{"id":"51e38c70-e595-4cb0-9db7-7d01c1df33ee","arxiv_id":"2506.04330","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Star clusters that assemble hierarchically with binaries, triples, or very massive stars produce 10^3 to 10^4 M_sun intermediate-mass black holes within 10 Myr, offering a path to supermassive black hole seeds.","lead":"This paper simulates the messy assembly of million-solar-mass star clusters, including binary and triple stars, and finds that stellar multiplicity dramatically boosts the formation of intermediate-mass black holes, reaching up to 10,000 solar masses within 10 million years. The results suggest that dense, early star clusters seen by JWST could already harbor supermassive black hole seeds above 10,000 solar masses.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's own §2.2.3 flags that collision mass conservation and 600 M⊙ track extrapolation are unquantified; the >10^4 M⊙ IMBH and JWST predictions depend on which of two opposing effects wins.","rationale":"The reader's weakest assumption is the same one I would flag: the collision and SMS prescriptions in §2.2.3 are load-bearing for the absolute IMBH masses and for the JWST extrapolation. The paper explicitly acknowledges that it cannot determine whether the competing effects of mass loss and radius inflation balance, and defers quantification to a future paper. That self-identified limitation is exactly where the central argument is least secure. The simulations are otherwise carefully constructed, and the isolated binary sample strengthens the qualitative role of multiplicity, but the 'up to 10^4 M⊙' and 'z~10 clusters host IMBHs above 10^4 M⊙' statements are quantitative and directly inherit this uncertainty. Since the reader already assigned a CONDITIONAL verdict, my read does not change that verdict; it reinforces it. No internal inconsistency or statistical artifact appears to be more important than this physical, unquantified prescription.","tokens_in":54322,"tokens_out":4436,"duration_ms":44736,"concrete_test":"Re-run HS450-C and HB150-A (and, if cost permits, all 12 hierarchical models) with three modified collision/SMS schemes: (a) 10% mass loss per star–star collision, (b) post-collision radii inflated by a factor following Suzuki et al. (2007) for one Kelvin–Helmholtz time before reverting to SEVN radii, and (c) both. Compare maximum IMBH mass, number of IMBHs above 300 M⊙ at 10 Myr, and the predicted M_bullet for the Adamo et al. (2024) clusters. If the >10^4 M⊙ merger disappears or the IMBH yield drops by more than a factor of ~2, the headline claim is not robust to the stated uncertainty.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Central quantitative results—IMBHs up to ~10^4 M⊙ and the inference that z~10 JWST clusters host M_bullet ≳ 10^4 M⊙—rest on the collision/SMS prescriptions in §2.2.3: collisions are 100% mass conserving, and collision products/SMSs are evolved with SEVN tracks extrapolated from the 600 M⊙ endpoint at equilibrium radii. The text itself states that realistic mass loss may reach ~10% per MS collision and that SMS collisions may lose loosely bound envelope mass, while post-collision inflation and rejuvenation are not modelled. Growth to 10^4 M⊙ requires ~10–600 collisions (§4.3.1), so even a 10% per-collision loss accumulates to a large suppression; inflated radii would act in the opposite direction. Because the two effects are not quantified, the sign and magnitude of the net error are unknown. Sections 6.1–6.2 extrapolate the fitted M_cl–M_bullet and Σ_h–M_bullet relations to the observed Cosmic Gems Arc and Firefly Sparkle clusters, so this uncertainty propagates directly into the paper's JWST claim. This is a genuine load-bearing weakness, not an inconsistency.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":54586,"tokens_out":5904,"duration_ms":55171,"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":[{"comment":"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.","section":"§2.2.3 and §4.3.1"},{"comment":"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.","section":"§6.2 and Table 8"}],"minor_comments":[{"comment":"The phrase 'as descried section 2.4' contains a typo and should read 'as described in section 2.4'.","section":"§4.1"},{"comment":"The cluster name 'NCC 1818' appears in the summary section and should be 'NGC 1818' for consistency with the earlier text.","section":"§5.4.3 and §8"},{"comment":"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.","section":"Fig. 3 caption"},{"comment":"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'.","section":"Table 5"},{"comment":"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.","section":"Data Availability Statement"}],"recommendation":"major_revision","confidential_remarks":"This is a strong technical contribution and the hierarchical-assembly IMBH channel is timely in light of JWST results. My main concern is that the headline quantitative claims rest on the acknowledged but unquantified collision/supermassive-star prescriptions. If the authors can provide a sensitivity test or an error budget, the paper would be suitable for publication. I would not reject, but major revision is needed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nThis 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.","headline":"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.","tokens_in":55150,"tokens_out":2340,"would_cite":true,"duration_ms":23750,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["intermediate-mass black holes","star cluster assembly","stellar collisions","binary stars","triple systems","tidal disruption events","gravitational waves","JWST high-redshift clusters"],"falsifier":"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.","tokens_in":1876,"feed_emoji":"🕳️","tokens_out":2944,"duration_ms":189645,"temperature":0.7,"pith_summary":"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 $10^4\\,M_\\odot$.","feed_headline":"Binaries and triples build 10,000-solar-mass black hole seeds","feed_subtitle":"Simulations tie JWST's dense early star clusters to supermassive black hole seed formation.","key_machinery":"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.","core_discovery":"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 $10^4\\,M_\\odot$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Paper I of this series; supplies the hierarchical assembly initial conditions and the single-star baseline results that this study extends.","marker":"Rantala et al. 2024"},{"why":"Companion study showing that hierarchical assembly drives close-to-equal-mass IMBH mergers; the present paper presents the detailed analysis of those simulations.","marker":"Rantala & Naab 2025"},{"why":"Introduces the SEVN stellar population synthesis code used here for single and binary stellar evolution and compact remnant formation.","marker":"Iorio et al. 2023"},{"why":"Provides the observed mass-dependent binary and triple fractions used to set the initial stellar multiplicity populations.","marker":"Offner et al. 2023"},{"why":"Reports the JWST Cosmic Gems Arc star clusters at $z\\sim10$, whose masses and surface densities anchor the paper's IMBH mass estimates.","marker":"Adamo et al. 2024"},{"why":"Reports the JWST Firefly Sparkle clusters at $z\\sim8.3$, used to place lower limits on the IMBHs hosted by unresolved early clusters.","marker":"Mowla et al. 2024"},{"why":"Provides the dynamical IMBH detection in $\\omega$ Centauri, used as a consistency check for the simulated scaling relations.","marker":"Häberle et al. 2024"}],"fun_headline_variants":["Binaries and triples forge 10,000-solar-mass black hole seeds","Stellar companions accelerate black hole seed growth in dense clusters","How binaries and triples jump-start supermassive black holes","Massive stars and companions seed supermassive black holes early","JWST clusters may host black hole seeds boosted by binaries"],"cache_read_input_tokens":57344,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Binaries and triples forge 10,000-solar-mass black hole seeds","Stellar companions accelerate black hole seed growth in dense clusters","How binaries and triples jump-start supermassive black holes","Massive stars and companions seed supermassive black holes early","JWST clusters may host black hole seeds boosted by binaries"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00063,"raw_usage":{"total_tokens":3049,"prompt_tokens":1225,"completion_tokens":1824,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":841,"completion_tokens_details":{"reasoning_tokens":1736}},"tokens_in":841,"tokens_out":1824,"duration_ms":12822,"temperature":1.0,"reasoning_tokens":1736,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:44:11.118911+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}