRAS bounds computed for a log-normal spectrum with sigma=1 can be transferred to the top-hat benchmarks (Fig. 3 caption).
    This transfer sets R_FIRAS approximately 10^-2 kpc, the central scale input of the argument.
  • domain assumption The PBH merger rate depends on f_PBH(1+xi_bar) (Sec. 4, citing Ref. [150]).
    Quoted from literature; used to conclude that small xi_bar makes clustering irrelevant.
  • domain assumption Local ansatz zeta=F(zeta_g) captures the relevant non-Gaussianity for single-field and curvaton models (Eq. 2.6).
    Excludes multi-field scenarios with a separate modulating field, which the paper explicitly flags as outside its claim.

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Cite this review

Pith. "Pith review of The Irrelevance of Primordial Black Hole Clustering in the LVK mass range." pith.science (2026). https://pith.science/paper/TAJMSYFK

@misc{pith2026250201617,
  author       = {Pith},
  title        = {Pith review of: The Irrelevance of Primordial Black Hole Clustering in the LVK mass range},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TAJMSYFK}},
  note         = {Machine review of arXiv:2502.01617}
}
read the original abstract

We show that in realistic models where primordial black holes are formed due to the collapse of sizeable inflationary perturbations, their initial spatial clustering beyond Poisson distribution does not play any role in the binary mergers, including sub-solar primordial black holes, responsible for the gravitational waves detectable by LIGO-Virgo-KAGRA. This is a consequence of the existing FIRAS CMB distortion constraints on the relevant scales. This conclusion might not hold for lighter masses potentially accessible by future gravitational wave observations.

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Forward citations

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Reviewed August 9, 2026 · model on record in the stance chip above.