Intermediate-Mass Black Hole Formation from Hierarchical Mergers in Galactic Nuclei

Amanda Newton; Bao-Minh Hoang; Frederic A. Rasio; Fulya K{\i}ro\u{g}lu; Sanaea C. Rose

REVIEW 4 major objections 5 minor 3 cited by

In dense galactic nuclei, repeated black-hole mergers and star collisions can assemble ~500-solar-mass intermediate-mass black holes, provided the initial stellar black holes are massive enough.

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2026-08-03 04:41 UTC pith:ZEK3EDK3

load-bearing objection A transparent extension of the Rose et al. collision model that plausibly shows a sharp threshold for ~500 Msun IMBH formation, but the fixed-background cusp makes the headline number an upper limit that could easily drop by an order of magnitude. the 4 major comments →

arxiv 2602.04176 v3 pith:ZEK3EDK3 submitted 2026-02-04 astro-ph.GA

Intermediate-Mass Black Hole Formation from Hierarchical Mergers in Galactic Nuclei

Amanda Newton , Sanaea C. Rose , Fulya K{\i}ro\u{g}lu , Bao-Minh Hoang , Frederic A. Rasio This is my paper

classification astro-ph.GA

keywords intermediate-mass black holesnuclear star clustersgravitational-wave captureblack hole-star collisionshierarchical mergersextreme-mass-ratio inspiralsblack hole spinsgalactic center

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.

This paper argues that the inner 0.1 parsec of a nuclear star cluster around a 4-million-solar-mass supermassive black hole can act as a black-hole assembly line. Using a semianalytic model that combines gravitational-wave captures between single black holes with direct collisions between black holes and stars, it shows that if the cluster's black holes start with the mass distribution predicted by one set of stellar-evolution models (up to ~100 solar masses), sequential mergers can build intermediate-mass black holes of roughly 500 solar masses. If the initial black holes are instead drawn from a more conservative distribution topping out near 25 solar masses, no intermediate-mass black holes form. The paper also finds that successive black-hole-star collisions spin black holes down to chi < 0.2, and that most intermediate-mass black holes above 200 solar masses eventually sink into the supermassive black hole as extreme-mass-ratio inspirals. The result matters because it provides a dynamical route to the intermediate-mass black holes and high-mass gravitational-wave events seen in recent detections.

Core claim

The central claim is that growth through gravitational-wave capture between single black holes dominates mass growth in nuclear star clusters, and that this channel is powerful enough to produce ~500 solar-mass IMBHs when the initial stellar black hole population extends to ~100 solar masses (with or without primordial mergers). The growth timescale for a black hole of mass m scales roughly as 1/m^2, so heavier black holes grow faster and most of the mass gain comes from BH-BH mergers rather than from accreting stars during direct collisions. Dynamical friction then limits in-situ growth to a few hundred solar masses: once a black hole becomes much heavier than its neighbors it sinks toward

What carries the argument

The machinery is a semianalytic Monte Carlo model that follows 1000 black holes for 10 Gyr, assigning at each timestep the probability per unit time for two channels: gravitational-wave capture between two single black holes (using the capture cross-section from two-body GW radiation, with prompt merger, remnant mass and spin from numerical-relativity fits, and recoil kicks) and direct collisions with 1-solar-mass stars (with Bondi-Hoyle mass capture limited by radiative feedback, and spin changes from prograde and retrograde accretion). A fixed power-law density cusp for stars (alpha=1.25) and a Bahcall-Wolf-like black-hole cusp provide the background, while two-body relaxation is modeled a

Load-bearing premise

The calculation holds the number-density profiles of stars and black holes fixed for the full 10 Gyr; if a growing IMBH scatters lighter objects out of the cusp, the fuel for further growth disappears and the reported masses and rates are too high.

What would settle it

A concrete check: measure or simulate the central cusp of a Milky Way-like nuclear star cluster and ask whether the stellar black hole mass distribution tops out below ~25 solar masses. If it does, this model predicts zero in-situ IMBHs; detecting a >100 solar mass black hole in such a nucleus would falsify the lower-limit branch. Alternatively, a long-duration EMRI survey of galactic nuclei that finds no events with mass ratio >5e-5 at the predicted rate of ~4 per Gyr per galaxy would falsify the IMBH-formation channel.

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

If this is right

If the initial BH mass distribution is the upper-limit one, ~500 solar-mass IMBHs form, comprising up to 14.3% of the final BH population; the lower-limit distribution forms none.
Most IMBHs above 200 solar masses (70% in one upper-limit run, 55% in the primordial-merger run) sink to the center and merge with the SMBH as extreme-mass-ratio inspirals, at rates of a few per Gyr for mass ratios above 5e-5.
BH-BH gravitational-wave capture merger rates are ~1e-8 per year per Milky Way-like galaxy for the upper-limit mass distribution and a few times 1e-9 for the lower-limit distribution; including primordial binaries boosts second-generation merger rates by up to an order of magnitude.
Successive BH-star collisions systematically spin BHs down: after many collisions, spins settle below chi ~ 0.2, so dynamically grown IMBHs that grew mostly by star collisions should be low-spin.
Because the GW capture timescale scales as 1/m^2, the initial BH mass distribution—not the presence of primordial binaries—is the main driver of final BH masses and merger rates.

Where Pith is reading between the lines

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

If the fixed background cusp is depleted by a growing IMBH scattering lighter stars and BHs to wider orbits—an effect the paper explicitly leaves out—the reported IMBH masses and EMRI rates are likely overestimates; a self-consistent cusp model is the natural next test.
The appendix's steep-cusp run (alpha=1.75) produces runaway growth to ~1e7 solar masses, suggesting that the collision channel could be far stronger in clusters that maintain a dense stellar cusp; the paper treats this as an upper limit, but it hints that observed IMBHs below ~1000 solar masses may be natural in the densest nuclei.
If the model is right, future gravitational-wave observatories should see a population of heavy EMRIs (mass ratio ~1e-4 to 1e-3) preferentially at late times, and high-generation mergers with chi_eff up to ~0.5; absence of such events after a long observing campaign would challenge the IMBH-in-NSC channel.
The lower-limit result suggests a sharp threshold in initial BH mass: clusters whose stellar BHs end below ~25-30 solar masses should form no IMBHs in situ, which could be tested by combining stellar-evolution prescriptions with observations of nuclear star clusters of different metallicities.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

A transparent extension of the Rose et al. collision model that plausibly shows a sharp threshold for ~500 Msun IMBH formation, but the fixed-background cusp makes the headline number an upper limit that could easily drop by an order of magnitude.

read the letter

Colleague — this one is worth a look, but read Table 1 with the appendix in hand. The real news is that they add GW capture between single BHs to the Rose et al. (2022) collision model, vary the initial BH mass/spin distributions, and get a sharp threshold: upper-limit initial masses give ~500 Msun IMBHs, lower-limit gives none. The qualitative result is plausible and the parameter study is clean. They also show successive BH-star collisions spin BHs down to chi<0.2, and that primordial merger products mainly boost the 2G merger rate. That last point is a genuinely useful prediction for the LIGO/LISA era.

What the paper does well: the recipes are standard, the equations are transparent, and the authors are unusually honest about the limitations. Section 5.4 explicitly flags the fixed-background cusp, and the appendix's alpha=1.75 run is effectively a control experiment that shows the model becomes unphysical without cusp depletion. That self-awareness helps, but it doesn't fix the problem.

The soft spot is exactly where the stress-test points. The GW-capture timescale uses a constant n_BH, but every capture consumes BHs from the finite inner-cluster population — with 371-535 mergers per 1000 initial BHs, that is not a small depletion. The fixed stellar cusp is similarly load-bearing: the appendix's runaway to ~1e7 Msun shows how sensitive mass growth is to the density profile. So the headline ~500 Msun is an upper limit, not a robust prediction. Cusp depletion could easily lower it by an order of magnitude. The lack of error bars — single 1000-BH runs per initial condition — makes that worse. I would have liked an ensemble or at least a sensitivity run with a self-consistently depleted cusp.

The circularity burden is minor. They use their own Hoang et al. and Rose et al. papers for initial conditions, but the claims are scenario predictions, not restatements of inputs. The 15% primordial merger fraction is an assumption, but it's traced to an earlier calculation and varied indirectly.

Bottom line: this is a solid extension of an established semianalytic framework, and the qualitative threshold behavior is likely correct. The quantitative numbers should not be quoted without the caveat. Send it to a serious referee — it deserves the time. I'd cite it for the parameter study and for the 2G merger rate effect, but I'd label the IMBH mass as an upper limit.

Referee Report

4 major / 5 minor

Summary. The paper presents a semianalytic model for the 10 Gyr dynamical evolution of 1000 stellar-mass BHs in the inner 0.1 pc of a Milky-Way-like nuclear star cluster surrounding a 4e6 Msun SMBH. The model combines BH-star direct collisions (with Bondi-Hoyle accretion and spin evolution), BH-BH gravitational-wave capture and prompt merger (with NR-based remnant masses, spins, and recoil kicks), two-body relaxation/random walks, and GW inspiral into the SMBH. Four initial-condition sets are compared: lower- and upper-limit BH mass distributions, each with and without a 15% primordial binary-merger component, plus a collision-only simulation. The central claim is that ~500 Msun IMBHs form in the upper-limit runs (max masses 407 and 526 Msun, IMBH fractions 7.8% and 14.3%), while the lower-limit runs form none, and that most IMBHs >200 Msun eventually become EMRIs. Merger rates are reported as ~1e-8 per galaxy per year for the upper-limit distributions and ~1e-9 for the lower limit, with secondary-generation merger rates boosted by primordial binaries.

Significance. If the result holds, the paper identifies a plausible dynamical assembly channel for IMBHs in NSCs and makes concrete, falsifiable predictions for LVK sources: generation-dependent effective spins, a population of low-spin BHs from successive star collisions, and EMRI rates tied to IMBH formation. The model is transparent and uses standard NR fitting recipes; the paper also explicitly explores the sensitivity to the initial BH mass function and to the presence of primordial binary merger products. The inclusion of a collision-only simulation and the candid caveats in Section 5.4 and the Appendix are strengths. The headline claim is a scenario prediction rather than a restatement of inputs, so the paper is not circular.

major comments (4)

[Section 5.4 and Eq. (6)] The GW-capture timescale in Eq. (6) uses a fixed BH number density n_BH from the Aharon & Perets cusp, which is never updated as mergers consume BHs. The gold and blue runs record 371 and 535 GW captures among the 1000 tracked BHs; if these are representative, the local BH reservoir is depleted by roughly half over 10 Gyr, so t_GW is systematically underestimated at late times. This is load-bearing for the headline ~500 Msun IMBHs because the most massive objects require many sequential captures. Please either demonstrate that the untracked BH population is large enough to keep n_BH constant, or run a depletion-corrected variant and quantify the change in maximum mass and in the IMBH fractions in Table 1.
[Appendix and Section 5.4] The fixed stellar background is similarly load-bearing. The Appendix's alpha=1.75 collision-only run produces a ~1e7 Msun BH, which the authors correctly call unrealistic because the stellar supply is finite. But the same finite-supply issue operates at alpha=1.25, just less severely: Eq. (17) uses n_star from Eq. (2) with no depletion by the growing BHs. The paper should provide a consistency check that the total stellar mass accreted over 10 Gyr is small compared to the stellar mass available in the relevant annuli. Without this, the maximum masses in Table 1 should be explicitly labeled as upper limits rather than robust predictions.
[Table 1 and Section 5.5] All numerical results are based on a single 1000-BH realization per initial condition. The low-count runs (34 and 40 mergers) have Poisson errors of order 15-20%, and the maximum BH mass is a tail statistic with large variance. The 7.8% and 14.3% IMBH fractions and the ~500 Msun maximum mass therefore lack error bars. Please provide multiple realizations (or a bootstrap/resampling analysis) and report the scatter in maximum mass, IMBH fraction, and merger rates before these claims are presented as quantitative.
[Section 4.1, Eq. (6)] The text states 'We take m2 in η to be the average of the initial mass distribution.' After several generations, the interacting pair includes evolved, high-mass BHs and a population that has itself been modified by mergers and collisions. Using a fixed initial average mass for m2 can change the capture cross section and rate for the most massive BHs. Please justify this approximation quantitatively or use the current mass distribution when evaluating t_GW.

minor comments (5)

[Section 2] Typo: 'Our rational for the value of α' should be 'rationale'.
[Section 4.2.1] Typo: 'g.g., Bondi & Hoyle' should be 'e.g.'.
[Section 3 / Figure 1 caption] Figure 1's caption refers to 'Section 2.1' but the initial-condition distributions are described in Section 3.
[Section 2] The BH number density profile is taken from a published figure (Aharon & Perets 2016, their Figure 1) but not given as an analytic fit. For reproducibility, please provide a tabulated fit or an equation for n_BH(r) in the inner 0.1 pc.
[Section 5.4] The statement '70% (55%) of BHs over >200 Msun became EMRIs' is clear in context, but the abstract's phrasing 'most IMBHs ≳200 Msun eventually sink...' could be made more precise by quoting the percentages.

Circularity Check

0 steps flagged

No significant circularity: central IMBH and merger-rate results are emergent simulation outputs under stated assumptions, not restatements of inputs.

full rationale

The paper's central claims (IMBHs of ~500 Msun in upper-limit runs, none in lower-limit runs; EMRI rates; merger-generation statistics) are emergent outputs of a stochastic semianalytic simulation, not definitions or fitted parameters. The input mass/spin distributions, the 15% primordial binary fraction, and the density profiles are adopted as external assumptions from Hoang et al. (2018), Kremer et al. (2020), Aharon & Perets (2016), and Genzel et al. (2010). Although Hoang and Rose are coauthors of the present work, the simulation machinery does not depend on those citations to define the target result; the cited prior work provides initial conditions and standard rate equations, not a uniqueness theorem or ansatz that forces the conclusions. The gold run (upper limit without primordial mergers) starts below 100 Msun and still reaches 407 Msun, so the IMBH claim does not reduce to seeding the initial population with IMBHs. The fixed-background cusp approximation (Eq. 2 and the Aharon & Perets 2016 BH profile) is explicitly flagged in Section 5.4 and the Appendix as a limitation; it makes the model sensitive to an assumed background, and the Appendix's alpha=1.75 runaway illustrates that sensitivity, but this is a modeling assumption rather than a circular derivation. No prediction is statistically forced by a fit, no quantity is defined in terms of the quantity it is supposed to predict, and the self-citations are not load-bearing in a circularity sense. The result is therefore a scenario prediction with stated limitations, not a restatement of its inputs.

Axiom & Free-Parameter Ledger

5 free parameters · 7 axioms · 0 invented entities

The model uses standard dynamical formulas plus several astrophysical assumptions: a fixed background cluster, a 15% primordial-merger fraction, and simplified spin prescriptions. The free parameters are scenario choices (density slope, mass distribution bracket, sample size) rather than fits to data.

free parameters (5)

Stellar density slope α = 1.25 (fiducial), 1.75 (appendix)
Chosen for the NSC stellar cusp (Eq. 2); α=1.25 is motivated as conservative, α=1.75 gives an upper limit in the appendix.
Primordial binary merger fraction = 15%
Taken from Hoang et al. 2018; the paper randomly pairs and merges 15% of BHs in the lower-limit case to mimic primordial mergers.
Initial BH mass distribution = upper: up to ~90 Msun (single) / ~175 Msun (with mergers); lower: up to ~15-25 Msun
Two bracketing scenarios from prior literature; the entire IMBH formation result hinges on this choice.
Accretion efficiency η = 0.1
Assumed radiative efficiency at ISCO used in the mass-growth prescription (Section 4.2.1).
Simulation sample size = 1000 BHs
Choice limiting statistics; single realization per initial-condition set.

axioms (7)

standard math GW capture cross-section formulas (Eqs. 3-6)
Used to set the BH-BH merger timescale; taken from Quinlan & Shapiro 1987 and O'Leary et al. 2009.
standard math NR fitting formulas for remnant mass, spin, and recoil kicks (Eqs. 7-15)
Borrowed from Barausse et al. 2012 and Holley-Bockelmann et al. 2008; not derived in this paper.
domain assumption BH cusp density profile of Aharon & Perets 2016 is representative
Normalizes n_BH in the inner 0.1 pc; the paper notes divergence outside this region is irrelevant.
domain assumption GW-capture binaries merge promptly before dynamical perturbations
Section 4.1: 'we assume the merger is prompt, within hours', following O'Leary et al. 2009.
ad hoc to paper Star and BH background density profiles are fixed for 10 Gyr
Section 5.4 caveat: cusp depletion by a growing IMBH is neglected; the appendix shows this can cause runaway growth.
domain assumption Spin-down during stellar collisions follows fully anti-aligned accretion (Volonteri et al. 2013)
Section 4.2.2; authors state this gives a lower limit on final BH spins.
domain assumption 15% of primordial BH binaries merge in the NSC (Hoang et al. 2018)
Used to build initial conditions with primordial merger products; held fixed across lower-limit cases.

pith-pipeline@v1.3.0-alltime-deepseek · 21208 in / 13420 out tokens · 135181 ms · 2026-08-03T04:41:06.728518+00:00 · methodology

0 comments

read the original abstract

Dense stellar environments like nuclear star clusters (NSCs) can dynamically assemble gravitational wave (GW) sources. We consider a population of single stellar mass black holes (BHs) in the inner $0.1$~pc of a NSC surrounding a $4 \times 10^6$~M$_\odot$ supermassive black hole (SMBH). Using a semianalytic model, we account for direct collisions between BHs and stars and GW capture between BHs. We explore the effect of the initial BH mass and spin distributions on their final properties and the production of GW sources. Specifically, we consider upper and lower limits for the BH initial mass distribution, and we account for the possibility that a subset of our initial population are the merger products of primordial BH binaries. We find that $\sim 500$ M$_{\odot}$ intermediate mass black holes (IMBHs) can form for our upper limit mass distribution, while our lower limit mass distribution forms none. Most IMBHs $\gtrsim 200$~M$_\odot$ eventually sink towards the center of the cluster and merge with the SMBH. We also find successive BH-star collisions can produce low-spinning BHs with $\chi \lesssim 0.2$. Our results have implications for LIGO-Virgo-KAGRA sources. We find that the overall merger rate depends primarily on the initial BH mass distribution and is $\gtrsim 10^{-9}$~yr$^{-1}$ per Milky Way-like galaxy for our range of initial conditions. However, primordial binaries can change the number of GW mergers with second and higher generation progenitor BHs by an order of magnitude.

Figures

Figures reproduced from arXiv: 2602.04176 by Amanda Newton, Bao-Minh Hoang, Frederic A. Rasio, Fulya K{\i}ro\u{g}lu, Sanaea C. Rose.

**Figure 1.** Figure 1: We show the distributions for the four initial BH mass distributions described in Section 2.1. Left: The lower limit refers to the conservative case of all initially low mass stellar BHs from Kremer et al. (2020); Belczynski et al. (2016), as well as the lower limit where 30% of the binary BHs have primordial mergers. Right: The upper limit describes a case that includes primordial binaries and single BHs.… view at source ↗

**Figure 2.** Figure 2: We show the resulting spins and the change in mass of 1000 BHs after 1 billion years with the Hoang et al. (2018) Single BHs (gold) initial conditions, as described in Section 5.1. All BHs underwent at least two collisions with stars, with a maximum number of collisions per BH of 194. As shown in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Left: We plot the distributions of initial mass in gray and final mass in color for the four initial conditions described in Section 3, as well as IC gold (Hoang et al. (2018) Single BHs) with only stellar collisions with density α = 1.25. Right: We plot the distributions of initial spin in gray and final spin in color for the four initial conditions described in Section 3 [PITH_FULL_IMAGE:figures/full_fi… view at source ↗

**Figure 4.** Figure 4: Contribution of each dynamical channel to that mass growth for each BH with Mf > Mi. The x-axis shows the amount of mass accreted over the entirety of the simulation from direct collisions with stars, which is limited by both feedback during the accretion process and environmental conditions like stellar density and velocity dispersion. The y-axis shows the growth through BH-BH gravitational wave capture… view at source ↗

**Figure 5.** Figure 5: For four simulations described in [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

**Figure 6.** Figure 6: For the four simulations described in [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗

**Figure 7.** Figure 7: For the four simulations described in [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗

**Figure 8.** Figure 8: For the four simulations in [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗

**Figure 9.** Figure 9: The initial (gray) and final (gold) distributions of mass and spin using gold (Hoang et al. (2018) single BHs) ICs and including only BH-star collisions with stellar density α = 1.75 [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗

**Figure 10.** Figure 10: The initial (gray) and final (gold) distributions of mass and spin using gold (Hoang et al. (2018) single BHs) ICs and including only BH-star collisions with stellar density α = 1.25 [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗

discussion (0)

Forward citations

Cited by 3 Pith papers

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astro-ph.GA 2026-05 unverdicted novelty 7.0

Machine learning regressors trained on Rapster simulations forecast that globular clusters rarely host black holes above 100 solar masses while a few nuclear star clusters may exceed this threshold.
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astro-ph.GA 2026-06 unverdicted novelty 6.0

N-body simulations demonstrate runaway GW BBH mergers in dense BH clusters (≥5×10^9 M⊙/pc³) produce ~10³ M⊙ IMBHs within 10 Myr.
Constraining the Galactic Center Dark Cluster with ELT/MICADO Observations
astro-ph.GA 2026-05 unverdicted novelty 4.0

ELT/MICADO is projected to enable the first quantitative constraints on the dark cluster of stellar compact objects in the Galactic Center through variability and astrometric searches.

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3 extracted references · 1 linked inside Pith · cited by 3 Pith papers

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