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The Impact of High-Mass Clusters on the Binary Black Hole Merger Rate over Cosmic Time

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Aidan Mai, Kyle Kremer, Fulya K{i}roğlu

Astrophysics of Galaxies · Submitted 14 Oct 2026 · 103 references

Pith's verdict: A solid extension of the CMC merger rate framework with new N=10^7 models, but the central linear relation between hierarchical fraction and Ms relies on scaling relations that are asserted rather than validated across metallicity.

T0 · AI review

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CitationAidan Mai, Fulya K{\i}ro\u{g}lu, Kyle Kremer. "The Impact of High-Mass Clusters on the Binary Black Hole Merger Rate over Cosmic Time." arXiv:2610.20698 (2026). https://pith.science/paper/NVPPZHZH
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The paper

The paper claims

The local binary black hole merger rate is dominated by low-mass mergers from young, high-metallicity clusters, not from massive old globulars.

The reading

Binary black hole mergers from dense star clusters have been thought to come mainly from old, metal-poor globular clusters, but a new cosmic integration of cluster formation and dynamics finds the opposite. The paper shows that the local merger rate is actually dominated by low-mass (primary mass $<20\,M_\odot$) mergers that originate from recently formed, high-metallicity clusters. While massive clusters are the primary sites for hierarchical (second-generation and higher) mergers, their overall contribution is heavily suppressed by the exponential cutoff of the cluster initial mass function. The fraction of hierarchical mergers is thus highly sensitive to the Schechter mass that sets the high-mass cutoff, offering a way to constrain cluster formation using gravitational-wave data.

What carries the argument

The central machinery is the analytic cluster formation model that convolves a Schechter initial cluster mass function $p(M)\propto M^{-2}\exp(-M/M_S)$ (with variable cutoff $M_S$) with delay-time distributions from the CMC N-body simulations. The model also includes a lognormal metallicity spread (with redshift-dependent mean from Madau & Fragos 2017) and a Gaussian virial radius distribution. Key scaling relations $N_{\rm BBH}\propto M_{\rm cl}^{1.4}$ for first-generation and $\propto M_{\rm cl}^{2.0}$ for hierarchical mergers bridge between the discrete model grid and continuous parameter space.

Extended reading notes

Core claim

The authors demonstrate that, when the full observed ranges of star cluster mass, metallicity, and virial radius are folded into a cosmological calculation using the Cluster Monte Carlo simulation suite, the local volumetric binary black hole merger rate of 8.4–22.1 Gpc$^{-3}$ yr$^{-1}$ is dominated by primary masses below $20\,M_\odot$. These mergers arise preferentially from clusters that formed recently (at low redshift) with near-solar metallicity. High-mass clusters ($M_{\rm cl}>10^{6.5}\,M_\odot$) are efficient at producing hierarchical mergers, but they are exponentially rare in the assumed Schechter initial mass function [$p(M)\propto M^{-2}\exp(-M/M_S)$], so they contribute little to the total rate. The resulting hierarchical merger fraction at $z=0$ ranges from 18% (for $M_S=10^{6.26}\,M_\odot$) to 31% (for $M_S\to\infty$). Because this fraction is a linear function of $\log M_S$, gravitational-wave measurements of the hierarchical fraction can independently probe the high-mass end of the cluster mass function.

Load-bearing premise

The paper assumes that the initial cluster mass function is a Schechter-like power law with an exponential cutoff at high mass ($p(M)\propto M^{-2}\exp(-M/M_S)$), but acknowledges that the shape of the high-mass tail is uncertain and some observations are consistent with an untruncated power law.

Editorial extensions

If this is right

  • The local dynamical BBH merger rate is dominated by mergers with primary masses $<20\,M_\odot$, so LVK observations of low-mass BBH events may be predominantly of dynamical origin.
  • The hierarchical merger fraction in the local universe can vary between 18% and 31% depending on the Schechter mass, providing a testable prediction for gravitational-wave data.
  • A Schechter mass $M_S=10^7\,M_\odot$ best matches the present-day mass function of Virgo globular clusters and also yields a hierarchical fraction (27%) that is intermediate between the canonical value and the untruncated limit.
  • The cluster formation model reproduces the observed present-day globular cluster number density ($\sim 2.3\,\mathrm{Mpc}^{-3}$) without fine-tuning, for all Schechter mass values considered.
  • High-mass clusters produce hierarchical mergers with masses up to $\sim 80\,M_\odot$ (low metallicity) and $\sim 40\,M_\odot$ (high metallicity), but their contribution to the overall rate is exponentially suppressed; thus most hierarchical mergers in the local universe may come from the relatively rare high-mass tail.

Reading between the lines

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

  • If future gravitational-wave observations find a higher hierarchical fraction than current estimates, it would imply a larger Schechter mass, pushing modelers to reconsider the universality of the exponential cutoff in the cluster mass function.
  • The dominance of young, metal-rich clusters in the local merger rate suggests that star cluster formation at low redshift ($z<1$) is disproportionately important for gravitational-wave source production, even though most clusters formed at higher redshift.
  • The paper's scaling relations ($\beta=1.4$ for first-generation, $\beta=2.0$ for hierarchical) could be tested by future high-resolution N-body simulations of clusters with explicitly varied initial conditions.
Original abstract

We present a cosmological calculation of the binary black hole merger rate from dense star clusters, using a $\texttt{Cluster Monte Carlo}$ ($\texttt{CMC}$) model grid spanning the full observed parameter space of cluster mass, metallicity, and virial radius. For the first time, we incorporate $N=10^{7}$-particle $\texttt{CMC}$ models into this calculation, including a new solar-metallicity model, extending the $\texttt{CMC Cluster Catalog}$ to the most massive clusters observed in the local Universe. Using an analytic cluster formation model which convolves the cluster formation rate with delay-time distributions from our model suite, we find that the local cluster BBH merger rate is dominated by low-mass ($M_1 \lesssim 20\,M_\odot$) mergers originating from recently formed, high-metallicity clusters. While high-mass clusters are the primary sites of hierarchical (second-generation and higher) mergers, their contribution to the overall merger rate is suppressed by the exponential cutoff of the cluster initial mass function. The hierarchical merger rate is therefore highly sensitive to the exact value of the Schechter mass, $M_S$, suggesting that gravitational wave constraints on the hierarchical merger fraction can independently constrain the high-mass end of the cluster initial mass function. Our cluster formation model reproduces the observed present-day GC number density without fine tuning, and we find $M_S = 10^7\,M_\odot$ provides the best match to the observed Virgo GC mass distribution. For this value, $27\%$ of local dynamical mergers are hierarchical, compared to $18\%$ for the canonical value of $M_S = 10^{6.26}\,M_\odot$. Our merger weights, which we make publicly available, can be used for direct comparison with LVK distributions and other formation models.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-16: 1 request, 6,719 input tokens.

What the abstract itself states, as probabilities: claims first 0.86 · states limitation 0.48 · human participants 0.01 · controlled comparison 0.28 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.88 · mentions code or data availability 0.95.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We present a cosmological calculation of the binary black hole merger rate from dense star clusters, using a $\texttt{Cluster Monte Carlo}$ ($\texttt{CMC}$) model grid spanning the full observed parameter space of cluster mass, metallicity, and virial radius. For the first time, we incorporate $N=10^{7}$-particle $\texttt{CMC}$ models into this calculation, including a new solar-metallicity model, extending the $\texttt{CMC Cluster Catalog}$ to the most massive clusters observed in the local Universe. Using an analytic cluster formation model which convolves the cluster formation rate with delay-time distributions from our model suite, we find that the local cluster BBH merger rate is dominated by low-mass ($M_1 \lesssim 20\,M_\odot$) mergers originating from recently formed, high-metallicity clusters. While high-mass clusters are the primary sites of hierarchical (second-generation and higher) mergers, their contribution to the overall merger rate is suppressed by the exponential cutoff of the cluster initial mass function. The hierarchical merger rate is therefore highly sensitive to the exact value of the Schechter mass, $M_S$, suggesting that gravitational wave constraints on the hierarchical merger fraction can independently constrain the high-mass end of the cluster initial mass function. Our cluster formation model reproduces the observed present-day GC number density without fine tuning, and we find $M_S = 10^7\,M_\odot$ provides the best match to the observed Virgo GC mass distribution. For this value, $27\%$ of local dynamical mergers are hierarchical, compared to $18\%$ for the canonical value of $M_S = 10^{6.26}\,M_\odot$. Our merger weights, which we make publicly available, can be used for direct comparison with LVK distributions and other formation models.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We present a cosmological calculation of the binary black hole merger rate from dense star clusters, using a $\texttt{Cluster Monte Carlo}$ ($\texttt{CMC}$) model grid spanning the full observed parameter space of cluster mass, metallicity, and virial radius. contribution 0.84 0.96 0.05 asserted 0.99
s1 For the first time, we incorporate $N=10^{7}$-particle $\texttt{CMC}$ models into this calculation, including a new solar-metallicity model, extending the $\texttt{CMC Cluster Catalog}$ to the most massive clusters observed in the local Universe. contribution 0.82 0.96 0.04 asserted 1.00
s2 Using an analytic cluster formation model which convolves the cluster formation rate with delay-time distributions from our model suite, we find that the local cluster BBH merger rate is dominated by low-mass ($M_1 \lesssim 20\,M_\odot$) mergers originating from recently formed, high-metallicity clusters. result 1.00 0.97 0.47 asserted 1.00
s3 While high-mass clusters are the primary sites of hierarchical (second-generation and higher) mergers, their contribution to the overall merger rate is suppressed by the exponential cutoff of the cluster initial mass function. result 0.91 0.85 0.20 asserted 1.00
s4 The hierarchical merger rate is therefore highly sensitive to the exact value of the Schechter mass, $M_S$, suggesting that gravitational wave constraints on the hierarchical merger fraction can independently constrain the high-mass end of the cluster initial mass function. interpretation 0.99 0.76 0.23 hedged possibility 0.84
s5 Our cluster formation model reproduces the observed present-day GC number density without fine tuning, and we find $M_S = 10^7\,M_\odot$ provides the best match to the observed Virgo GC mass distribution. result 0.98 0.96 0.12 asserted 1.00
s6 For this value, $27\%$ of local dynamical mergers are hierarchical, compared to $18\%$ for the canonical value of $M_S = 10^{6.26}\,M_\odot$. result 1.00 0.95 0.11 asserted 1.00
s7 Our merger weights, which we make publicly available, can be used for direct comparison with LVK distributions and other formation models. contribution 0.66 0.87 0.03 asserted 0.91

What kind of paper the classifier reports this is: simulation study 0.96 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Primary vs. secondary mass for all BBH mergers in colossus-blue and colossus-red, which are identical N = 107 models except for differing metallicity: Z/Z⊙ = 0.1 and 1.0, respectively. Each merger is colored by hierarchical generation. While both models are roughly similar in terms of total number of mergers and 2G+ fraction, the bottom-heavy high-metallicity BH mass function of colossus-red results in much lower component masses than colossus-blue, producing no pair-instability￾gap events even at third generation. We also show observed component masses for eight select LVK events…
Figure 2. Contributions of different cluster and merger types to the dynamical BBH merger rate over redshift, obtained using the cosmic integration process described in Section 3. Within our formation model, low-mass (M1 < 20 M⊙) mergers from young, high-metallicity clusters dominate the merger rate at low redshifts (z < 2). The contribution of high-mass clusters beyond the assumed Schechter mass, MS = 106.26 M⊙, is suppressed due to the exponential term in Equation 2. denoting the fraction of star formation mass used for massive star cluster formation, where p(M) is the Schechter function…
Figure 3. Cumulative present-day cluster mass functions (blue) for observed Virgo GCs, and for our analytic forma￾tion model with varying values of the Schechter mass. We find MS = 107 M⊙ best matches the Virgo sample. We also show the same mass functions weighted by total number of BBH mergers (orange). Roughly independent of Schechter mass, half of all dynamical BBH mergers come from the heav￾iest ∼ 15% of dense clusters, whose cutoff mass increases with the chosen Schechter mass. of 2.29 Mpc−3 for old globular clusters in the local Uni￾verse, consistent with empirical estimates which…
Figure 4. Top row: Contributions of different merger masses to the dynamical BBH merger rate over redshift (same as top right panel of
Figure 5. Fraction of 2G+ mergers in the total dynamical merger rate for different z, as a function of Schechter mass. The 2G+ fraction is approximately linear with log(MS/M⊙) until MS = 107 M⊙. At high z, clusters have not lived long enough to create a substantial 2G BH population, so the 2G+ merger fraction is low. We shade the region past log(MS/M⊙) > 7 to show the end of our model coverage in cluster mass. In dotted black, we show the linear fit for z = 0 given in Equation 14. 0 2 4 6 rh [pc] 0.0 0.2 0.4 0.6 0.8 1.0 Cumulative fraction Virgo GCs µr = 2 pc µr = 3 pc µr = 4 pc
Figure 6. Comparison between cumulative present-day rh functions for observed Virgo GCs, and for our analytic for￾mation model with varying values of µr. We find that a value of µr = 3 pc best matches the Virgo rh distribution
Figure 8. Fraction of 2G+ mergers in the total dynamical merger rate for different z, as a function of mean virial radius µr. For all redshifts, the 2G+ fraction decreases linearly with µr. In dotted black, we show the linear fit to z = 0 given in Equation 15. The 2G+ fractions for µr = 2, 3, and 4 pc are 17.6%, 14.9%, and 12.5%, respectively. In
Figure 10. Mass ratio versus primary mass contours (50%, 90%) for dynamical BBH mergers at z = 0.2, using the same contour colors as
Figure 11. dR/dm1 distributions at z = 0.2 for three variations of the Schechter mass, compared with the B-Spline model from GWTC-4.0. We apply smoothing to our rate curves to capture global trends independent of our particular BH mass function. The overall shape of the dR/dm1 distributions remain consistent when increasing MS, except for a stronger tertiary peak at 80 M⊙ due to increasing 2G+ fraction at high MS. Several features of our mass distributions are qualitatively comparable to the B-Spline model: a low-mass peak at 10 M⊙, a secondary peak at 40 M⊙ due to our implemented…

Showing 9 of the paper's 11 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2610.20698 by the authors.

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