Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

A flexible six-component fit to 259 black hole mergers finds a geometric mass ladder but no matching spin-alignment ladder, challenging earlier claims of aligned high-mass remnants.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 23:22 UTC pith:M66IY5YU

load-bearing objection A flexible SPL+5G mixture fit that usefully challenges previous aligned high-mass claims, but the per-component isotropy result is likely a weak-constraint artifact, not a physical discovery. the 4 major comments →

arxiv 2607.15431 v1 pith:M66IY5YU submitted 2026-07-16 astro-ph.HE

Uncovering Hierarchical Sub-Population of Binary Black Holes

Muhammad Zeeshan , Richard O'Shaughnessy This is my paper
classification astro-ph.HE
keywords binary black holesgravitational-wave population inferencehierarchical black hole mergersmass spectrumblack hole spin alignmentmixture modelsisotropic spinsstochastic gravitational-wave background
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to establish that the binary black hole population separates into six distinct components — a smooth low-mass power law and five Gaussian mass peaks — and that once each component is allowed its own spin magnitude and tilt distribution, only the lowest-mass component is preferentially aligned. The recovered peaks form a roughly geometric mass ladder around 9.6, 17.5, 27.7, 52.8, and 84.5 solar masses, with adjacent ratios near 1.6–1.9, the pattern expected if black holes repeatedly merge into heavier black holes. The spin behavior, however, does not follow the ladder: all higher-mass components are consistent with isotropic orientations, with spin magnitude rising only mildly with mass. If correct, earlier evidence for aligned high-mass black holes was an artifact of restrictive model assumptions, and any hierarchical-formation picture must explain why merger remnants do not display the large aligned spins vacuum general relativity predicts.

Core claim

On its own terms, the paper reports that a population model with a smoothed power-law component and five Gaussian components, each with independently fitted rate, mass, spin magnitude, and spin-tilt mixture, fits the 259 observed binary black hole mergers and recovers a hierarchical mass structure: peak masses at roughly 9.6, 17.5, 27.7, 52.8, and 84.5 solar masses with adjacent ratios of about 1.6 to 1.9. The spin structure is the key result: the lowest-mass Gaussian is low-spin and aligned, consistent with isolated binary evolution, while every higher-mass component is consistent with isotropic spin directions, with average spin magnitude increasing toward the most massive component. The p

What carries the argument

The central object is the multi-component mixture model: a smoothed power-law component plus five truncated-Gaussian mass components (SPL+5G), where each component has its own rate, mass mean and width, spin-magnitude distribution, and spin-tilt distribution (a truncated Gaussian mixed with an isotropic component), with only the redshift evolution shared. Because the spin and tilt parameters are not tied across components, the model does not force higher-mass components to inherit the aligned, high-spin character of merger remnants; that freedom is what allows the paper to separate the mass ladder from the spin ladder. A secondary mechanism is the coagulation-style bookkeeping that converts

Load-bearing premise

The load-bearing premise is that 259 events split across six components give enough per-component spin-tilt information to tell isotropy from alignment; if the per-component posteriors are simply broad, 'consistent with isotropic' could mean 'not yet measured,' a limitation the paper itself acknowledges.

What would settle it

Take the highest-signal subset of mergers in the 25–40 solar mass range with well-measured spin tilts and compute the posterior on the isotropic fraction of the tilt mixture; if that fraction is concentrated well below 1 (say, with 95% credible interval excluding 1), the claim that all but the lowest-mass component are isotropic is falsified. Equally direct: compute the posterior predictive distribution of effective spin versus mass; a significant excess of positive effective spin in the high-mass components would be the aligned signature the paper says is absent.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The mass peaks form a near-geometric sequence with ratios of about 1.6–1.9, and the comparable merger rates of the second and third components suggest a near-equilibrium cascade, while higher-mass components appear 'starved' — a pattern consistent with repeated mergers.
  • Because the higher-mass components are consistent with isotropic spins, a naive hierarchical formation picture in which remnants retain large aligned spins is in tension with the data.
  • Previous inferences favoring aligned high-mass black holes likely depended on model assumptions that fixed post-merger spins, not on the data themselves.
  • A straw-man axion spin-down model can reproduce the suppressed high-mass spins, but its simplest version predicts a stochastic gravitational-wave background close to current upper limits, so the naive model may be ruled out in the next few years.

Where Pith is reading between the lines

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

  • If the isotropic spin conclusion survives with more data, the mass ladder alone cannot prove hierarchical formation: a formation channel that produces a ladder of mass peaks without high remnant spins (such as gas-damped accretion or axion spin-down) would look the same in mass but different in spin.
  • A natural next test is to restrict the analysis to events with well-measured spin tilts and check whether the per-component isotropic fractions tighten around one; if they instead move toward alignment as the sample grows, the claimed absence of spin–mass correlation would evaporate.
  • The axion scenario makes a frequency-specific prediction: a stochastic background peaked near roughly 48 Hz for an axion mass near 10^-13 eV, which future cross-correlation searches at that frequency could confirm or exclude independently of population modeling.
  • The same component-wise fitting logic could be applied to neutron star–black hole mergers, where a hierarchical mass ladder is less expected; a null result there would sharpen the interpretation that the ladder is specific to repeated black hole mergers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper fits a mixture population model to 259 binary black holes from GWTC-5, consisting of one smoothed power-law component and five Gaussian mass components (SPL+5G), each with independently modeled spin magnitude and spin-tilt distributions and a shared redshift evolution. The authors report that the recovered mass peaks form a roughly geometric ladder, that the highest-mass Gaussian is consistent with isotropic spin tilts, and that the previously claimed aligned high-mass population disappears once the model allows per-component spin orientations to vary freely. They interpret the mass ladder as evidence for hierarchical formation but note that the expected spin-mass correlation is absent, and they propose a speculative axion spindown model as a straw-man reconciliation, deriving a stochastic gravitational-wave background estimate that is in tension with current limits.

Significance. If the central claim were established, this would be an important update to the BBH population literature: it would challenge earlier inferences that high-mass components are preferentially aligned, and it would sharpen the discussion of hierarchical formation by separating mass structure from spin structure. The analysis uses a standard hierarchical Bayesian likelihood and a publicly available inference tool (GWKokab), and the model is transparently specified with prior ranges in Table I. The paper is also candid about several limitations, including the need for more observations and the bookkeeping nature of some derived quantities. However, the significance is substantially weakened by the lack of quantitative validation that the per-component spin-tilt data can actually distinguish isotropy from alignment, and by the fact that the mass ladder is partly a consequence of the chosen prior windows. The axion model is clearly speculative and the stochastic-background estimate is a conditional consistency check rather than a robust prediction.

major comments (4)
  1. [Section III.B, Fig. 4, Table II] The headline claim that all components except G1 are 'consistent with isotropic spins' is presented as a physical finding, but the paper provides no measure of how strongly the data constrain each component's tilt distribution. With 259 events split across six overlapping mass components and the flexible tilt model in Table I (mixture fraction zeta in U(0,1), tilt width sigma_cos in U(0.01,4)), broad posteriors can reflect lack of information rather than isotropy. The paper itself concedes that 'a firm resolution of this discrepancy requires more observations' and that G5 is 'less well constrained.' To make the claim load-bearing, the authors should report per-component effective sample sizes and posterior distributions of zeta (or an equivalent aligned fraction), and ideally perform injection-recovery tests in which simulated populations with aligned high-mass components are fit to veri
  2. [Section III.A, Table I, Table III] The 'roughly geometric ladder' of Gaussian mass peaks is in part imposed by the prior. Table I restricts the Gaussian means to disjoint, observationally motivated bands: G1 in U(5,13), G2 in U(13,25), G3 in U(25,45), G4 in U(45,65), G5 in U(65,90). The adjacent peak-mass ratios in Table III therefore cannot fall below unity and will naturally be close to the ratios of the band centers. The text acknowledges this in Section III.A ('in part by construction'), but the conclusion that the data unveil a hierarchical mass ladder is still presented as a discovery. The authors should demonstrate that the likelihood, rather than the prior, selects the specific ladder: for example, by comparing prior and posterior distributions of the means, rerunning with wider or overlapping windows, or fitting an ordered-Gaussian model without disjoint bands. As written, the hierarchical-mass conclusion is part
  3. [Section IV.B, Table IV] The effective interaction four-volumes K_eff defined in Table IV are bookkeeping ratios of the fitted component rates (e.g., R_G2/R_G1^2). The statement that these values are 'roughly consistent with one another, suggesting a similar origin as starved hierarchical mergers' is not supported: compatibility among ratios of the same inferred rates is not evidence for a hierarchical origin unless there is an independent prediction for a common K_eff and an explicit comparison that accounts for posterior correlations and selection effects. The discussion should either be framed as a purely descriptive consistency check or replaced with a posterior predictive comparison of the coagulation model's predicted rates to the inferred component rates. As it stands, the hierarchical-origin language overstates what Table IV can establish.
  4. [Appendix B, Eq. (B3), Section V] The stochastic gravitational-wave background estimate is presented as a prediction ('it would predict a stochastic background...'), but it is derived by combining the inferred G1 rate with an assumed extracted energy per remnant within a model whose parameters are explicitly hand-tuned to reproduce the observed mass-spin behavior. The resulting Omega_GW ~ (1.3-1.8)e-9 xi_z is then compared with O4a limits. This is a conditional consistency check, not an independent prediction: the axion model is built partly to produce the spin behavior already inferred, and the uncertainty in the rate is large (Table II gives R_G1 ~ 15.6 +13.25/-5.48 Gpc^-3 yr^-1). The authors should either propagate the full uncertainty and clearly state that this is a toy-model consistency check, or soften the claim that the model can be 'ruled out completely in the next few years.'
minor comments (5)
  1. [Section II.B, after Eq. (8)] The phrase that spin tilts are 'identically but not independently distributed' is unclear. Please specify the dependence structure between the two component spins (e.g., whether they share a draw and are therefore perfectly correlated, or are independent) and define whether the secondary tilt is tied to the primary in the same way as the spin magnitude.
  2. [Table I] The note that secondary spin magnitude and tilt parameters are tied to primary parameters unless listed separately is ambiguous because no separate secondary parameters appear in the table. Please state explicitly how secondary spin and tilt are derived from the primary component parameters.
  3. [Section III.A, Fig. 1] The text says 'The labels and contours indicate the contributions from each of the five Gaussian components which dominate the overall population,' but the color/line coding of the components in Figs. 1 and 2 is not defined in the captions. Please add a legend or explicit color mapping.
  4. [Section III.B, Fig. 4] Typo: 'miss-alignment' should be 'misalignment' in the right-panel caption. Also 'subtantially' should be 'substantially' in Section III.B.
  5. [Section II.A and III.A] Minor wording issues: 'the more more flexible model' should be 'the more flexible model'; the heading 'C. Redshift distribution conventional' is unclear and should be rewritten.

Circularity Check

2 steps flagged

Mass ladder is partly imposed by disjoint Gaussian priors; K_eff table is bookkeeping of fitted rates, so the hierarchical packaging is partially circular, while the spin-isotropy claim is an underpowered fit rather than a tautology.

specific steps
  1. self definitional [Section III.A and Table I / Table III; Abstract]
    "Gaussian masses µ_m,gg,1 U(5,13)M⊙ ... µ_m,gg,5 U(65,90)M⊙ [Table I]. Though flexible, our multi component model builds in strong prior knowledge on the location of each gaussian component, motivated by observations to date. ... our multi-component model has a roughly hierarchical spectrum of gaussian mass peaks [Abstract]."

    The model is defined as a powerlaw plus five 'successively higher mass gaussians' whose means are confined to disjoint ordered prior windows (5–13, 13–25, 25–45, 45–65, 65–90 M⊙). The recovered peaks in Table III (9.56, 17.53, 27.69, 52.79, 84.46 M⊙) therefore lie in those windows by construction; the existence of a monotonically increasing mass ladder is built into the parameterization, not discovered. The exact spacings retain some posterior content, but the 'roughly hierarchical spectrum' is largely a restatement of the prior layout.

  2. fitted input called prediction [Section IV.B and Table IV]
    "To assess the self-consistency of high-mass mergers as a “starved” hierarchical cascade, we assume R_c = K R_a R_b where K has units of 4-volume. Table IV reports this factor... For the higher-generation mergers, we find these factors are roughly consistent with one another, suggesting a similar origin as starved hierarchical mergers. ... These quantities are bookkeeping factors used to compare possible hierarchical transition paths, not literal microscopic cross sections."

    Table IV defines K_eff as RG2/RG1^2, RG3/(RG1 RG2), RG4/RG2^2, RG5/(RG3 RG4), etc. Each entry is simply a ratio of the same six component rates already fitted to the data. With K_eff defined this way, the relation R_c = K R_a R_b is an identity for every channel; comparing the resulting numbers across channels tests only whether the fitted rates are mutually proportional, not whether the hierarchical cascade is real. The paper's own caption calls them 'bookkeeping factors,' so the 'suggesting a similar origin' conclusion is a reinterpretation of fitted rates, not an independent test.

full rationale

The core population fit is a legitimate hierarchical-Bayesian analysis of 259 BBH against an explicit SPL+5G model, and using the authors' own GWKokab framework is not circular. The genuinely load-bearing spin-orientation result (G2–G5 consistent with isotropic) is an inference from the fit, not a construction, though the paper concedes its own power limitation ('A firm resolution of this discrepancy requires more observations'), so any weakness there is statistical rather than circular. What does reduce by construction is the hierarchical mass ladder: the five Gaussian means are assigned disjoint ordered prior windows, so a rising sequence of peaks is built into the parameterization; Table III's ratios are posterior summaries of those fitted locations. The K_eff table is explicitly bookkeeping: each entry is a ratio of the same fitted component rates, so the 'roughly consistent' starved-cascade conclusion repackages the fit rather than testing it. The axion/SGWB estimate is a hand-tuned straw-man extrapolation and is not a circular prediction. On balance, the paper contains partial circularity in its 'hierarchical' packaging, but the spin claim itself is an underpowered inference rather than a tautology, giving a score of 6.

Axiom & Free-Parameter Ledger

13 free parameters · 10 axioms · 1 invented entities

The central fit pulls in the standard hierarchical-likelihood machinery and public PE/injection products; the paper's own contribution is the mixture model, whose component locations are strongly guided by priors. The interpretive 'prediction' (axion SGWB) depends on hand-tuned parameters and fitted rates.

free parameters (13)
  • Power-law mass indices alpha_spl, beta_spl and smoothing deltas delta_m1,spl, delta_m2,spl = posterior fit
    Define SPL component shape; fitted to GWTC-5 events (Eq. 7, Table I).
  • Minimum masses m1,min, m2,min = posterior fit
    Common lower mass cutoff for all components, free in fit (Table I).
  • Redshift evolution index kappa = posterior fit
    Single shared redshift power-law (Eq. 6), fitted.
  • Component merger rates R_spl, R_G1...R_G5 = posterior fit
    Six independent rates; posteriors shown in Fig. 6 and Table II.
  • Gaussian mass means mu_m,gg,1..5 = posterior fit
    Prior windows at 5-13, 13-25, 25-45, 45-65, 65-90 Msun; fitted (Table I).
  • Gaussian mass widths sigma_m,gg,1..5 = posterior fit
    Uniform priors 0-10 or 0-15 Msun; fitted (Table I).
  • Spin magnitude locations mu_chi for SPL and G1..G5 = posterior fit
    Truncated Gaussian spin magnitudes; fitted per component (Table I).
  • Spin magnitude widths sigma_chi for SPL and G1..G5 = posterior fit
    Widths of spin magnitude; fitted per component (Table I).
  • Tilt mixture fractions zeta for k=0..5 = posterior fit
    Mixture weight between Gaussian cos(tilt) and isotropic; fitted per component.
  • Tilt locations mu_cos(theta) for k=0..5 = posterior fit
    Centers of truncated Gaussian tilt component; fitted per component.
  • Tilt widths sigma_cos(theta) for k=0..5 = posterior fit
    Widths of truncated Gaussian tilt component; fitted per component.
  • Axion toy-model kernel parameters (a, b_low, M_turn, DeltaM, sigma_low, sigma_high, A_loc, epsilon_int) = hand-set (Appendix A, Table V)
    Used in constructive hierarchical simulation; hand-tuned, not inferred from GW data.
  • Axion residency parameters (m_a, tau0, alpha_ref, p_tau, t_res, M_disrupt, DeltaM_disrupt, Lambda_cap, t_enc,ref, p_enc, = hand-set (Appendix B, Table VI)
    Used in straw-man axion spindown model; developed as fine-tuned demonstration.
axioms (10)
  • standard math Inhomogeneous Poisson process likelihood for GW detections
    Eq. 2; standard hierarchical likelihood.
  • domain assumption Selection function P_det from injection sets captures detectability
    Eq. 5; relies on injected search sensitivity estimates.
  • domain assumption Published PE samples with MIXED priors are unbiased after weighting
    Section II.C; weights standardize heterogeneous PE priors.
  • ad hoc to paper The true BBH population is well described by one SPL plus five independent 1-D Gaussian mass components with shared redshift evolution
    Eq. 6-8; model choice, not derived from physics.
  • ad hoc to paper Gaussian mass components are separable in m1,m2
    Eq. 8; no correlated 2-D Gaussian in the mass plane.
  • ad hoc to paper Spin tilts are a mixture of a truncated Gaussian and an isotropic distribution
    Section II.B; parameterization affects isotropy claims.
  • ad hoc to paper m_max fixed at 300 Msun for all components
    Section II.B; ad hoc high-mass cutoff.
  • ad hoc to paper Prior windows for Gaussian means are observationally motivated and centered in disjoint mass bands
    Table I; shapes the hierarchical appearance of the recovered mass ladder.
  • domain assumption Remnant mass/spin fits from vacuum GR describe hierarchical merger products
    Used in Section IV and Appendix A to compare expected hierarchical spin.
  • ad hoc to paper Axion superradiance spin-down model with specific parameters
    Appendix A/B; proposed straw-man explanation, not yet independently verified.
invented entities (1)
  • Axion scalar boson with m_a ~ 1.55e-13 eV and associated boson cloud independent evidence
    purpose: Spin down black-hole remnants to explain low spins and isotropic orientations in high-mass components.
    Makes falsifiable prediction of stochastic GW background near f~48 Hz with amplitude close to current LVK limits; also constrained by scalar-cloud searches. But parameters are hand-tuned.

pith-pipeline@v1.3.0-alltime-deepseek · 18858 in / 13196 out tokens · 134605 ms · 2026-08-01T23:22:54.502184+00:00 · methodology

0 comments
read the original abstract

Enabled by improved instruments with increasing sensitivity, the ongoing gravitational wave census now contains 259 binary black holes, numerous enough to unveil trends, substructure, and subpopulations which may provide key clues to their underlying formation mechanisms. In this work, motivated by evidence for multiple formation channels including hierarchical formation, we build a natively multi component mixture model for the binary black hole population, in which each component has an independently recovered rate, mass, spin, and spin misalignment model. (The components share a common redshift distribution.) Using a model carefully tuned to avoid parameter degeneracies, a powerlaw model plus five successively higher mass gaussians, we recover overall merger rates versus mass and trends versus redshift which are consistent with previously published results. Too, we recover previously identified overall trends versus spin: preferential alignment and low spin at low mass; large spin and isotropic spins at high mass. Critically, however, our multi-component model disagrees with previously published results, finding all components except the lowest mass are consistent with isotropy. Too, our multi-component model has a roughly hierarchical spectrum of gaussian mass peaks, but without the expected correlations between spin and mass expected from naked hierarchical formation

Figures

Figures reproduced from arXiv: 2607.15431 by Muhammad Zeeshan, Richard O'Shaughnessy.

Figure 1
Figure 1. Figure 1: FIG. 1: The left panel shows the inferred merger-rate density in the ( [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The left (right) figure shows the marginal distributions of primary (secondary) mass. The solid gray line represent [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The left figure shows the posterior distributions of the full model for primary and secondary masses. The solid (dashed) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The left figure shows the posterior distributions of the spin magnitudes of each component. The solid (dashed) lines [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The left figure shows the distribution of the spin magnitude conditioned on primary mass, [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: This plot shows the recovery for the rate of each component of the full model with units in Gpc [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The redshift distribution comparison between the [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Monte Carlo simulation of hierarchical BH formation with axionic spindown. [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Binary merger rate versus component masses expected from a Monte Carlo simulation of hierarchial binary formation, [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A Four-dimensional Model-agnostic Probe into the Astrophysical Origins of Binary Black Hole Subpopulations

    astro-ph.HE 2026-07 conditional novelty 7.0

    A GPU-accelerated binned Gaussian process yields the first model-agnostic 4D BBH population in (m1, q, χeff, χp), revealing four mass-based subpopulations and new spin-mass-ratio correlations.

Reference graph

Works this paper leans on

118 extracted references · 85 linked inside Pith · cited by 1 Pith paper

  1. [1]

    We denote the normalized tracer distribution by pk(m, χ) for single black holes in generationS k, and let Nk denote its scalar abundance

    T racer populations and abundance weights The Monte Carlo implementation uses tracer particles for the shape of each single-black-hole generation, but it carries the scalar abundance of each generation sepa- rately. We denote the normalized tracer distribution by pk(m, χ) for single black holes in generationS k, and let Nk denote its scalar abundance. The...

  2. [2]

    Thus Sk ←S j +S k−1, j= 1,

    Generation recurrence The finite-generation recurrence follows the Doctor- style coagulation construction used in the demonstration code: to build generationS k, the newest available genera- tionS k−1 is paired against all earlier resident generations Sj with 1≤j≤k−1. Thus Sk ←S j +S k−1, j= 1, . . . , k−1.(A3) For each channel (j, ℓ), withℓ=k−1, the diff...

  3. [3]

    Even with a large globalqexponent, the reservoir abundanceN 1 can makeS 1 +S 3 orS 1 +S 4 dominate the high-mass event budget

    High-mass locality kernel A separable power-law pairing kernel, Ksep(m1, m2)∝(m 1 +m 2)aqb, q= min(m1, m2) max(m1, m2) , (A8) does not by itself suppress the abundant mixed channels involvingS 1. Even with a large globalqexponent, the reservoir abundanceN 1 can makeS 1 +S 3 orS 1 +S 4 dominate the high-mass event budget. The constructive model therefore u...

  4. [4]

    The axion model is applied as a post-merger spin-residency map for retained single black holes

    Axion spin map The interaction model above determines the mass- pairing and channel weights. The axion model is applied as a post-merger spin-residency map for retained single black holes. The Regge floor is approximated by the leadingm=ℓ= 1 scalar trajectory, α(M;m a) = 0.0748 M 10M⊙ ma 10−12 eV ,(A13) χR(M;m a) = 4α 1 + 4α2 .(A14) 13 TABLE V: Parameters...

  5. [5]

    B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Aber- nathy, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, et al., Phys. Rev. Lett116, 061102 (2016), 1602.03837

    Pith/arXiv arXiv 2016

  6. [6]

    The LIGO Scientific Collaboration, The Virgo Collabo- ration, B. P. Abbott, R. Abbott, T. D. Abbott, F. Ac- ernese, K. Ackley, C. Adams, T. Adams, P. Addesso, et al., Phys. Rev. X9, 031040 (2019)

    2019

  7. [7]

    Abbott, T

    R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, N. Adhikari, R. X. Adhikari, V. B. Adya, C. Affeldt, D. Agarwal, et al., Physical Review X13, 041039 (2023), 2111.03606

    Pith/arXiv arXiv 2023

  8. [8]

    The LIGO Scientific Collaboration, the Virgo Collabo- ration, the KAGRA Collaboration, A. G. Abac, R. Ab- bott, H. Abe, F. Acernese, K. Ackley, C. Adamcewicz, S. Adhicary, et al., Submitted to ApJ, available as arXiv:2508.18082 arXiv:2508.18082 (2025), 2508.18082

    Pith/arXiv arXiv 2025

  9. [9]

    The LIGO Scientific Collaboration, the Virgo Collabo- ration, and the KAGRA Collaboration, arXiv e-prints arXiv:2605.27090 (2026), 2605.27090

    Pith/arXiv arXiv 2026

  10. [10]

    LIGO Scientific Collaboration, J. Aasi, B. P. Abbott, R. Abbott, T. Abbott, M. R. Abernathy, K. Ackley, C. Adams, T. Adams, P. Addesso, et al., Classical and Quantum Gravity32, 074001 (2015), 1411.4547

    Pith/arXiv arXiv 2015

  11. [11]

    Acernese, M

    F. Acernese, M. Agathos, K. Agatsuma, D. Aisa, N. Allemandou, A. Allocca, J. Amarni, P. Astone, G. Balestri, G. Ballardin, et al., Classical and Quan- tum Gravity32, 024001 (2015), 1408.3978

    Pith/arXiv arXiv 2015

  12. [12]

    Akutsu, M

    T. Akutsu, M. Ando, K. Arai, Y. Arai, S. Araki, A. Araya, N. Aritomi, Y. Aso, S. Bae, Y. Bae, et al., Progress of Theoretical and Experimental Physics2021, 05A101 (2021), 2005.05574

    arXiv 2021

  13. [13]

    The LIGO Scientific Collaboration, The Virgo Col- laboration, B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Ad- desso, et al., Astrophysical Journal882, L24 (2019), 1811.12940

    Pith/arXiv arXiv 2019

  14. [14]

    Abbott, T

    The LIGO Scientific Collaboration, the Virgo Col- laboration, R. Abbott, T. D. Abbott, S. Abraham, F. Acernese, K. Ackley, A. Adams, C. Adams, V. B. 15 Adya, et al., Astrophysical Journal913, L7 (2021), 2010.14533

    Pith/arXiv arXiv 2021

  15. [15]

    The LIGO Scientific Collaboration, the Virgo Collabo- ration, the KAGRA Collaboration, A. G. Abac, R. Ab- bott, H. Abe, F. Acernese, K. Ackley, C. Adamcewicz, S. Adhicary, et al., Submitted to ApJ, available as arxiv:2508.18083 arXiv:2508.18083 (2025), 2508.18083, URLhttps://dcc.ligo.org/P2400004

    Pith/arXiv arXiv 2025

  16. [17]

    Ab- bott, T

    The LIGO Scientific Collaboration, The Virgo Collab- oration, The KAGRA Scientific Collaboration, R. Ab- bott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, N. Adhikari, R. X. Adhikari, et al. (LIGO Scientific Col- laboration, Virgo Collaboration, and KAGRA Collabo- ration), Phys. Rev. X13, 011048 (2023), URLhttps: //link.aps.org/doi/10.1103/PhysRevX.13.011048

    doi:10.1103/physrevx.13.011048 2023

  17. [18]

    Mandel and F

    I. Mandel and F. S. Broekgaarden, Living Reviews in Relativity25, 1 (2022), 2107.14239

    Pith/arXiv arXiv 2022

  18. [19]

    Gerosa and M

    D. Gerosa and M. Fishbach, Nature Astronomy5, 749 (2021), 2105.03439

    Pith/arXiv arXiv 2021

  19. [20]

    J. M. Ezquiaga and D. E. Holz, Phys. Rev. Lett129, 061102 (2022), 2202.08240

    Pith/arXiv arXiv 2022

  20. [21]

    Mastrogiovanni, D

    S. Mastrogiovanni, D. Laghi, R. Gray, G. C. Santoro, A. Ghosh, C. Karathanasis, K. Leyde, D. A. Steer, S. Perri` es, and G. Pierra, Phys. Rev. D108, 042002 (2023), 2305.10488

    Pith/arXiv arXiv 2023

  21. [22]

    Mali and R

    U. Mali and R. Essick, Astrophysical Journal980, 85 (2025), 2410.07416

    arXiv 2025

  22. [23]

    Gennari, T

    V. Gennari, T. Bertheas, and N. Tamanini, arXiv e- prints arXiv:2604.14290 (2026), 2604.14290

    Pith/arXiv arXiv 2026

  23. [24]

    Scarpa, S

    A. Scarpa, S. Mastrogiovanni, F. Santoliquido, and M. Arca-Sedda, arXiv e-prints arXiv:2603.15332 (2026), 2603.15332

    arXiv 2026

  24. [25]

    Tagliazucchi, M

    M. Tagliazucchi, M. Moresco, N. Borghi, and C. Cia- petti, A&A709, A197 (2026), 2601.03347

    Pith/arXiv arXiv 2026

  25. [26]

    A. G. Abac, I. Abouelfettouh, F. Acernese, K. Ackley, C. Adamcewicz, S. Adhicary, D. Adhikari, N. Adhikari, R. X. Adhikari, V. K. Adkins, et al., Astrophysical Jour- nal993, L25 (2025), 2507.08219

    Pith/arXiv arXiv 2025

  26. [27]

    Delfavero, S

    V. Delfavero, S. Ray, H. E. Cook, K. Nathaniel, B. McKernan, K. E. S. Ford, J. Postiglione, E. McPike, and R. O’Shaughnessy, Submitted to PRL arXiv:2508.13412 (2025), 2508.13412

    Pith/arXiv arXiv 2025

  27. [28]

    Liu and D

    B. Liu and D. Lai, arXiv e-prints arXiv:2511.13820 (2025), 2511.13820

    Pith/arXiv arXiv 2025

  28. [29]

    Li, S.-P

    Y.-J. Li, S.-P. Tang, L.-Q. Xue, and Y.-Z. Fan, arXiv e-prints arXiv:2507.17551 (2025), 2507.17551

    arXiv 2025

  29. [30]

    The LIGO Scientific Collaboration, the Virgo Collabo- ration, the KAGRA Collaboration, A. G. Abac, R. Ab- bott, H. Abe, F. Acernese, K. Ackley, C. Adamcewicz, S. Adhicary, et al., Submitted to ApJL (2025), URL https://dcc.ligo.org/P2500402

    2025

  30. [32]

    Passenger, S

    L. Passenger, S. Banagiri, E. Thrane, P. D. Lasky, A. Borchers, M. Fishbach, and C. S. Ye, Astrophysical Journal999, 236 (2026), 2510.14363

    arXiv 2026

  31. [33]

    Borchers, C

    A. Borchers, C. S. Ye, and M. Fishbach, Astrophysical Journal987, 146 (2025), 2503.21278

    Pith/arXiv arXiv 2025

  32. [34]

    A. M. Farah, A. Vijaykumar, and M. Fishbach, Astro- physical Journal1001, L40 (2026), 2601.03456

    Pith/arXiv arXiv 2026

  33. [35]

    Vijaykumar, A

    A. Vijaykumar, A. M. Farah, and M. Fishbach, Astro- physical Journal999, L30 (2026), 2601.03457

    arXiv 2026

  34. [36]

    C. S. Ye, M. Fishbach, K. Kremer, and M. Reina- Campos, Astrophysical Journal997, 267 (2026), 2507.07183

    Pith/arXiv arXiv 2026

  35. [37]

    Banagiri, E

    S. Banagiri, E. Thrane, and P. D. Lasky, arXiv e-prints arXiv:2509.15646 (2025), 2509.15646

    Pith/arXiv arXiv 2025

  36. [38]

    Li, Y.-Z

    Y.-J. Li, Y.-Z. Wang, S.-P. Tang, and Y.-Z. Fan, arXiv e-prints arXiv:2509.23897 (2025), 2509.23897

    Pith/arXiv arXiv 2025

  37. [39]

    H. Tong, T. A. Callister, M. Fishbach, E. Thrane, F. Antonini, S. Stevenson, I. M. Romero-Shaw, and F. Dosopoulou, arXiv e-prints arXiv:2511.05316 (2025), 2511.05316

    arXiv 2025

  38. [40]

    Tiwari, arXiv e-prints arXiv:2510.25579 (2025), 2510.25579

    V. Tiwari, arXiv e-prints arXiv:2510.25579 (2025), 2510.25579

    Pith/arXiv arXiv 2025

  39. [41]

    Antonini, I

    F. Antonini, I. M. Romero-Shaw, and T. Callister, Phys. Rev. Lett134, 011401 (2025), 2406.19044

    Pith/arXiv arXiv 2025

  40. [43]

    Sadiq, T

    J. Sadiq, T. Dent, and M. Gieles, Astrophysical Journal 960, 65 (2024), 2307.12092

    Pith/arXiv arXiv 2024

  41. [44]

    Zeeshan and R

    M. Zeeshan and R. O’Shaughnessy, Available as LIGO- P2500714 at dcc.ligo.org (2025)

    2025

  42. [45]

    Sigurdsson and E

    S. Sigurdsson and E. S. Phinney, ApJS99, 609 (1995), astro-ph/9412078

    Pith/arXiv arXiv 1995

  43. [46]

    R. A. W. Elson, S. Sigurdsson, M. Davies, J. Hurley, and G. Gilmore, MNRAS300, 857 (1998)

    1998

  44. [47]

    M. C. Miller and D. P. Hamilton, MNRAS330, 232 (2002), astro-ph/0106188

    Pith/arXiv arXiv 2002

  45. [49]

    Kimball, C

    C. Kimball, C. Talbot, C. P. L. Berry, M. Zevin, E. Thrane, V. Kalogera, R. Buscicchio, M. Carney, T. Dent, H. Middleton, et al., Astrophysical Journal 915, L35 (2021), 2011.05332

    Pith/arXiv arXiv 2021

  46. [50]

    Li and X.-L

    G.-P. Li and X.-L. Fan, Astrophysical Journal981, 177 (2025), 2411.09195

    Pith/arXiv arXiv 2025

  47. [51]

    Gayathri, V

    G. Gayathri, V. Iorio, H. Tagawa, D. Wysocki, J. An- glin, I. Bartos, S. Bhaumik, Z. Haiman, M. Mapelli, R. O’Shaughnessy, and L. Xue, Submitted to Astron- omy and Astrophysics, available as arXiv:2509.09647 arXiv:2509.09647 (2025), 2509.09647, URLhttps:// dcc.ligo.org/LIGO-P2500484

    Pith/arXiv arXiv 2025

  48. [52]

    Fragione, B

    G. Fragione, B. Kocsis, F. A. Rasio, and J. Silk, Astro- physical Journal927, 231 (2022), 2107.04639

    Pith/arXiv arXiv 2022

  49. [53]

    L. Xue, H. Tagawa, Z. Haiman, and I. Bartos, Phys. Rev. D112, 063034 (2025), 2504.19570

    arXiv 2025

  50. [54]

    Mould, D

    M. Mould, D. Gerosa, and S. R. Taylor, Phys. Rev. D 106, 103013 (2022), 2203.03651

    Pith/arXiv arXiv 2022

  51. [55]

    Kıro˘ glu, K

    F. Kıro˘ glu, K. Kremer, and F. A. Rasio, Astrophysical Journal994, L37 (2025), 2509.05415

    arXiv 2025

  52. [56]

    S. A. Popa and S. E. de Mink, Astrophysical Journal 995, L76 (2025), 2509.00154

    Pith/arXiv arXiv 2025

  53. [58]

    Wysocki, J

    D. Wysocki, J. Lange, and R. O’Shaughnessy, Phys. Rev. D100, 043012 (2019), 1805.06442

    Pith/arXiv arXiv 2019

  54. [59]

    Wysocki and R

    D. Wysocki and R. O’Shaughnessy,Popmodels o3a aps april 2021 presentation,https://gitlab.com/dwysock i/pop-models-o3a-aps-april-2021(2021)

    2021

  55. [60]

    Tiwari, Astrophysical Journal928, 155 (2022), 2111.13991

    V. Tiwari, Astrophysical Journal928, 155 (2022), 2111.13991

    Pith/arXiv arXiv 2022

  56. [61]

    Galaudage, arXiv e-prints arXiv:2605.25994 (2026), 2605.25994

    S. Galaudage, arXiv e-prints arXiv:2605.25994 (2026), 2605.25994. 16

    Pith/arXiv arXiv 2026

  57. [62]

    Qiu Cheng, A

    A. Qiu Cheng, A. Toubiana, S. Biscoveanu, and J. Gair, arXiv e-prints arXiv:2605.25980 (2026), 2605.25980

    Pith/arXiv arXiv 2026

  58. [63]

    Rinaldi and W

    S. Rinaldi and W. Del Pozzo, MNRAS509, 5454 (2022), 2109.05960

    Pith/arXiv arXiv 2022

  59. [64]

    Toubiana, D

    A. Toubiana, D. Gerosa, M. Mould, S. Rinaldi, M. Arca Sedda, T. Bruel, R. Buscicchio, J. Gair, L. Paiella, F. Santoliquido, et al., Phys. Rev. D113, 083006 (2026), 2507.13249

    Pith/arXiv arXiv 2026

  60. [65]

    Padhyegurjar and S

    S. Padhyegurjar and S. Mukherjee, arXiv e-prints arXiv:2606.00234 (2026), 2606.00234

    Pith/arXiv arXiv 2026

  61. [66]

    Hussain, M

    A. Hussain, M. Isi, and A. Zimmerman, arXiv e-prints arXiv:2605.24281 (2026), 2605.24281

    Pith/arXiv arXiv 2026

  62. [67]

    Pierra, S

    G. Pierra, S. Mastrogiovanni, and S. Perri` es, arXiv e- prints arXiv:2406.01679 (2024), 2406.01679

    Pith/arXiv arXiv 2024

  63. [68]

    A. Ray, S. Mukherjee, M. Zevin, and V. Kalogera, arXiv e-prints arXiv:2603.17987 (2026), 2603.17987

    Pith/arXiv arXiv 2026

  64. [69]

    Tiwari, arXiv e-prints arXiv:2606.18081 (2026), 2606.18081

    V. Tiwari, arXiv e-prints arXiv:2606.18081 (2026), 2606.18081

    Pith/arXiv arXiv 2026

  65. [70]

    Tiwari and S

    V. Tiwari and S. Fairhurst, Astrophysical Journal913, L19 (2021), 2011.04502

    Pith/arXiv arXiv 2021

  66. [71]

    Padhyegurjar and S

    S. Padhyegurjar and S. Mukherjee, arXiv e-prints arXiv:2606.02318 (2026), 2606.02318

    Pith/arXiv arXiv 2026

  67. [72]

    Afroz and S

    S. Afroz and S. Mukherjee, arXiv e-prints arXiv:2509.09123 (2025), 2509.09123

    Pith/arXiv arXiv 2025

  68. [73]

    Afroz and S

    S. Afroz and S. Mukherjee, Phys. Rev. D112, 023531 (2025), 2411.07304

    Pith/arXiv arXiv 2025

  69. [74]

    Y. B. Ginat, F. Antonini, E. Flanagan, and M. Gieles, arXiv e-prints arXiv:2604.07456 (2026), 2604.07456

    Pith/arXiv arXiv 2026

  70. [75]

    Qazalbash, M

    M. Qazalbash, M. Zeeshan, and R. O’Shaughnessy, GWKokab: A jax-based gravitational-wave population inference toolkit for parametric models(2024), URL https://github.com/kokabsc/gwkokab

    2024

  71. [76]

    Qazalbash, M

    M. Qazalbash, M. Zeeshan, and R. O’Shaughnessy, Phys. Rev. D113, 103003 (2026), 2509.13638

    Pith/arXiv arXiv 2026

  72. [77]

    Mandel, W

    I. Mandel, W. M. Farr, and J. R. Gair, MNRAS486, 1086 (2019), 1809.02063

    Pith/arXiv arXiv 2019

  73. [78]

    T. J. Loredo, inBayesian Inference and Maximum En- tropy Methods in Science and Engineering: 24th In- ternational Workshop on Bayesian Inference and Max- imum Entropy Methods in Science and Engineering, edited by R. Fischer, R. Preuss, and U. V. Toussaint (2004), vol. 735 ofAmerican Institute of Physics Con- ference Series, pp. 195–206, astro-ph/0409387

    Pith/arXiv arXiv 2004

  74. [79]

    W. M. Farr, J. R. Gair, I. Mandel, and C. Cutler, Phys. Rev. D91, 023005 (2015), URLhttps://link.aps.o rg/doi/10.1103/PhysRevD.91.023005

    doi:10.1103/physrevd.91.023005 2015

  75. [80]

    T. A. Callister, arXiv e-prints arXiv:2104.09508 (2021), 2104.09508

    Pith/arXiv arXiv 2021

  76. [81]

    Heinzel, M

    J. Heinzel, M. Mould, S. ´Alvarez-L´ opez, and S. Vitale, Phys. Rev. D111, 063043 (2025), 2406.16813

    Pith/arXiv arXiv 2025

  77. [83]

    The LIGO Scientific Collaboration, the Virgo Collabo- ration, and the KAGRA Collaboration, arXiv e-prints arXiv:2605.27226 (2026), 2605.27226

    Pith/arXiv arXiv 2026

  78. [84]

    Abbott, T

    The LIGO Scientific Collaboration, the Virgo Collabo- ration, R. Abbott, T. D. Abbott, S. Abraham, F. Acer- nese, K. Ackley, A. Adams, C. Adams, R. X. Adhikari, et al., Physical Review X11, 021053 (2021), 2010.14527

    Pith/arXiv arXiv 2021

  79. [85]

    Abbott, T

    R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, N. Adhikari, R. X. Adhikari, V. B. Adya, C. Affeldt, D. Agarwal, et al., Phys. Rev. D109, 022001 (2024)

    2024

  80. [86]

    The LIGO Scientific Collaboration, the Virgo Collabo- ration, and the KAGRA Collaboration, arXiv e-prints arXiv:2605.27225 (2026), 2605.27225

    Pith/arXiv arXiv 2026

Showing first 80 references.