FROST-CLUSTERS -- II. Massive stars, binaries and triples boost supermassive black hole seed formation in assembling star clusters

Antti Rantala; Gast\'on J. Escobar; Giuliano Iorio; Natalia Lah\'en; Thorsten Naab

REVIEW 2 major objections 5 minor 1 cited by

FROST-CLUSTERS -- II. Massive stars, binaries and triples boost supermassive black hole seed formation in assembling star clusters

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Hierarchical assembly of star clusters, seeded by binaries, triples, and massive single stars, forms intermediate-mass black holes up to $10^4\,M_\odot$ within 10 Myr.

desk verdict Solid, honest N-body study: binaries/triples and 450 Msun stars boost collisional IMBH formation to ~10^4 Msun, but the headline numbers rest on collision prescriptions the authors flag as unquantified. read the letter →

arxiv 2506.04330 v4 pith:6A25GZAX submitted 2025-06-04 astro-ph.GA

Antti Rantala , Natalia Lahén , Thorsten Naab , Gastón J. Escobar , Giuliano Iorio This is my paper

classification astro-ph.GA

keywords intermediate-massblackholesstarclusterassemblystellarcollisionsbinarystarstriplesystemstidaldisruptioneventsgravitationalwavesJWSThigh-redshiftclusters

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 argues that the messy way massive star clusters actually form—by many sub-clusters crashing together—turns stellar multiplicity into a black-hole-seed factory. Its simulations of roughly million-solar-mass, low-metallicity clusters with initial binaries, triples, and single stars up to 450 solar masses produce intermediate-mass black holes (IMBHs) with masses up to about $10^4\,M_\odot$ within the first 10 Myr. The growth happens through runaway stellar collisions, tidal disruption events, and black-hole mergers. This matters because such IMBHs are plausible seeds for the supermassive black holes seen at very high redshifts, and the paper concludes that the dense clusters observed by JWST at $z\sim8$–$10$ should host IMBHs above $10^4\,M_\odot$.

What carries the argument

The central mechanism is the collisional runaway growth of a very massive star in a sub-cluster, described by $dm/dt\propto m^\beta$ with $\beta>0$, so that each collision makes the next collision more likely. What unlocks this runaway is the initial binary and triple population, which provides large interaction cross sections and removes the high-velocity-dispersion suppression of collisions seen in single-star models. The simulations couple the direct $N$-body integrator BIFROST with the SEVN stellar-evolution module, including post-Newtonian dynamics, mass-conserving stellar collisions, tidal disruption events with 50% mass accretion, black-hole mergers with gravitational-wave recoil kicks, and binary and triple stellar evolution with safeguards against spurious evolution during strong few-body encounters.

What would settle it

A direct rerun of the same hierarchical initial conditions with a 10% mass-loss per stellar collision—while keeping all other physics fixed—would falsify the predicted $10^4\,M_\odot$ seeds if the maximum IMBH mass dropped below roughly $10^3\,M_\odot$. Observationally, if the dense $z\sim10$ clusters seen by JWST show no signs of compact massive remnants above $10^4\,M_\odot$ over a large sample, the channel would need substantial revision.

Watch

Extended reading notes

Core claim

The central claim is that in hierarchically assembling clusters of total mass $M_{\rm cl}\sim10^6\,M_\odot$ with a standard Kroupa initial mass function at $Z=0.01\,Z_\odot$, an initial stellar multiplicity population—or a higher stellar mass cutoff of 450 $M_\odot$—removes the velocity-dispersion bottleneck that suppresses collisional growth in single-star clusters. The result is a collisional runaway producing up to about ten IMBHs per assembling region, with the most massive reaching $M_\bullet\gtrsim10^4\,M_\odot$ within 10 Myr. Most of the IMBH mass is acquired through stellar collisions, with tidal disruption events contributing at least about 15% and gravitational-wave-driven black-hole mergers providing a smaller but important late contribution, including close-to-equal-mass IMBH-IMBH mergers. The IMBH masses follow approximate power-law relations with their host clusters, $M_\bullet\propto M_{\rm cl}$, $M_\bullet\propto\Sigma_h^{3/2}$, and $M_\bullet\propto\sigma^3$, and applying these relations to JWST's dense $z\sim8$–$10$ star clusters implies that those systems should host IMBHs well above $10^4\,M_\odot$.

Load-bearing premise

The model assumes stellar collisions conserve mass and that collision products and supermassive stars can be modeled by extrapolating the 600-solar-mass stellar tracks; if collisions lose even about 10% of the mass per event, or if inflated radii change collision rates, the predicted IMBH masses and numbers would shift substantially.

Editorial extensions

If this is right

With initial binaries, collisional IMBH formation is no longer suppressed in massive, high-velocity-dispersion clusters, and each sub-cluster above about $10^5\,M_\odot$ has an 80–100% chance of forming a black hole above $100\,M_\odot$.
Hierarchical assembly naturally produces close-to-equal-mass IMBH binaries, and several of these merge within 10 Myr; the most massive simulated merger forms a $10^4\,M_\odot$ black hole and would be detectable by LISA at $z>10$.
Final assembled clusters show outer density slopes close to $\rho\propto r^{-3}$, shallow central stellar cusps, and rotation velocities of roughly 2–4 km/s, providing quantitative baselines for interpreting young massive clusters.
The simulated IMBH–cluster scaling relations are shallower than the galactic $M_\bullet$–$\sigma$ relation, suggesting that the tight scaling relations seen locally are established later by gas accretion and mergers rather than at seed formation.
If the inferred IMBHs exist in the dense early clusters seen by JWST, those clusters would be plausible birthplaces for supermassive black hole seeds that could grow to the masses of early AGN.

Reading between the lines

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

The paper's scaling relations imply that fully assembled young clusters can hide IMBHs heavier than their current surface densities suggest, because cluster mergers dilute the half-mass surface density by roughly an order of magnitude relative to the denser birth sub-clusters.
A natural testable extension is to rerun the same hierarchical setups with 10% mass loss per stellar collision; this would likely lower the peak IMBH masses but could still leave $10^3$–$10^4\,M_\odot$ seeds from tidal disruption events and IMBH-IMBH mergers.
The same machinery could be applied to assembling regions more massive than $10^6\,M_\odot$ to see whether seeds reach the $10^5\,M_\odot$ heavy-seed regime sometimes invoked for the most luminous high-redshift AGN.
An observational consequence the authors leave implicit is that the predicted per-cluster TDE rate of about $5\times10^{-5}$ yr$^{-1}$ means that stacking a sufficiently large sample of high-redshift dense clusters should produce rare nuclear flare counterparts alongside gravitational-wave events.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Antti and coauthors have done a careful, transparent job. What's new: this is the first time initial binaries, triples, and 450 Msun single stars have been put through hierarchical cluster assembly simulations at 1.8e6 particles with post-Newtonian dynamics and a full stellar evolution coupling. The result—multiplicity boosts the number and mass of collisionally formed IMBHs, up to ~10 per cluster and 10^4 Msun in 10 Myr—is internally consistent and the numerical set-up is thorough. The comparison against isolated clusters and the literature is thoughtful, and the paper is honest about what it does not model.

The soft spot is real and centrally located. Section 2.2.3 states collisions are mass-conserving, collision products are evolved using 600 Msun tracks extrapolated upward, and temporary post-collision radius inflation is ignored. The authors note realistic mass loss may be ~10% per main-sequence collision and that SMS collisions may shed loosely bound envelope mass, while inflated radii would increase collision rates. Because the two effects push in opposite directions and are not quantified, the net error on the final IMBH masses is unknown in sign. Growth to 10^4 Msun requires on the order of 10-600 collisions, so even 10% per-collision mass loss would accumulate to severe suppression. This directly affects the paper's JWST claim that z~10 clusters host M_bullet > 10^4 Msun. The authors acknowledge the issue and promise improved prescriptions in FROST-CLUSTERS III, which is fair, but without a sensitivity study the headline numbers remain conditional.

The scaling relations fitted to simulation output and applied to independent JWST observations are not circular; the observed cluster properties were not used to set simulation parameters. The extrapolation is nonetheless long: the Cosmic Gems Arc clusters are several times more massive than any simulated final cluster. Minor complaints: data availability is 'on reasonable request,' and the paper would be stronger with released simulation outputs and a public version of the binary stellar evolution interface.

This paper deserves a serious referee. The central mechanism is plausible, the numerics are state of the art, and the implications for SMBH seeds and LISA are concrete. A referee should ask for a sensitivity analysis around collision mass loss and radius inflation, and ideally a data release. I would not desk-reject this.

Referee Report

2 major / 5 minor

Summary. This manuscript presents a large set of direct-summation N-body simulations of the hierarchical assembly of massive star clusters (M_cl ~ 10^6 Msun, N = 1.8e6), using the BIFROST code with post-Newtonian dynamics coupled to the SEVN stellar evolution code. The authors compare models with single stars only, with initial binaries, with initial triples, and with massive singles up to 450 Msun. They find that stellar multiplicity and a high mass limit boost the number and mass of IMBHs formed through stellar collisions, TDEs, and BH mergers, reaching masses up to ~10^4 Msun within 10 Myr. They derive scaling relations between IMBH mass and host cluster mass, surface density, and velocity dispersion, and apply these to JWST-detected z~8-10 clusters to infer IMBHs with M_bullet ≳ 10^4 Msun.

Significance. The paper's technical machinery is impressive: 1.8e6-particle direct N-body integrations with PN terms, a full binary/triple stellar evolution interface, explicit TDE and GW-merger criteria, and stability checks. The central qualitative result—that initial stellar multiplicity removes the velocity-dispersion bottleneck for collisional IMBH formation and that hierarchical assembly produces multiple IMBHs per cluster, enabling near-equal-mass IMBH-IMBH GW mergers within 10 Myr—is well supported and of high astrophysical interest. The paper is also careful to identify the main simplifications in the collision and supermassive-star modeling. However, the quantitative headline (IMBHs up to 10^4 Msun and JWST cluster predictions of M_bullet ≳ 10^4 Msun) rests on assumptions whose net effect is explicitly acknowledged to be unquantified.

major comments (2)

[§2.2.3 and §4.3.1] The simplified modeling of stellar collisions and supermassive stars is load-bearing for the paper's central quantitative claim. The text states that collisions are mass conserving, that no temporary radius inflation or rejuvenation is modeled, and that collision products above 600 Msun use extrapolated tracks; it also states that realistic mass loss may reach ~10% per main-sequence collision and that the two competing effects (mass loss vs. inflation) cannot currently be balanced (§2.2.3). §4.3.1 reports that ~10–600 collisions are required to build a 10^3 Msun progenitor, so a 10% per-collision loss would compound to a large suppression, while inflated radii would increase collision rates. Because the sign of the net error is unknown, the reported IMBH masses and the JWST-based inference in §6.2 could shift by a large factor. I ask for a quantitative sensitivity test—for example, rerunning at least one flagship hierarchical model (e.g., HS450-C or HB150-A) with a 10% per-collision mass loss and a simple post-collision inflation prescription—or an explicit analytic error budget bracketing the effect of both assumptions.
[§6.2 and Table 8] The translation of the scaling relations into the JWST prediction is under-specified. Table 8 lists four different M_cl–M_bullet relations with slopes ranging from 0.66 (single stars) to 1.21 (massive singles); for the adopted Cosmic Gems Arc cluster mass M_cl = 2×10^6 Msun these relations give M_bullet between ~9×10^3 Msun and ~8×10^4 Msun, a factor ~9 spread. The text quotes M_bullet ~ 3.6×10^4 Msun as the 'fiducial' value without stating which model relation is used, and the quoted predictions carry no uncertainty from the fit scatter or model choice. In addition, the fitting range (sub-clusters with M_cl ≲ 2.5×10^5 Msun) is extrapolated by an order of magnitude in mass and two orders of magnitude in surface density to the observed clusters. Please specify the adopted relation, show the prediction band, and discuss the extrapolation assumption explicitly.

minor comments (5)

[§4.1] The phrase 'as descried section 2.4' contains a typo and should read 'as described in section 2.4'.
[§5.4.3 and §8] The cluster name 'NCC 1818' appears in the summary section and should be 'NGC 1818' for consistency with the earlier text.
[Fig. 3 caption] The caption refers to 'thick line-style' without defining what the thick line style indicates; please clarify this in the caption or in the text.
[Table 5] The two columns labeled 'max M_bullet [Msun]' and 'max M_bullet (t=10Myr)' are ambiguous; please relabel them as 'maximum M_bullet over the run' and 'M_bullet at t = 10 Myr'.
[Data Availability Statement] The data availability statement says data will be shared 'on reasonable request' but does not specify a repository or timeline; given the complexity of the simulations, a public release of initial conditions and analysis scripts would strengthen reproducibility.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: fitted scaling relations are extrapolated to independent JWST observations, and self-citations provide supporting methodology rather than forcing the central result.

full rationale

The central quantitative results are produced by direct BIFROST+SEVN N-body simulations with initial conditions, collision criteria, and stellar-evolution prescriptions described in the paper and in Paper I. The self-citations to Paper I and Rantala & Naab (2025) supply the code methodology, the hierarchical initial-condition setup, prior isolated single-star comparison runs, and a previous simulation sample; they do not by themselves force the new binary/triple/massive-single outcomes, which are presented in new runs in Secs. 3–4 and Table 5. The M_cl–M_bullet, Sigma_h–M_bullet, and sigma–M_bullet relations in Sec. 6 (Eqs. 6–8, Table 8) are least-squares fits to the authors' own simulation sub-clusters and final clusters. Applying these fitted relations to the Cosmic Gems Arc and Firefly Sparkle clusters (Adamo et al. 2024; Mowla et al. 2024) is an extrapolation to independent observational data: the observed cluster masses and surface densities were not used to construct the simulations or to set the fit parameters, so the JWST inference is not statistically forced by the fit. The paper's own Sec. 2.2.3 identifies the mass-conserving collision assumption and the extrapolated 600 M_sun tracks as limitations that can shift the magnitude of the IMBH masses, but these are explicit model assumptions rather than circular reductions. No equation in the paper is equivalent to its own input by construction, and no central claim is justified solely by a self-citation.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central results depend on several chosen parameters and domain assumptions, most notably the collision/M-SMS prescriptions, the TDE accretion fraction, the tidal cutoff, and the adoption of local Universe multiplicity fractions for high redshift. These are all stated in the paper, but they are not independently calibrated.

free parameters (4)

TDE accretion fraction = 0.5
Set to 50% of disrupted stellar mass accreted onto the BH; affects IMBH mass budget. The paper estimates 10% or 100% would change TDE contribution from ~3% to ~26% (Section 4.3.3).
Tidal cutoff radius = 37 pc
Applied to final cluster mass and half-mass radius to correct for missing external tidal field; changes half-mass radii by factor ~2.5 and masses by factor ~1.4 (Section 5.1.1), and is used for the scaling relations and JWST comparison.
Cluster size normalization f_h = 1/8
Scale factor in the mass-size relation (Eq. 1) chosen to match observed embedded cluster birth radii; sets the central densities that drive collision rates.
Collision mass-loss fraction = 0
Collisions are treated as mass conserving (Section 2.2.3). The paper acknowledges this may overestimate IMBH masses, while neglected radius inflation underestimates collision rates.

assumptions (5)

domain assumption Plummer models with the adopted mass-size relation (Eq. 1) and f_h=1/8 capture the structure of high-redshift embedded star clusters.
Initial cluster properties determine collision rates; the paper cites hydrodynamical simulations and observed embedded clusters, but the mapping to z~10 is uncertain (Section 2.3).
domain assumption Local Universe stellar multiplicity fractions (Offner et al. 2023; Table 2) apply to high-redshift, low-metallicity star formation.
High-z binary fractions are unconstrained; the paper adopts local observations (Section 2.4).
domain assumption PARSEC/SEVN stellar tracks and Hurley et al. (2002) binary evolution prescriptions adequately model low-metallicity massive stars and their remnants.
The evolution of stars up to 600 M_sun and the collision products relies on these tracks; stellar wind mass loss at low Z is uncertain (Section 2.2).
domain assumption No external tidal field is included; stars with E>0 removed at 200 pc.
This simplification is stated (Section 2.1) and corrected post hoc with a tidal cutoff at 37 pc, but the early cluster evolution may be affected.
domain assumption The initial conditions generation procedure does not significantly bias the binary/triple populations.
They ensure stability of triples and check for chance binaries (Section 2.4), but the procedure is still a realization of a complex sampling.

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Cite this review

Pith. "Pith review of FROST-CLUSTERS -- II. Massive stars, binaries and triples boost supermassive black hole seed formation in assembling star clusters." pith.science (2026). https://pith.science/paper/6A25GZAX

@misc{pith2026250604330,
  author       = {Pith},
  title        = {Pith review of: FROST-CLUSTERS -- II. Massive stars, binaries and triples boost supermassive black hole seed formation in assembling star clusters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6A25GZAX}},
  note         = {Machine review of arXiv:2506.04330}
}

abstract

Observations and high-resolution hydrodynamical simulations indicate that massive star clusters form through a complex hierarchical assembly. We use simulations including post-Newtonian dynamics (the BIFROST code) and stellar evolution (the SEVN module) to investigate this collisional assembly. With a full initial stellar mass function, we study the effect of initial binary, triple and massive single stars (450 $M_\odot$) on the assembly, structure, and kinematics of massive ($M_\mathrm{cl}\sim10^6 M_\odot$, $N=1.8 \times 10^6$) star clusters. Simultaneously, intermediate mass black holes (IMBHs), potential seeds for supermassive black holes, can form and grow in our models by stellar collisions, tidal disruption events (TDEs) and black hole (BH) mergers. At a fixed cluster mass, stellar multiplicity or a high mass limit increase the numbers (up to $\sim$ 10) and masses (up to $10^4 M_\odot$) of the formed IMBHs within the first 10 Myr of cluster evolution. The TDE rates peak at $\Gamma_\mathrm{tde}\sim 5 \times 10^{-5}$ yr$^{-1}$ after IMBH formation at $\sim 2$ Myr. In all simulations, we find gravitational wave driven mergers involving stellar BHs and IMBHs. Initial multiplicity or a high mass limit also result in IMBH-IMBH mergers. The IMBH masses correlate with the initial cluster masses, surface densities and velocity dispersions approximately as $M_\bullet \propto M_\mathrm{cl}$, $M_\bullet\propto\Sigma_\mathrm{h}^\mathrm{3/2}$ and $M_\bullet\propto\sigma^\mathrm{3}$. Our results suggest the dense $z\sim10$ star clusters recently observed by the James Webb Space Telescope host IMBHs with masses above $M_\bullet \gtrsim 10^4 M_\odot$.

Figures

Figures reproduced from arXiv: 2506.04330 by the authors.

**Figure 1.** The treatment of gravitational dynamics and binary stellar evolution for a binary star with a semi-major axis 𝑎 in close interactions with other stars. In weak encounters with a large impact parameter (𝑏3 > 𝑟strong) the dynamics of the binary is mildly perturbed and no special treatment for the binary evolution is required. If the encounter is close enough (𝑏2 < 𝑟strong) binary stellar evolution is temporarily chan… view at source ↗

**Figure 2.** The maximum collisionally grown stellar masses 𝑚max in the isolated setups with initial binaries (small crosses) within 5 Myr. The single star models of Paper I are displayed as the shaded background. The highest 𝑚max for each cluster mass 𝑀cl linearly increases (dotted line) until 𝑀cl ∼ 105 M⊙. At higher cluster masses, the stellar collisions are suppressed in the single star models. In the binary models, the colli… view at source ↗

**Figure 3.** The mass assembly histories of selected IMBHs and their VMS progenitors that reached masses above ≳ 500 M⊙ during the first 5 Myr of their evolution. The collisional growth is rapid and usually truly runaway (𝛽 > 0) between 0.5 Myr ≲ 𝑡 ≲ 2.5 Myr. First stellar collisions occur earlier in models with initial stellar multiplicity (binaries and triples). Most collisions occur in the main sequence phase of the stars and… view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: Left panel: the total VMS stellar collision rates (𝑚1 + 𝑚2 > 150 M⊙) in the hierarchical simulations in time bins of 0.25 Myr. The peak VMS collision rates Γcoll ∼ 6 × 10−4–1.4 × 10−3 yr−1 are reached near the end of the lifetimes of the collisionally grown stars at 2 …

**Figure 5.** Figure 5: The 3D mass density profiles for stars and stellar BHs of the assembled hierarchical clusters at 𝑡 = 10 Myr. BHs in the IMBH mass range are excluded from the analysis. Double power-law fits of the profiles are provided in [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: The line-of-sight (LOS) velocity profiles 𝑉los of the hierarchically assembled clusters at 𝑡 = 10 Myr within < 2𝑅e measured in a narrow slit of |𝑧 | < 1 pc. For the analysis the cluster is oriented along its angular momentum vector within 𝑅e. is brought into the main c…

**Figure 8.** Figure 8: All the three cluster models are almost spherical or mildly axisymmetric at their centres (𝑟 ≲ 0.3 pc) with 𝑏/𝑎 ≳ 0.98 and 0.95 ≲ 𝑐/𝑎 ≲ 0.98. One of the clusters remains axisymmetric 𝑏/𝑎 ∼ 1 until 𝑟 = 4–5 pc while the two other models have either a relative flat shape …

**Figure 7.** Figure 7: The three hierarchical single star clusters in different line-styles separated into their initially central, accreted and total components in their density 𝜌 (top panel), rotation velocity 𝑉tan (middle panel) and velocity anisotropy 𝛽 (bottom panel). The vertical lines…

**Figure 9.** Figure 9: The time evolution of binary fractions 𝑓bin (top panels) and triple fractions 𝑓trip (bottom panels) in the hierarchical cluster assembly models with singles, massive singles, binaries and triples. In the models without initial stellar multiplicity the binary and triple…

**Figure 10.** Figure 10: The final 𝑡 = 10 Myr radial profiles for total binary 𝑓bin and triple fractions 𝑓trip in the assembled clusters of setups with initial binary and triple systems. form dynamically from initially unbound single stars (binaries) and in single-binary interactions (triples…

**Figure 11.** Figure 11: Left panel: the star cluster mass 𝑀cl vs the IMBH mass 𝑀• (above 𝑀• ≳ 150 M⊙) for our individual sub-clusters (small symbols) that formed at least a single IMBH in the hierarchical setups as well as the final assembled clusters (large symbols). Most massive IMBHs form…

**Figure 12.** Figure 12: The 1D star cluster velocity dispersions 𝜎1D measured within 𝑅e for the individual sub-clusters (small symbols) and the final assembled clusters (large symbols). The power-law 𝑀•–𝜎 scaling relation is less steep, approximately 𝑀• ∝ 𝜎3 for our cluster models compared t…

**Figure 13.** Figure 13 [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]

**Figure 14.** Figure 14: The primary and secondary masses of (IM)BH binaries at 𝑡 = 10 Myr (left panel) and the GW mergers in the simulations at 𝑡 < 10 Myr (right panel) in the hierarchical cluster assembly models. The Advanced LIGO / Advanced Virgo GW mergers in the GWTC-3 catalogue are show…

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write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.stat...

Pith tools

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

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