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Lower bound on the primordial black hole merger rate

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper establishes a conservative lower bound on the merger rate of primordial black hole binaries, combining a maximally suppressed initial-binary population with a conservative population of re-hardened binaries, and shows the floor…

desk verdict Solid incremental extension with a real new disruption mechanism, but the 'lower bound' label is not fully earned because Rp uses alpha=1 rather than the stated conservative alpha→∞ limit. read the letter →

arxiv 1908.09752 v3 pith:BPD2MKB4 submitted 2019-08-26 astro-ph.CO astro-ph.HEhep-ph

Ville Vaskonen , Hardi Veermäe This is my paper · ORCID
classification astro-ph.COastro-ph.HEhep-ph
keywords primordialblackholesmergerrategravitationalwavesdarkmatterbinarydisruptiongravothermalcollapselowerboundLIGO/Virgoconstraints
topics Dark Matter
open problems 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 tries to settle what the merger rate of primordial black hole (PBH) binaries can be at the very least. The authors construct a lower bound by combining two populations: initial binaries whose merger rate is maximally suppressed by assuming that every binary living in a gravitationally unstable clump is disrupted, and "perturbed" binaries that are re-hardened by three-body encounters and can still merge today. The sum of the two populations has a floor, and that floor exceeds the merger rate inferred from current gravitational-wave observations whenever more than about 4% of dark matter is made of $\sim 10\,M_\odot$ PBHs. Because the bound is deliberately conservative, the authors conclude that PBHs cannot constitute all of the dark matter in the $1\text{--}100\,M_\odot$ range, and that the observed binary black hole events can be explained by a narrow PBH mass function peaked near $20\,M_\odot$ with abundance $0.1\text{--}4\%$.

What carries the argument

The load-bearing construction is a two-term merger rate, $R=P_{\rm np}R_{\rm np}+R_p$. $P_{\rm np}$ is computed from a halo mass function for PBHs: haloes smaller than a critical size $N_c \simeq 1500\,f_{\rm PBH}^{10/7}(\ln\Lambda)^{4/7}$ are assumed to undergo gravothermal collapse and disrupt every initial binary they contain, while substructure is assumed to survive absorption, which maximizes the disrupted fraction. $R_p$ is built from the energy distribution of binaries re-formed in three-body encounters, with the transfer kernel $K(E|E')=(\alpha/E')\,e^{-\alpha(E/E'-1)}$ and angular momentum distribution $\mathrm{d}P/\mathrm{d}j=\gamma j^{\gamma-1}$; the paper uses $\gamma=2$ and $\alpha=1$ to get a smaller rate. These two pieces turn the uncertain history of early binaries into a definite floor.

What would settle it

A cosmological N-body simulation that follows PBH structure formation with $f_{\rm PBH}=0.1$, $m=20\,M_\odot$, resolves binaries and three-body encounters until $z=0$, and measures both the energy transfer kernel $K(E|E')$ and the effective $\gamma$ of re-formed binaries would settle the question. If that measured perturbed-binary merger rate falls below the gravitational-wave observed band at $f_{\rm PBH}\simeq 0.04$, the claimed lower bound is wrong.

Watch

Extended reading notes

Core claim

The central claim is that even in the worst case for primordial black hole binaries—every early binary that could be disrupted is disrupted—the total present-day merger rate cannot be pushed below the rate indicated by gravitational-wave observations unless the PBH abundance is small. The paper's lower bound is $R = P_{\rm np}R_{\rm np}+R_p$, where $P_{\rm np}$ is the smallest plausible fraction of initial binaries that survive unperturbed, $R_{\rm np}$ their merger rate, and $R_p$ a conservative rate from binaries whose orbits were changed by encounters with other PBHs. With $P_{\rm np}$ estimated from the fraction of PBHs in haloes small enough to undergo gravothermal core collapse within a Hubble time, and $R_p$ evaluated from an energy-transfer kernel and angular momentum distribution calibrated by the authors' earlier N-body work, the bound exceeds the observed rate when $f_{\rm PBH}\gtrsim 0.04$ for $O(10\,M_\odot)$ masses. The authors therefore state that PBH dark matter is ruled out in $1\text{--}100\,M_\odot$, and that future detector sensitivity can reach PBH masses from $10^{-2}$ to $10^3\,M_\odot$.

Load-bearing premise

The load-bearing premise is that $\alpha=1$ and $\gamma=2$, drawn from the authors' N-body calibration, bound the true perturbed-binary rate from below; if real perturbed binaries are less hardened or more circular than that calibration, $R_p$ is overestimated and the lower-bound claim collapses.

Editorial extensions

If this is right

  • Current gravitational-wave observations already exclude PBHs as all of the dark matter for masses from about 1 to 100 $M_\odot$.
  • The exclusion is robust to the history of early binaries: even maximally disrupting initial binaries leaves a merger floor from re-hardened binaries that still exceeds the observed rate above $f_{\rm PBH}\simeq 4\%$.
  • The allowed window for explaining the observed binary black hole events with PBHs is a narrow mass function around $20\,M_\odot$ with abundance roughly $0.1\text{--}4\%$.
  • At design sensitivity, gravitational-wave detectors could probe PBH abundances across the much wider mass range $10^{-2}\text{--}10^3\,M_\odot$.
  • If PBHs were initially clustered, the lower bound strengthens: clustering suppresses initial binaries more while increasing the rate of perturbed binaries.

Reading between the lines

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

  • The paper's two-population decomposition suggests a template for future detectors: fitting the mass and redshift dependence of the merger rate could separate the unperturbed initial binaries from the re-formed population, since they scale differently with $f_{\rm PBH}$ and time.
  • The bound is phrased for monochromatic mass functions; extending it to broad mass functions requires tracking mass segregation, and the paper's own discussion suggests heavy PBHs dominate perturbed binaries, a testable prediction for the mass ratios of PBH merger events.
  • The halo-stability criterion $N_c$ could be checked independently with idealized cluster simulations: the sharp claim is that every halo with $N<N_c$ disrupts all its binaries within a Hubble time, and a single counterexample would weaken only that part of the bound, while the perturbed-binary floor would remain.
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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 paper derives a purported lower bound on the present merger rate of primordial black hole (PBH) binaries by combining a maximally suppressed merger rate of primordial binaries, Pnp Rnp, with an estimated contribution from binaries that have been perturbed by encounters in early PBH haloes, Rp. The suppression factor Pnp is computed from the fraction of PBHs in haloes and subhaloes that are gravitationally unstable within a Hubble time, and Rp is modeled with energy-transfer and angular-momentum kernels calibrated from the authors' earlier N-body work. Using LIGO/Virgo O1+O2 observations, the authors obtain 95% constraints on the PBH abundance for a monochromatic mass function, exclude f_PBH = 1 in the 1-100 solar mass range, and project the reach of LIGO/Virgo design sensitivity.

Significance. If the lower-bound construction were rigorously established, the paper would be an important step: it would convert rate-based PBH constraints into conservative statements that are less sensitive to uncertain binary-disruption physics, and it provides concrete projections for LIGO/Virgo design sensitivity. The authors are explicit about the conservative direction of many assumptions, and the analytic model for the perturbed-binary contribution is a useful contribution. However, the central 'lower bound' label is not currently justified because the most conservative limits of the Rp parameters are not used in the final constraints.

major comments (3)
  1. [Section IV, Eq. (20) and Section VI] The parameters used for Rp are not the conservative extremes. The text states that the limiting case alpha -> infinity gives a lower bound on the merger rate because binding energies do not change, and Fig. 2 shows that Rp decreases with increasing alpha; nevertheless the constraints in Section VI use alpha = 1. Likewise, footnote 5 notes that gamma -> infinity (circular orbits) gives a much smaller Rp, yet the constraints use gamma = 2. Since the claimed lower bound is R = Pnp Rnp + Rp, using non-extremal alpha and gamma can overestimate Rp and thereby invalidate the bound, even if the numerical impact on the final constraints is small. The authors should either prove that alpha = 1 and gamma = 2 are guaranteed to bound the true rate from below, or recompute the constraints with the stated extremal limits and relabel the result as an estimate rather than a lower bound.
  2. [Section IV, Eqs. (18)-(20)] The calibration of alpha ~ 1 and the angular-momentum distribution in Eq. (19) comes exclusively from the authors' own N-body simulations in Ref. [17]. The paper does not provide independent validation or systematic uncertainties for these kernels, and the conclusion concedes that the analytic results 'should be fully and rigorously tested with numerical simulations.' Because the bound depends on the assumed forms of K(E|E') and P(j), the absence of a robustness test weakens the claim that the final rate is a lower bound. I recommend a sensitivity study varying alpha and gamma, or an explicit comparison with independent N-body results.
  3. [Section III, Eq. (10)] The construction of Pnp assumes that all subhaloes survive and that every binary in a halo with N <= Nc is perturbed with certainty. These assumptions are conservative for suppressing Rnp, but the total rate also includes Rp, so the treatment should clarify how the subhalo-survival assumption affects the perturbed-binary population. In addition, the formula evaluates the halo distribution at zc without an explicit derivation of how Nc(z) evolves with redshift; a more detailed derivation would help the reader verify the bookkeeping in Eq. (10).
minor comments (5)
  1. [Section VII] The phrase 'in a the Hubble time' should read 'within a Hubble time'.
  2. [Section VI, Eq. (23)] The text states that rho(m1,m2,z)2 is the signal-to-noise ratio, but the quantity entering the expected number of events should be the squared SNR; please clarify the notation.
  3. [Section III, Eq. (10)] The definitions and normalizations of \bar p_N and \tilde p_N are implicit; please state explicitly that they are normalized as in Eq. (11) and clarify the meaning of the inequality sign in the text.
  4. [Section V, Eq. (22)] The clustering rescaling in Eq. (22) is plausible but is stated without a detailed derivation; a short derivation or a more explicit reference would strengthen the argument that the conclusions persist for clustered PBH models.
  5. [General] Several quantitative inputs (alpha, gamma, suppression factor, and the kernel K) are taken from Ref. [17]; please list precisely which quantities are imported from that paper and which are derived here, so that the self-citation burden is transparent to the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the lower-bound construction combines an analytic survival fraction with N-body-calibrated perturbed-binary kernels; the alpha=1 vs alpha->infinity choice is a rigor concern, not a definitional reduction.

full rationale

The central result R = Pnp Rnp + Rp is not fitted to the LIGO/Virgo rate. Rnp in Eq. (3) is the standard unperturbed rate from Ref. [17] (same authors, but based on independent analytic/N-body work that does not contain the present bound); Pnp is a new analytic estimate, Eqs. (9)-(10), of the fraction of initial binaries that avoid disruption by unstable haloes; and Rp is built from Eqs. (18)-(20), with the angular-momentum and energy kernels calibrated by N-body simulations in Ref. [17]. These are independent inputs, so the self-citations are real evidence under the stated rules and do not make the argument circular. The one notable weakness is not circularity: Section IV states 'Binding energies do not change in the limiting case alpha -> infinity, which gives a lower bound on the merger rate,' while Section VI computes constraints with 'alpha = 1 and gamma = 2 that yield a smaller rate.' If alpha can exceed 1, the claimed lower-bound status of Rp is not established; the conclusion also concedes that the analytic models 'should be fully and rigorously tested with numerical simulations.' This is a robustness/correctness caveat, not a reduction of the prediction to its inputs. No circular step was identified.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central bound adds no new entities and has no data-fit parameters beyond conservative choices; it rests instead on analytic models of early structure formation and binary dynamics, many calibrated in the authors' earlier paper [17].

free parameters (3)
  • alpha = 1
    Energy-transfer parameter in K(E|E') of Eq. (20); set to 1 following Ref. [17] for the constraint, though the text states alpha to infinity gives the lower bound on Rp.
  • gamma = 2
    Exponent of the angular momentum distribution in Eq. (19), set to 2 to give a smaller Rp in the constraints; the allowed range is [1,2].
  • q = 20
    Multiplicative factor in the condition tau_i > q tp(y) in Eq. (13), chosen by hand to avoid binaries that emit more than 10% of binding energy in GWs; Rp depends weakly on q.
assumptions (5)
  • domain assumption Initial PBH spatial distribution is Poisson at scales relevant to binary formation.
    The halo mass function p_N in Eq. (9) and binary formation estimates assume initially Poisson-distributed PBHs; clustering is treated only as a rescaling in Sec. V.
  • domain assumption Press-Schechter-like halo mass function p_N(z) describes the distribution of early PBH haloes.
    Used in Eq. (9) and (10) to compute the fraction of binaries in haloes; adopted from Refs. [24,29,30].
  • domain assumption Gravothermal core collapse timescale satisfies t_cc >= 18 t_r with the relaxation time in Eq. (7).
    This relation defines N_c in Eq. (8) and determines which haloes disrupt their binaries; taken from globular cluster theory [19,20,27].
  • ad hoc to paper All binaries inside haloes with N <= N_c are perturbed with certainty and all subhaloes survive.
    Conservative assumption for maximal suppression of initial binaries, stated in Sec. III and used in Eq. (10); if false the suppression would be smaller, which would only make the lower bound more conservative.
  • domain assumption Heggie-Hills law: hard binaries are, on average, hardened by close encounters.
    Used in Sec. II to argue that perturbed initial binaries are not ionized but get longer coalescence times; standard result from Refs. [22,23].

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Cite this review

Pith. "Pith review of Lower bound on the primordial black hole merger rate." pith.science (2026). https://pith.science/paper/BPD2MKB4

@misc{pith2026190809752,
  author       = {Pith},
  title        = {Pith review of: Lower bound on the primordial black hole merger rate},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BPD2MKB4}},
  note         = {Machine review of arXiv:1908.09752}
}
abstract

We derive a lower bound on the merger rate of primordial black hole (PBH) binaries by estimating the maximal fraction of binaries that were perturbed between formation in the early Universe and merger, and computing a conservative merger rate of perturbed binaries. This implies robust constraints on the PBH abundance in the range $1-100 M_\odot$. We further show that LIGO/Virgo design sensitivity has the potential to reach the PBH mass range of $10^{-2}-10^3 M_\odot$. The constraint from the merger rate of perturbed binaries is stronger if PBHs are initially spatially clustered.

Figures

Figures reproduced from arXiv: 1908.09752 by the authors.

Figure 1
Figure 1. FIG. 1. Lower bound on the suppression factor of the merger [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The blue lines show the lower bound on the merger [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Constraints on the PBH abundance for monochro [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Pith tools

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