Pith. sign in

REVIEW 3 major objections 4 minor 3 cited by

Inferring the pair-instability mass gap from gravitational wave data

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read Using 69 gravitational-wave events, the paper finds that the effective spin of merging black holes changes sharply at a primary mass of $46^{+7}_{-5}\,M_\odot$: below this mass the spins are narrow and mildly aligned, while above it they…

desk verdict A careful non-parametric re-analysis that confirms the authors' earlier parametric claim of a spin transition near 45 M_sun, but the headline mass depends on an untested factorization assumption and the evidence remains statistical, not bulletproof. read the letter →

arxiv 2506.09154 v2 pith:YXCDYYVM submitted 2025-06-10 astro-ph.HE astro-ph.SR

Fabio Antonini , Thomas Callister , Fani Dosopoulou , Isobel Romero-Shaw , Debatri Chattopadhyay This is my paper
classification astro-ph.HEastro-ph.SR
keywords gravitationalwavesbinaryblackholeseffectiveinspiralspinpair-instabilitymassgaphierarchicalBayesianinferenceGaussianprocesssecond-generationGWTC-3
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 asks whether gravitational-wave data can reveal the pair-instability mass gap, the mass range where stellar theory predicts black holes should not form, by tracking how the effective spin $\chi_{\rm eff}$ of merging black holes changes with primary mass. Analysing 69 binary black hole mergers from GWTC-3 with flexible, non-parametric Gaussian-process population models, it finds a transition at $46^{+7}_{-5}\,M_\odot$. Above that mass, the $\chi_{\rm eff}$ distribution broadens and is consistent with symmetry about zero, with median $-0.03^{+0.36}_{-0.59}$ at 90% credibility; below it, the fraction of second-generation black holes is constrained to be $\lesssim 10\%$. The authors argue this is model-independent support for a high-mass, high-spin subpopulation produced by previous mergers in dense star clusters, which would naturally fill the pair-instability gap. If correct, the result links gravitational-wave observations to stellar evolution, cluster dynamics, and a potential standard mass scale for cosmology.

What carries the argument

The analysis is carried by hierarchical Bayesian inference on 69 GWTC-3 mergers, with the population factorised so that the effective spin distribution $p(\chi_{\rm eff}|m_1)$ is the primary object of study. The central non-parametric tool is a Gaussian process prior placed on functions of $\log m_1$ (and on $\chi_{\rm eff}$ for the high-mass component), which lets the data choose the shape of the spin-mass relation rather than imposing a fixed functional form. The key physical template is the predicted uniform $\chi_{\rm eff}$ distribution for second-generation mergers, $\mathrm{CDF}(\leq \chi_{\rm eff}) \simeq 0.5 + \chi_{\rm eff}$ with $|\chi_{\rm eff}| < 0.47$, and the transition mass $\tilde m$ separates a low-mass Gaussian spin component from the high-mass component whose inferred distribution is compared against this template.

What would settle it

Take a future catalogue with a few hundred high-mass mergers and measure the median $\chi_{\rm eff}$ for primaries above $50\,M_\odot$ together with the lower bound $\chi_{\rm eff,min}$. If $\mathrm{CDF}(\chi_{\rm eff}=0)$ is found to be significantly above 0.5, or if $\chi_{\rm eff,min}$ is constrained to be positive at very high credibility, the symmetric second-generation-merger interpretation would be falsified; similarly, if the transition mass moves by more than its 90% uncertainty when the factorisation is relaxed to allow dependence on $m_2$ and $z$, the central claim fails.

Watch

Extended reading notes

Core claim

The central discovery is a mass-dependent transition in the effective spin distribution of binary black holes. At primary masses below roughly $46\,M_\odot$, the $\chi_{\rm eff}$ distribution is narrow, centred at a small positive value (mean $\mu_\chi = 0.05^{+0.03}_{-0.03}$), and the contribution from an isotropically spinning population is below about 15%, with the fraction of second-generation black holes constrained to be $\lesssim 10\%$. Above that transition, the distribution broadens ($\log \sigma > -1$ at 90% credibility), and the median effective spin is $-0.03^{+0.36}_{-0.59}$, consistent with symmetry about zero. The paper interprets this as evidence that the pair-instability mass gap is populated not by ordinary stellar collapse but by second-generation black holes formed in earlier mergers, whose spins are expected to be isotropically oriented and bounded at $|\chi_{\rm eff}| \simeq 0.47$. The inferred upper edge of the high-mass spin distribution is $\chi_{\rm eff,max} = 0.57^{+0.21}_{-0.19}$, while the lower edge remains weakly constrained, so asymmetric distributions skewed toward positive spins are not excluded.

Load-bearing premise

The load-bearing assumption is that, once the merger rate is factorised as in Eq. (2), the effective spin distribution depends on primary mass alone and not on the companion mass or the redshift of the merger; if spin-mass correlations vary with redshift or companion mass, the inferred transition at $46\,M_\odot$ and the high-mass spin distribution could be biased.

Editorial extensions

If this is right

  • The inferred spin transition at roughly $46\,M_\odot$ provides model-independent evidence that the pair-instability mass gap starts near the lower end of its predicted range and is being repopulated by merger remnants.
  • Below the transition, at most about 10% of mergers can involve second-generation black holes, placing a quantitative limit on the dynamical contribution to the low-mass merger rate.
  • The high-mass $\chi_{\rm eff}$ distribution being consistent with symmetry about zero supports dense cluster origins over strongly aligned channels such as AGN disks or chemically homogeneous evolution as the dominant source of the highest-mass mergers.
  • Because the transition mass is stable across parametric and non-parametric models, it could serve as a physically motivated, redshift-invariant mass scale for spectral siren cosmology, provided metallicity evolution does not shift the gap with redshift.
  • Upcoming gravitational-wave catalogues will sharpen the constraints on the lower edge of the high-mass spin distribution and determine whether the apparent positive skew in the lower tail is physical or a selection effect.

Reading between the lines

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

  • If the transition mass marks the onset of second-generation mergers, the mass-ratio distribution above roughly $45\,M_\odot$ should skew toward more unequal binaries; the paper's exploratory analysis of a mass-dependent mass-ratio slope is inconclusive, making this a targeted prediction for future data.
  • The factorisation $p(\chi_{\rm eff}|m_1)$ assumes the effective spin depends on primary mass alone; if spin-alignment efficiency changes with redshift or companion mass, the inferred transition mass could shift, and a future analysis allowing $p(\chi_{\rm eff}|m_1,m_2,z)$ would test this.
  • Because stellar evolution uncertainties smear the theoretical edges of the pair-instability gap by tens of solar masses, part of the observed broadening above $46\,M_\odot$ could reflect gap smearing rather than repopulation; discriminating the two requires joint modelling of the mass and spin distributions with stellar-population priors.
  • The symmetry test based on $\mathrm{CDF}(\chi_{\rm eff}=0)=0.5$ and $\mathrm{CDF}(-\chi_{\rm eff})=1-\mathrm{CDF}(\chi_{\rm eff})$ is prior-sensitive in the tails of the distribution, so a larger sample of high-mass events is needed to confirm whether the distribution is truly symmetric or mildly positive-skewed.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper performs hierarchical Bayesian inference on 69 binary black hole (BBH) mergers from GWTC-3, modeling the conditional distribution p(chi_eff|m1) with Gaussian-process (GP) and parametric models while accounting for selection effects using LVK injections. The headline result is a transition at m_tilde = 46(+7,-5) Msun above which the chi_eff distribution broadens and becomes consistent with being symmetric about zero, with a median of -0.03(+0.36,-0.59) at 90% credibility. The authors interpret the high-mass population as consistent with second-generation BHs formed in dense clusters that repopulate the pair-instability gap, and they constrain the fraction of such systems below the transition to <~10%. The paper also compares the inferred chi_eff distribution with predictions from cBHBd cluster simulations.

Significance. If the result is robust, it would provide one of the first data-driven, non-parametric indications of a mass-dependent spin transition in BBHs and support the existence of a high-mass, high-spin population consistent with hierarchical merger remnants. The analysis uses standard hierarchical likelihoods, public GWTC-3 posteriors and injections, tracks effective sample sizes, and checks robustness across several model classes, which are genuine strengths. The consistency of the inferred transition mass across models and the explicit comparison to cluster simulations are also valuable. The main limitations are that the headline 'transition' is not formally tested against a no-transition model, and the factorization in Eq. (2) is an unverified assumption that could in principle produce an apparent mass-dependent spin distribution. The statistical evidence is suggestive rather than decisive, and the symmetry claim in the abstract is weaker than it may first appear.

major comments (3)
  1. [Sec. II, Eq. (2)] The factorized rate model R(m1,m2,chi_eff;z) = ... p(m2|m1) p(chi_eff|m1) assumes p(chi_eff|m1,m2,z) = p(chi_eff|m1). The central inference is the mass dependence of this conditional distribution. Because high-mass BBHs are detected at higher redshifts on average, a redshift-dependent spin distribution (e.g., metallicity-driven spin evolution) or a spin-mass-ratio correlation could masquerade as a transition in m1 under this assumption. The authors do not test the factorization, e.g., by adding z- or q-dependent spin terms to Eq. (2) or by recovering an injected population that violates it. Without such a test, the inferred m_tilde = 46(+7,-5) Msun is the location of a transition within a model that already imposes this conditional independence, and the headline claim is only as strong as the factorization. I request an injection-recovery experiment or a sensitivity analysis that relaxes the factorization.
  2. [Secs. IIIA and IIIC] The paper states that the data 'strongly support' a mass-spin correlation and that the rise in zeta(m1) is 'statistically significant,' but no model comparison against a mass-independent p(chi_eff) is reported. The separation between the 95% upper bound of zeta below ~40 Msun and the 90% lower bound above it is a useful diagnostic, but it does not by itself quantify the evidence for a transition; the claim relies on posterior intervals that also depend on prior choices. A Bayes factor, posterior predictive check, or a nested no-transition model would make the central claim falsifiable. At a minimum, the language should be calibrated to the reported bounds (e.g., zeta > 0.1 at 90% credibility above 40 Msun).
  3. [Sec. IIIC2, Fig. 5] The symmetry claim in the abstract is weaker than it may appear. In Fig. 5, the lower tail of the inferred high-mass chi_eff distribution differs noticeably between Model 1 and Model 2, reflecting prior sensitivity in the region of negative chi_eff. The posterior median of -0.03 with 90% interval [-0.59, +0.36] is consistent with a symmetric distribution, but it also admits substantially asymmetric, positive-skewed distributions, as the paper itself acknowledges. The abstract's phrase 'becomes consistent with being symmetric around zero' should be framed as 'is not inconsistent with symmetry,' and the paper should not present symmetry as a positive detection.
minor comments (4)
  1. [Data and code availability] The statement that the codes and data products are 'available under reasonable request' is not ideal for a methods-driven paper; consider depositing the hierarchical inference code and derived data products in a public repository.
  2. [Sec. V] There is a typo: 'a part that the data remain consistent' should be 'apart from the fact that the data remain consistent' (or similar).
  3. [Fig. 1 caption] The caption uses 'purple line' twice: once for the non-parametric Model 1/2 pair and once for the beta_q GP model. Please clarify the color legend.
  4. [Abstract and Table S1] The phrase 'model-independent support' overstates the case; the GP models still assume Eq. (2) and specific kernel/prior choices. In Table S1, the prior 'ln l_betaq N(-0, 5, 1)' contains a typo and should presumably be N(-0.5, 1).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the claimed high-mass broadening of chi_eff is recovered by flexible Gaussian-process models that do not impose the uniform-distribution prior from Eq. (1).

full rationale

The paper's central claim, that the chi_eff distribution broadens above roughly 45 Msun, is derived from several models. The only input taken from the authors' own prior work is the uniform chi_eff prediction with w=0.47 (Eq. 1 from Ref. [57]), used in the mixture model Eq. (8). That input is not the source of the central detection: Sec. IIIB infers a mass-dependent Gaussian mean and variance, and Sec. IIIC2 uses a non-parametric GP for the high-mass chi_eff distribution (Eq. 13), neither of which assumes Eq. (1). The same transition mass around 45 Msun is recovered across all models, so the broadening is an empirical feature of the data rather than a consequence of the assumed uniform component. The interpretation that the broad high-mass population consists of second-generation BHs is presented as a consistency check and is explicitly cross-checked against cBHBd cluster models (Fig. 7), with the paper acknowledging alternative channels such as AGN disks and chemically homogeneous evolution. The factorization in Eq. (2) is a simplifying assumption that could bias the inference if spins correlate with mass ratio or redshift, but this is a model-misspecification risk, not a circular reduction: no fitted parameter is renamed as a prediction, and no equation is defined in terms of the result it is used to derive. The paper also directly tests the symmetry hypothesis it had previously imposed, stating that it 'seek[s] to more carefully test this hypothesis' rather than assuming it. There is no uniqueness theorem imported from the authors and no known result merely renamed. The self-citations to Ref. [57] and cBHBd are load-bearing only as physical predictions, and they are supported by independent flexible-model analyses and external benchmarks (GWTC-3 injections), so they do not constitute circularity.

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

The central inference depends on standard Bayesian machinery, a domain assumption about the rate factorization, and the LVK injection-based selection function. No new particles, forces, or physical entities are introduced. The paper's own Eq. (1) from Ref. [57] is used as a comparison prediction, not as a free parameter.

free parameters (4)
  • t_m (transition mass) = 46+7-5 Msun
    Free parameter in parametric models (Eq. 12), and derived from GP models; the central scale of the claimed transition.
  • GP kernel amplitude and length scale for chi_eff = Posterior distributions (Table S1)
    Hyperparameters governing smoothness and amplitude of the non-parametric spin distribution; results are sensitive to them in the negative tail.
  • Mixture fraction GP parameters a_zeta, l_zeta = Posterior distributions
    Control the mass-dependent fraction of the isotropic spin component in Eq. (9).
  • chi_eff,max and chi_eff,min (uniform bounds) = 0.57+0.21-0.19 and -0.53+0.52-0.42
    Inferred bounds of the high-mass uniform distribution in Eq. (12); the lower bound is weakly constrained.
assumptions (3)
  • domain assumption The selection function from LVK injections accurately represents detection efficiency.
    Used in Eq. (S3) to compute the detection efficiency xi(Lambda); if injection campaigns are incomplete or incorrect, the population inference is biased.
  • domain assumption The rate factorization in Eq. (2) holds: p(chi_eff|m1) is independent of m2 and z given m1.
    This is the structural assumption of the hierarchical model, stated in Section II. Redshift evolution of spin or mass-ratio dependence could bias the inferred p(chi_eff|m1).
  • ad hoc to paper Gaussian process priors with chosen kernels and length scales provide adequate flexibility.
    The GP priors are chosen in Appendix D; the paper shows some prior sensitivity in the negative chi_eff tail. The choice of kernel and length scale bounds is not uniquely determined by theory.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Inferring the pair-instability mass gap from gravitational wave data." pith.science (2026). https://pith.science/paper/YXCDYYVM

@misc{pith2026250609154,
  author       = {Pith},
  title        = {Pith review of: Inferring the pair-instability mass gap from gravitational wave data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YXCDYYVM}},
  note         = {Machine review of arXiv:2506.09154}
}
abstract

We use hierarchical Bayesian inference with non-parametric Gaussian process models to investigate the effective inspiral spin parameter, $\chi_{\rm eff}$, as a function of primary black hole mass in the third gravitational-wave transient catalog (GWTC-3). Our analysis reveals a transition in the population at a primary mass of $46^{+7}_{-5}\,M_\odot$. Beyond this mass, the $\chi_{\rm eff}$ distribution broadens, becomes consistent with being symmetric around zero, and has a median of $-0.03^{+0.36}_{-0.59}$ (90\% credibility). These results are consistent with the presence of a pair-instability mass gap that is repopulated by black holes that are the remnant of a previous merger, formed in dense star clusters. However, asymmetric distributions skewed toward positive $\chi_{\rm eff}$ are not excluded by current data. Below the inferred transition mass, we constrain the fraction of second-generation black holes to be $\lesssim 10\%$. These results provide model-independent support for a high-mass and high-spin population of black holes in the data, consistent with earlier work using parametric models. Imminent gravitational-wave data releases will be essential to sharpen constraints on spin symmetry and clarify the origin of the black holes.

Figures

Figures reproduced from arXiv: 2506.09154 by the authors.

Figure 1
Figure 1. FIG. 1: Posteriors of the transitional mass [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. It exhibits a clear trend with primary BH mass. At m1 ≲ 5 M⊙, ζ(m1) is unconstrained, spanning the full range from 0 to 1. This indicates that the data are not informative in this regime: this is expected, given the low number of detections and the difficulty in dis￾tinguishing spin contributions at small masses. Between m1 ≈ 10 M⊙ and 40 M⊙, the 95% credible interval for ζ(m1) remains consistent with 0 and constrai… view at source ↗
Figure 3
Figure 3. FIG. 3: The lower and upper bounds of the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The distribution of [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Posterior distributions of the [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Comparison between the inferred [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

Discussion (0). Sign in to comment.

Forward citations

Cited by 3 Pith papers

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

  1. Signatures of a subpopulation of hierarchical mergers in the GWTC-4 gravitational-wave dataset

    gr-qc 2026-01 unverdicted novelty 6.0 of 10

    Using a joint effective-spin and precession-spin model on 155 gravitational-wave events, the authors infer that the hierarchical (second-generation) merger fraction rises sharply above ~46 M_sun and peaks again near 1...

  2. Spectral siren cosmology from gravitational-wave observations in GWTC-4.0

    astro-ph.CO 2025-09 conditional novelty 5.0 of 10

    Spectral siren analysis of 152 GWTC-4.0 black hole mergers with a Gaussian Process mass model, combined with GW170817, gives H0 = 69 (+7/-6) km/s/Mpc at 10% precision.

  3. Binary Black Hole Phase Space Discovers the Signature of Pair Instability Supernovae Mass Gap

    astro-ph.HE 2025-09 reject novelty 4.0 of 10

    Applying a phase-space overlap method to GWTC-4, the paper claims first-generation black holes are truncated near 45.5 solar masses, but the cutoff follows from the assumed exponential mass prior rather than from the data.

Reference graph

Works this paper leans on

105 extracted references · 7 canonical work pages · cited by 3 Pith papers

  1. [57]

    Pierra, S

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

    arXiv 2024

  2. [1]

    As in Ref

    Parametric: Independent minimum and maximum bounds We first proceed parametrically. As in Ref. [57], we adopt an effective spin model that transitions from a Gaussian to a uniform distribution below and above˜m, respectively. We extend this model, however, by regard- ing the upper ( χeff,max) and lower truncation bounds (χeff,min) of the uniform distribut...

  3. [2]

    (12) in which the spin distri- bution at high masses is described via a GP prior: p(χeff | m1) = ( N (χeff ; µ, σ) m1 < ˜m eΘ(χeff )/ R 1 −1 eΘ(χeff ) dχeff m1 ≥ ˜m

    Non-parametric: Gaussian process effective spin distributions As a further check on these conclusions, we consider a more flexible version of Eq. (12) in which the spin distri- bution at high masses is described via a GP prior: p(χeff | m1) = ( N (χeff ; µ, σ) m1 < ˜m eΘ(χeff )/ R 1 −1 eΘ(χeff ) dχeff m1 ≥ ˜m . (13) Here, the functionΘ(χeff ) is generated...

  4. [3]

    However, we observe no- ticeable differences in the lower5% credible bound out- side the range0 ≲ χeff ≲ 0.7, where Model 2 leads to sig- nificantly lower probability values

    and −0.02+0.34 −0.59 (Model 2). However, we observe no- ticeable differences in the lower5% credible bound out- side the range0 ≲ χeff ≲ 0.7, where Model 2 leads to sig- nificantly lower probability values. This indicates that the current gravitational-wave data provide only weak constraints on the shape of theχeff distribution in these regions. Consequen...

  5. [4]

    Abbott et al., Physical Review X 9, 031040 (2019), 1811.12907

    arXiv 2019

  6. [5]

    LIGO Scientific Collaboration and Virgo Collaboration, ApJL 882, L24 (2019), 1811.12940

    arXiv 2019

  7. [7]

    Abbott et al., Physical Review X13, 041039 (2023), 2111.03606

    arXiv 2023

  8. [8]

    Abbott et al., Physical Review X11, 021053 (2021), 2010.14527

    arXiv 2021

Show all 105 references
  1. [9]

    Spera, M

    M. Spera, M. Mapelli, and A. Bressan, Monthly Notices of the Royal Astronomical Society 451, 4086 (2015), ISSN 0035-8711, URL http://adsabs.harvard.edu/abs/2015MNRAS.451. 4086Shttp://mnras.oxfordjournals.org/lookup/ doi/10.1093/mnras/stv1161

    2015 doi

  2. [10]

    Belczynski, D

    K. Belczynski, D. E. Holz, T. Bulik, and R. O’Shaughnessy, Nature534, 512 (2016), ISSN 0028- 0836, URL http://arxiv.org/abs/1602.04531http: //dx.doi.org/10.1038/nature18322http://www. nature.com/doifinder/10.1038/nature18322

    2016 arXiv

  3. [11]

    Stevenson, F

    S. Stevenson, F. Ohme, and S. Fairhurst, The Astro- physical Journal810, 58 (2015), ISSN 1538-4357, URL http://adslabs.org/adsabs/abs/2015ApJ...810... 58S/

    2015

  4. [12]

    P.Marchant, N.Langer, P.Podsiadlowski, T.M.Tauris, and T. J. Moriya, A&A588 (2016)

    2016

  5. [13]

    Farrell, J

    E. Farrell, J. H. Groh, R. Hirschi, L. Murphy, E. Kaiser, S. Ekström, C. Georgy, and G. Meynet, MNRAS502, L40 (2021), 2009.06585

    2021 arXiv

  6. [14]

    Belczynski, R

    K. Belczynski, R. Hirschi, E. A. Kaiser, J. Liu, J. Casares, Y. Lu, R. O’Shaughnessy, A. Heger, S. Justham, and R. Soria, Astrophys. J. 890, 113 (2020), 1911.12357

    2020 arXiv

  7. [15]

    S. E. Woosley, The Astrophysical Jour- nal 836, 244 (2017), ISSN 1538-4357, URL 12 https://arxiv.org/pdf/1608.08939.pdfhttp: //arxiv.org/abs/1608.08939http://dx.doi.org/ 10.3847/1538-4357/836/2/244http://stacks. iop.org/0004-637X/836/i=2/a=244?key=crossref. 26431d83a9818abda1d...

    2017 arXiv

  8. [16]

    Spera and M

    M. Spera and M. Mapelli, MNRAS470, 4739 (2017), 1706.06109

    2017 arXiv

  9. [18]

    D. D. Hendriks, L. A. C. van Son, M. Renzo, R. G. Izzard, and R. Farmer, MNRAS 526, 4130 (2023), 2309.09339

    2023 arXiv

  10. [19]

    Farmer, M

    R. Farmer, M. Renzo, S. E. de Mink, P. Marchant, and S. Justham, Astrophys. J.887, 53 (2019), 1910.12874

    2019 arXiv

  11. [20]

    Leung, K

    S.-C. Leung, K. Nomoto, and S. Blinnikov, Astrophys. J. 887, 72 (2019), 1901.11136

    2019 arXiv

  12. [21]

    Abbott et al., ApJL913, L7 (2021), 2010.14533

    2021 arXiv

  13. [22]

    Edelman, Z

    B. Edelman, Z. Doctor, and B. Farr, ApJL913, L23 (2021), 2104.07783

    2021 arXiv

  14. [24]

    A. Ray, I. M. Hernandez, S. Mohite, J. Creighton, and S. Kapadia, Astrophys. J.957, 37 (2023), 2304.08046

    2023 arXiv

  15. [25]

    T. A. Callister and W. M. Farr, Physical Review X14, 021005 (2024), 2302.07289

    2024 arXiv

  16. [26]

    Li, Y.-Z

    Y.-J. Li, Y.-Z. Wang, S.-P. Tang, and Y.-Z. Fan, Phys. Rev. Lett. 133, 051401 (2024), 2303.02973

    2024 arXiv

  17. [27]

    E. R. J. Winch, J. S. Vink, E. R. Higgins, and G. N. Sabhahitf, MNRAS529, 2980 (2024), 2401.17327

    2024 arXiv

  18. [28]

    R. M. O’Leary, F. A. Rasio, J. M. Fregeau, N. Ivanova, and R. O’Shaughnessy, Astrophys. J.637, 937 (2006), astro-ph/0508224

    2006 arXiv

  19. [29]

    Antonini and F

    F. Antonini and F. A. Rasio, Astrophys. J.831, 187 (2016), 1606.04889

    2016 arXiv

  20. [30]

    C. L. Rodriguez, M. Morscher, B. Pattabiraman, S. Chatterjee, C.-J. Haster, and F. A. Rasio, Physical Review Letters 115, 051101 (2015), ISSN 0031-9007, URL http://adslabs.org/adsabs/abs/ 2015PhRvL.115e1101R/

    2015

  21. [31]

    Gerosa and E

    D. Gerosa and E. Berti, Phys. Rev. D 95, 124046 (2017), 1703.06223

    2017 arXiv

  22. [32]

    Y. Yang, I. Bartos, V. Gayathri, K. E. S. Ford, Z. Haiman, S. Klimenko, B. Kocsis, S. Márka, Z. Márka, B. McKernan, et al., Phys. Rev. Lett. 123, 181101 (2019), 1906.09281

    2019 arXiv

  23. [33]

    Antonini, M

    F. Antonini, M. Gieles, and A. Gualandris, MNRAS 486, 5008 (2019), 1811.03640

    2019 arXiv

  24. [34]

    Fragione and B

    G. Fragione and B. Kocsis, MNRAS486, 4781 (2019), 1903.03112

    2019 arXiv

  25. [35]

    Dall’Amico, M

    M. Dall’Amico, M. Mapelli, U. N. Di Carlo, Y. Bouf- fanais, S. Rastello, F. Santoliquido, A. Ballone, and M. Arca Sedda, MNRAS508, 3045 (2021), 2105.12757

    2021 arXiv

  26. [36]

    Arca Sedda, M

    M.Mapelli, M.Dall’Amico, Y.Bouffanais, N.Giacobbo, M. Arca Sedda, M. C. Artale, A. Ballone, U. N. Di Carlo, G. Iorio, F. Santoliquido, et al., MNRAS505, 339 (2021), 2103.05016

    2021 arXiv

  27. [37]

    Tagawa, Z

    H. Tagawa, Z. Haiman, I. Bartos, B. Kocsis, and K. Omukai, MNRAS507, 3362 (2021), 2104.09510

    2021 arXiv

  28. [38]

    Fishbach, D

    M. Fishbach, D. E. Holz, and B. Farr, ApJL840, L24 (2017), 1703.06869

    2017 arXiv

  29. [39]

    Baibhav, D

    V. Baibhav, D. Gerosa, E. Berti, K. W. K. Wong, T. Helfer, and M. Mould, Phys. Rev. D102, 043002 (2020), 2004.00650

    2020 arXiv

  30. [40]

    Ajith, M

    P. Ajith, M. Hannam, S. Husa, Y. Chen, B. Brüg- mann, N. Dorband, D. Müller, F. Ohme, D. Pollney, C.Reisswig, etal., Phys.Rev.Lett. 106, 241101(2011), 0909.2867

    2011 arXiv

  31. [41]

    Vitale, R

    S. Vitale, R. Lynch, J. Veitch, V. Raymond, and R. Sturani, Physical Review Letters112 (2014), ISSN 10797114

    2014

  32. [42]

    W. M. Farr, S. Stevenson, M. C. Miller, I. Mandel, B. Farr, and A. Vecchio, Nature548, 426 (2017), ISSN 0028-0836, 1476-4687, URL http://www.nature.com/ doifinder/10.1038/nature23453

    2017 doi

  33. [43]

    Golomb and C

    J. Golomb and C. Talbot, Phys. Rev. D108, 103009 (2023), 2210.12287

    2023 arXiv

  34. [44]

    Vitale, R

    S. Vitale, R. Lynch, R. Sturani, and P. Graff, Classical and Quantum Gravity34, 03LT01 (2017), 1503.04307

    2017 arXiv

  35. [45]

    C. L. Rodriguez, M. Zevin, C. Pankow, V. Kalogera, and F. A. Rasio, ApJL, In Press (2016), URL http: //arxiv.org/abs/1609.05916

    2016 arXiv

  36. [46]

    Schmidt, F

    P. Schmidt, F. Ohme, and M. Hannam, Phys. Rev. D 91, 024043 (2015), 1408.1810

    2015 arXiv

  37. [47]

    Zevin, S

    M. Zevin, S. S. Bavera, C. P. L. Berry, V. Kalogera, T. Fragos, P. Marchant, C. L. Rodriguez, F. Antonini, D. E. Holz, and C. Pankow, Astrophys. J. 910, 152 (2021), 2011.10057

    2021 arXiv

  38. [48]

    Gerosa and M

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

    2021 arXiv

  39. [49]

    T. A. Callister, C.-J. Haster, K. K. Y. Ng, S. Vitale, and W. M. Farr, ApJL922, L5 (2021), 2106.00521

    2021 arXiv

  40. [50]

    T. A. Callister, S. J. Miller, K. Chatziioannou, and W. M. Farr, ApJL937, L13 (2022), 2205.08574

    2022 arXiv

  41. [51]

    I. M. Romero-Shaw, P. D. Lasky, and E. Thrane, arXiv e-prints arXiv:2206.14695 (2022), 2206.14695

    2022 arXiv

  42. [52]

    C. Hoy, S. Fairhurst, M. Hannam, and V. Tiwari, As- trophys. J. 928, 75 (2022), 2110.13542

    2022 arXiv

  43. [53]

    C. Hoy, S. Fairhurst, and I. Mandel, Phys. Rev. D111, 023037 (2025), 2408.03410

    2025 arXiv

  44. [54]

    Kimball, C

    C. Kimball, C. Talbot, C. P. L. Berry, M. Carney, M. Zevin, E. Thrane, and V. Kalogera, Astrophys. J. 900, 177 (2020), 2005.00023

    2020 arXiv

  45. [55]

    Liu and D

    B. Liu and D. Lai, MNRAS 502, 2049 (2021), 2009.10068

    2021 arXiv

  46. [56]

    Franciolini and P

    G. Franciolini and P. Pani, Phys. Rev. D105, 123024 (2022), 2201.13098

    2022 arXiv

  47. [58]

    Heinzel, M

    J. Heinzel, M. Mould, S. Álvarez-López, and S. Vitale, Phys. Rev. D 111, 063043 (2025), 2406.16813

    2025 arXiv

  48. [59]

    Heinzel, M

    J. Heinzel, M. Mould, and S. Vitale, arXiv e-prints arXiv:2406.16844 (2024), 2406.16844

    2024 arXiv

  49. [60]

    F.Antonini, I.M.Romero-Shaw, andT.Callister, Phys. Rev. Lett. 134, 011401 (2025), 2406.19044

    2025 arXiv

  50. [61]

    Baibhav, E

    V. Baibhav, E. Berti, D. Gerosa, M. Mould, and K. W. K. Wong, Phys. Rev. D 104, 084002 (2021), 2105.12140

    2021 arXiv

  51. [62]

    Fragione, A

    G. Fragione, A. Loeb, and F. A. Rasio, ApJL895, L15 (2020), 2002.11278

    2020 arXiv

  52. [63]

    Tagawa, B

    H. Tagawa, B. Kocsis, Z. Haiman, I. Bartos, K. Omukai, and J. Samsing, Astrophys. J. 908, 194 (2021), 2012.00011

    2021 arXiv

  53. [65]

    The LIGO Scientific Collaboration, Virgo Collabora- tion, and KAGRA Collaboration,GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo Dur- ing the Second Part of the Third Observing Run — 13 O1+O2+O3 Search Sensitivity Estimates(2021), URL https://doi.org/10.5281/ze...

    2021 doi

  54. [66]

    Fishbach, D

    M. Fishbach, D. E. Holz, and W. M. Farr, The Astro- physical Journal Letters863, L41 (2018), URLhttps: //dx.doi.org/10.3847/2041-8213/aad800

    2018 doi

  55. [67]

    Callister, M

    T. Callister, M. Fishbach, D. E. Holz, and W. M. Farr, The Astrophysical Journal Letters 896, L32 (2020), URLhttps://dx.doi.org/10.3847/ 2041-8213/ab9743

    2020

  56. [68]

    Godfrey, B

    J. Godfrey, B. Edelman, and B. Farr, arXiv e-prints arXiv:2304.01288 (2023), 2304.01288

    2023 arXiv

  57. [69]

    I.MandelandS.E.DeMink, MNRAS 458, 2634(2016)

    2016

  58. [70]

    Bartos, B

    I. Bartos, B. Kocsis, Z. Haiman, and S. Márka, Astro- phys. J. 835, 165 (2017), 1602.03831

    2017 arXiv

  59. [71]

    Sadiq, T

    J. Sadiq, T. Dent, and A. Lorenzo-Medina, arXiv e- prints arXiv:2506.02250 (2025), 2506.02250

    2025

  60. [72]

    Biscoveanu, T

    S. Biscoveanu, T. A. Callister, C.-J. Haster, K. K. Y. Ng, S. Vitale, and W. M. Farr, ApJL932, L19 (2022), 2204.01578

    2022 arXiv

  61. [73]

    Heinzel, S

    J. Heinzel, S. Biscoveanu, and S. Vitale, Phys. Rev. D 109, 103006 (2024), 2312.00993

    2024 arXiv

  62. [74]

    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, 011048 (2023), 2111.03634

    2023 arXiv

  63. [75]

    Li, Y.-Z

    Y.-J. Li, Y.-Z. Wang, S.-P. Tang, T. Chen, and Y.-Z. Fan, arXiv e-prints arXiv:2501.09495 (2025), 2501.09495

    2025 arXiv

  64. [76]

    Hut, Astronomy and Astrophysics (1981), URL http://adsabs.harvard.edu/full/1981A&A....99

    P. Hut, Astronomy and Astrophysics (1981), URL http://adsabs.harvard.edu/full/1981A&A....99. .126H

    1981

  65. [77]

    F. R. N. Schneider, P. Podsiadlowski, and E. Laplace, ApJL 950, L9 (2023), 2305.02380

    2023 arXiv

  66. [78]

    Chatterjee, C

    S. Chatterjee, C. L. Rodriguez, and F. A. Rasio, Astro- phys. J. 834, 68 (2017), 1603.00884

    2017 arXiv

  67. [79]

    C. L. Rodriguez, P. Amaro-Seoane, S. Chatterjee, and F. A. Rasio, Phys. Rev. Lett. 120, 151101 (2018), 1712.04937

    2018 arXiv

  68. [80]

    Rastello, P

    S. Rastello, P. Amaro-Seoane, M. Arca-Sedda, R. Capuzzo-Dolcetta, G. Fragione, and I. Tosta e Melo, MNRAS 483, 1233 (2019), 1811.10628

    2019 arXiv

  69. [81]

    U. N. Di Carlo, M. Mapelli, M. Pasquato, S. Rastello, A. Ballone, M. Dall’Amico, N. Giacobbo, G. Iorio, M. Spera, S. Torniamenti, et al., MNRAS507, 5132 (2021), 2105.01085

    2021 arXiv

  70. [82]

    Banerjee, MNRAS503, 3371 (2021), 2011.07000

    S. Banerjee, MNRAS503, 3371 (2021), 2011.07000

    2021 arXiv

  71. [83]

    Antonini and M

    F. Antonini and M. Gieles, MNRAS492, 2936 (2020), 1906.11855

    2020 arXiv

  72. [84]

    Antonini, M

    F. Antonini, M. Gieles, F. Dosopoulou, and D. Chat- topadhyay, MNRAS522, 466 (2023), 2208.01081

    2023 arXiv

  73. [85]

    McKernan and K

    B. McKernan and K. E. S. Ford, MNRAS531, 3479 (2024), 2309.15213

    2024 arXiv

  74. [86]

    Mandel, MNRAS456, 578 (2016)

    I. Mandel, MNRAS456, 578 (2016)

    2016

  75. [87]

    Fuller and L

    J. Fuller and L. Ma, ApJL881, L1 (2019), 1907.03714

    2019 arXiv

  76. [88]

    Riley, I

    J. Riley, I. Mandel, P. Marchant, E. Butler, K. Nathaniel, C. Neijssel, S. Shortt, and A. Vigna- Gómez, MNRAS505, 663 (2021), 2010.00002

    2021 arXiv

  77. [89]

    Dall’Amico, M

    M. Dall’Amico, M. Mapelli, G. Iorio, G. Costa, S. Char- lot, E. Korb, C. Sgalletta, and M. Lecroq, A&A 695, A221 (2025), 2501.04778

    2025 arXiv

  78. [90]

    H. Tong, M. Fishbach, and E. Thrane, arXiv e-prints arXiv:2502.10780 (2025), 2502.10780

    2025 arXiv

  79. [91]

    Belczynski, A

    K. Belczynski, A. Heger, W. Gladysz, A. J. Ruiter, S. Woosley, G. Wiktorowicz, H. Y. Chen, T. Bulik, R. O’Shaughnessy, D. E. Holz, et al., A&A 594, A97 (2016), 1607.03116

    2016 arXiv

  80. [92]

    Farmer, M

    R. Farmer, M. Renzo, S. E. de Mink, M. Fishbach, and S. Justham, ApJL902, L36 (2020), 2006.06678

    2020 arXiv

  81. [93]

    Y. Shen, B. Guo, R. J. deBoer, E. Li, Z. Li, Y. Li, X. Tang, D. Pang, S. Adhikari, C. Basu, et al., Astro- phys. J. 945, 41 (2023)

    2023

  82. [94]

    Marchant and T

    P. Marchant and T. J. Moriya, A&A 640, L18 (2020), 2007.06220

    2020 arXiv

  83. [95]

    S. E. Woosley and A. Heger, ApJL912, L31 (2021), 2103.07933

    2021 arXiv

  84. [96]

    Marchant, M

    P. Marchant, M. Renzo, R. Farmer, K. M. W. Pappas, R. E. Taam, S. E. de Mink, and V. Kalogera, Astrophys. J. 882, 36 (2019), 1810.13412

    2019 arXiv

  85. [97]

    Greenfield, T

    P. Greenfield, T. Robitaille, E. Tollerud, T. Aldcroft, K. Barbary, P. Barrett, E. Bray, N. Crighton, A. Con- ley, S. Conseil, et al.,Astropy: Community Python li- brary for astronomy, Astrophysics Source Code Library, record ascl:1304.002 (2013)

    2013

  86. [98]

    Ashton, M

    G. Ashton, M. Hübner, P. D. Lasky, C. Talbot, K. Ack- ley, S. Biscoveanu, Q. Chu, A. Divakarla, P. J. Easter, B. Goncharov, et al., ApJS241, 27 (2019), 1811.02042

    2019 arXiv

  87. [99]

    Bradbury, R

    J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. Van- derPlas, S. Wanderman-Milne, et al.,JAX: composable transformations of Python+NumPy programs (2018), URL http://github.com/google/jax

    2018

  88. [100]

    C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gom- mers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, et al., Nature (London)585, 357 (2020), 2006.10256

    2020 arXiv

  89. [101]

    D. Phan, N. Pradhan, and M. Jankowiak, arXiv e-prints arXiv:1912.11554 (2019), 1912.11554

    2019 arXiv

  90. [102]

    Bingham, J

    E. Bingham, J. P. Chen, M. Jankowiak, F. Obermeyer, N. Pradhan, T. Karaletsos, R. Singh, P. A. Szerlip, P. Horsfall, and N. D. Goodman, J. Mach. Learn. Res. 20, 28:1 (2019), URL http://jmlr.org/papers/v20/ 18-403.html

    2019

  91. [103]

    Virtanen, R

    P. Virtanen, R. Gommers, T. E. Oliphant, M. Haber- land, T. Reddy, D. Cournapeau, E. Burovski, P. Peter- son, W. Weckesser, J. Bright, et al., Nature Methods 17, 261 (2020)

    2020

  92. [104]

    Tiwari and S

    V. Tiwari and S. Fairhurst, ApJL 913, L19 (2021), 2011.04502

    2021 arXiv

  93. [105]

    Galaudage and A

    S. Galaudage and A. Lamberts, A&A694, A186 (2025), 2407.17561

    2025 arXiv

  94. [106]

    Mandel, W

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

    2019 arXiv

  95. [107]

    Abbott et al., Phys. Rev. D 109, 022001 (2024), 2108.01045

    2024 arXiv

  96. [108]

    Essick and W

    R. Essick and W. Farr, arXiv e-prints arXiv:2204.00461 (2022), 2204.00461

    2022 arXiv

  97. [109]

    hierar- chical mergers,

    In the literature, such events are often termed "hierar- chical mergers," implying growth through multiple suc- cessive mergers. However, we avoid this terminology, as theoretical models predict that the vast majority of BHs in the mass gap originate from a single prior merger...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.