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arxiv: 2605.21593 · v1 · pith:TAJ6XRH4new · submitted 2026-05-20 · 🌌 astro-ph.GA · astro-ph.CO· astro-ph.HE· gr-qc

Predicting intermediate-mass black hole formation in star clusters with machine learning

Konstantinos Kritos , Digvijay Wadekar , Emanuele Berti This is my paper

Pith reviewed 2026-05-22 09:16 UTC · model grok-4.3

classification 🌌 astro-ph.GA astro-ph.COastro-ph.HEgr-qc
keywords intermediate-mass black holesstar clustersmachine learningglobular clustersnuclear star clustersblack hole mergerscollisional runaway
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The pith

Machine learning on cluster simulations predicts globular clusters rarely host black holes heavier than about 100 solar masses.

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

The paper trains neural network and random forest models on synthetic star cluster catalogs to map observable properties like total mass and half-mass radius to the mass of the heaviest black hole formed by repeated mergers. When applied to real nearby clusters, the models indicate that globular clusters have a very low chance of containing such black holes above roughly 100 solar masses. Nuclear star clusters show a few cases with higher predicted masses. This approach offers a fast way to forecast intermediate-mass black hole populations without running full simulations for each observed system. Discrepancies with kinematic observations point to formation channels involving gas or star accretion beyond mergers.

Core claim

By training neural network and random forest regressors on synthetic catalogs generated with the Rapster cluster evolution code, we map observable cluster properties such as total mass and half-mass radius onto the mass of the heaviest black hole built up through repeated mergers. Applying these models to nearby globular and nuclear star clusters forecasts the intermediate-mass black hole population that each system may host, with globular clusters unlikely to contain black holes more massive than ~100 M⊙ and an occupation fraction near 0.02, although they can produce remnants within the upper mass gap with masses approaching 100 M⊙. Among nuclear star clusters, a handful of cases including,

What carries the argument

Neural network and random forest regressors that predict the mass of the heaviest black hole from observable cluster properties, combined with a normalizing flow to quantify the likelihood of initial conditions favoring collisional runaway in the first few million years.

If this is right

  • Globular clusters are unlikely to contain black holes more massive than ∼100 M⊙, with an occupation fraction near 0.02.
  • Globular clusters can produce black hole remnants in the upper mass gap with masses approaching 100 M⊙.
  • A handful of nuclear star clusters including NGC 5102 and NGC 5206 yield predicted central black hole masses above 100 M⊙.
  • Where observationally claimed masses exceed the predictions, the assembly history involved processes beyond hierarchical mergers, most plausibly accretion of gas and stars.
  • Normalizing flows can quantify for individual globular clusters the likelihood that their initial conditions were favorable to a collisional runaway during the first few million years after formation.

Where Pith is reading between the lines

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

  • The trained models could be applied to larger catalogs from upcoming surveys to identify promising targets for follow-up observations of potential intermediate-mass black holes.
  • Systematic differences between model predictions and kinematic mass estimates in nuclear clusters may help isolate the relative importance of accretion versus merger channels.
  • Future gravitational-wave observations of black hole mergers could provide an independent test of the mass distributions predicted for clusters with different observable properties.
  • Refining the regressors with additional simulation suites that vary stellar evolution or initial mass functions would test how sensitive the occupation fractions are to those inputs.

Load-bearing premise

The synthetic catalogs generated with the Rapster cluster evolution code accurately capture the physics and parameter space of black hole formation through repeated mergers in real star clusters, allowing the trained models to generalize to observed systems.

What would settle it

A kinematic or other direct detection of a central black hole with mass well above 100 solar masses in a typical globular cluster would contradict the low occupation fraction and mass limit forecasts.

Figures

Figures reproduced from arXiv: 2605.21593 by Digvijay Wadekar, Emanuele Berti, Konstantinos Kritos.

Figure 1
Figure 1. Figure 1: FIG. 1. Seven realizations of star cluster simulations with [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Probability [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Suite of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The predicted local population of the heaviest BH [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Logarithm of the heaviest BH mass ( [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. We use ML algorithms to predict the heaviest BH mass from the final ( [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Same as the left panel of Fig [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Same as in Fig [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Predicted maximum BH mass for a subset of GCs. The black squares with error bars correspond to predictions from the [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Predictions of the most massive BH mass formed through hierarchical mergers in nearby young massive clusters. We [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Initial cluster properties (denoted by a subscript [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. P–P plot for the normalizing-flow reconstruction of [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
read the original abstract

Whether intermediate-mass black holes reside in nearby star clusters has remained contested for decades. We address this question by training neural network and random forest regressors on synthetic catalogs generated with the {\sc Rapster} cluster evolution code, mapping observable cluster properties such as total mass and half-mass radius onto the mass of the heaviest black hole built up through repeated mergers. Applying these models to nearby globular and nuclear star clusters, we forecast the intermediate-mass black hole population that each system may host. Globular clusters are unlikely to contain black holes more massive than $\sim 100\,M_\odot$, with an occupation fraction near 0.02, although they can produce remnants within the upper mass gap with masses approaching $100\,M_\odot$. Among nuclear star clusters, a handful of cases, including NGC 5102 and NGC 5206, yield predicted central black hole masses above $100\,M_\odot$, which we contrast with kinematically inferred estimates. Where the observationally claimed masses exceed our predictions, the implication is that the assembly history involved processes beyond hierarchical mergers, most plausibly accretion of gas and stars. Finally, we employ a normalizing flow to quantify, for individual globular clusters, the likelihood that their initial conditions were favorable to a collisional runaway during the first few million years after formation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to predict the presence and masses of intermediate-mass black holes (IMBHs) in star clusters using machine learning models trained on synthetic data from the Rapster cluster evolution code. By mapping observable properties such as total cluster mass and half-mass radius to the mass of the heaviest black hole formed via repeated mergers, the authors apply the trained neural network and random forest regressors to observed globular clusters (GCs) and nuclear star clusters (NSCs). They conclude that GCs are unlikely to host black holes more massive than approximately 100 solar masses, with an occupation fraction around 0.02, though they may produce remnants approaching the upper mass gap. For NSCs, a few systems including NGC 5102 and NGC 5206 are predicted to have central black holes exceeding 100 solar masses. The work also uses a normalizing flow to evaluate the probability that initial conditions in GCs favored a collisional runaway process shortly after formation.

Significance. If the results hold, this study provides a practical framework for estimating IMBH masses in observed clusters based on readily available observables, bypassing the need for computationally expensive individual simulations. The predicted low occupation fraction in globular clusters and the identification of potential IMBH hosts in nuclear clusters offer concrete, falsifiable predictions that can guide observational campaigns and help differentiate between hierarchical merger and other formation channels such as gas accretion. The incorporation of a normalizing flow for assessing collisional runaway likelihood adds a probabilistic dimension to the analysis of cluster initial conditions. These elements collectively advance the field by linking simulation-based insights directly to observational data in a scalable manner.

major comments (2)
  1. [Methods and Results (Rapster catalog generation and model application)] The central predictions, including the occupation fraction near 0.02 for black holes above ~100 M⊙ in globular clusters and the specific mass forecasts for NGC 5102 and NGC 5206, rest on the fidelity of the Rapster synthetic catalogs. The manuscript should provide explicit comparisons of Rapster's treatment of dynamical friction, binary hardening, merger retention, and natal kicks against other codes or observational benchmarks (e.g., in the methods or results section describing the training data generation) to quantify potential systematic biases that could alter the reported occupation fraction and mass predictions.
  2. [Results (application to observed clusters and discussion of discrepancies)] The application of the trained models to observed clusters and the implication that discrepancies with kinematic mass estimates indicate processes beyond hierarchical mergers (such as gas accretion) requires quantified uncertainties. The paper should report ML prediction uncertainties (e.g., via cross-validation metrics, ensemble variance, or error propagation) and assess sensitivity to Rapster physics parameters, as these are load-bearing for the contrast drawn with observations in the nuclear star cluster cases.
minor comments (2)
  1. [Methods] Clarify the exact ranges of total mass, half-mass radius, and other initial conditions used to generate the Rapster training catalogs, as this directly affects the applicability of the models to the observed sample of globular and nuclear clusters.
  2. [Methods (ML model training)] Include performance metrics (such as R² or mean absolute error) for both training and held-out test sets in the description of the neural network and random forest regressors to demonstrate generalization and lack of overfitting.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments and positive evaluation of the work's significance. We address each major comment point by point below and indicate the revisions that will be incorporated into the next version of the manuscript.

read point-by-point responses

  1. Referee: [Methods and Results (Rapster catalog generation and model application)] The central predictions, including the occupation fraction near 0.02 for black holes above ~100 M⊙ in globular clusters and the specific mass forecasts for NGC 5102 and NGC 5206, rest on the fidelity of the Rapster synthetic catalogs. The manuscript should provide explicit comparisons of Rapster's treatment of dynamical friction, binary hardening, merger retention, and natal kicks against other codes or observational benchmarks (e.g., in the methods or results section describing the training data generation) to quantify potential systematic biases that could alter the reported occupation fraction and mass predictions.

    Authors: We agree that additional context on Rapster's physical prescriptions would strengthen the presentation. In the revised manuscript we will insert a concise subsection in the Methods describing Rapster's implementations of dynamical friction, binary hardening, merger retention, and natal kicks, with direct comparisons to the corresponding treatments in CMC and MOCCA together with references to existing benchmark studies. This addition will allow readers to assess the magnitude of any systematic effects on the derived occupation fraction and mass predictions. revision: yes

  2. Referee: [Results (application to observed clusters and discussion of discrepancies)] The application of the trained models to observed clusters and the implication that discrepancies with kinematic mass estimates indicate processes beyond hierarchical mergers (such as gas accretion) requires quantified uncertainties. The paper should report ML prediction uncertainties (e.g., via cross-validation metrics, ensemble variance, or error propagation) and assess sensitivity to Rapster physics parameters, as these are load-bearing for the contrast drawn with observations in the nuclear star cluster cases.

    Authors: We will expand the Results section to report ensemble variances from both the random-forest and neural-network models as well as additional cross-validation metrics. We will also add a short sensitivity analysis that varies key Rapster parameters (primarily natal-kick dispersion and binary-hardening efficiency) and shows the resulting spread in predicted IMBH masses for the nuclear star clusters. These quantified uncertainties will be used to contextualize the comparison with kinematic estimates for NGC 5102 and NGC 5206. revision: yes

Circularity Check

0 steps flagged

No significant circularity: ML predictions trained on independent Rapster synthetic catalogs

full rationale

The derivation trains neural network and random forest regressors on synthetic catalogs produced by the external Rapster cluster evolution code, then maps observable properties (total mass, half-mass radius) to heaviest merger-built BH mass and applies the models to real globular and nuclear star clusters. This chain does not reduce any prediction to a fitted parameter defined from the target observations, nor does it rely on self-definitional loops, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via self-citation. The normalizing-flow step for initial conditions likewise operates on the same independent synthetic data. The central claims therefore remain self-contained against external benchmarks and receive no circularity penalty.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the fidelity of the Rapster simulation code for generating training data and on the assumption that observable properties suffice to predict the heaviest black hole mass without additional hidden variables.

axioms (1)
  • domain assumption Rapster accurately models repeated black hole mergers and cluster evolution across the relevant mass and density range
    This underpins the synthetic catalogs used to train the regressors and normalizing flow.

pith-pipeline@v0.9.0 · 5777 in / 1289 out tokens · 51021 ms · 2026-05-22T09:16:42.397865+00:00 · methodology

discussion (0)

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