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In massive star clusters, the mass of an intermediate-mass black hole made by runaway stellar collisions is set almost entirely by the cluster's half-mass surface density and metallicity, through a simple fitting formula; the channel turns

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2026-08-03 10:58 UTC pith:36I3O4WX

load-bearing objection A genuinely wide simulation grid mapping IMBH masses against cluster density and metallicity, but the headline masses and cosmic rates are anchored to an extrapolated wind law that the authors themselves show could change by a factor of four. the 3 major comments →

arxiv 2601.07917 v2 pith:36I3O4WX submitted 2026-01-12 astro-ph.GA

FROST-CLUSTERS -- III. Metallicity-dependent intermediate mass black hole formation by runaway collisions in dense star clusters

Antti Rantala , Thorsten Naab , Natalia Lah\'en , Klaus Reuter , Markus Rampp , Martyna Chru\'sli\'nska , Basti\'an Reinoso This is my paper
classification astro-ph.GA
keywords intermediate-mass black holesrunaway stellar collisionsstar cluster surface densitymetallicitystellar windsN-body simulationsblack hole seedscosmic IMBH formation rate
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.

Runaway stellar collisions in young massive star clusters can build extremely massive stars that collapse into intermediate-mass black holes (IMBHs) of roughly 300–6000 solar masses, according to more than 1400 direct N-body simulations. The paper claims that for clusters heavier than about 10^4 solar masses, the maximum IMBH mass is set almost entirely by two host-cluster properties—half-mass surface density and metallicity—through a fitting formula. The channel works in low-metallicity, dense environments and is suppressed in the high-metallicity, low-density conditions typical of the local Universe. If correct, the fitting formulae let galaxy-formation models seed supermassive black holes from star clusters, and the derived cosmic IMBH formation rate—peaking at redshift 2–4, with half forming below z~1.5–3—would challenge the idea that all local IMBHs are failed high-redshift seeds.

Core claim

The paper establishes that in massive star clusters (M_cl ≳ 10^4 Msun) the maximum mass of an intermediate-mass black hole produced by runaway collisions is a function of only the cluster's half-mass surface density Sigma_h and its metallicity Z, stated as M_bullet = M_bullet(Sigma_h, Z) with a Heaviside threshold and piecewise metallicity-dependent coefficients. IMBHs with masses of roughly 300–6000 Msun form when Z is below about 0.2–0.3 Zsun and Sigma_h above about 3e4 Msun/pc^2; above that metallicity, strong stellar winds quench the runaway growth, and below that density collisions are too rare. The paper additionally gives a hierarchical-assembly extension showing that extremely dense

What carries the argument

The load-bearing machinery is the competition between collisional mass gain and stellar wind mass loss for stars grown beyond about 600 solar masses. The paper uses an extrapolated wind-loss law with no cap, plus a mass-loss term in each collision (f_loss = C q/(q+1)^2, C=0.3), and metallicity-dependent radii, to produce the plateau in stellar mass that sets the IMBH mass. The central object that carries the argument is the fitting formula M_bullet = theta_H(Sigma_h - Sigma_crit(Z)) [A(Z) log10 Sigma_h + B(Z) (log10 Sigma_h)^2 + C(Z)], with coefficients tabulated in three metallicity ranges; a second linear formula handles extrapolation to higher Sigma_h. This converts a two-dimensional simu

Load-bearing premise

The entire result rests on the adopted wind mass-loss law for stars above about 600 solar masses—an extrapolation without an upper cap—because replacing it with a standard cap changes the final black-hole masses by a factor of about four and shifts the channel into the supermassive-star regime.

What would settle it

Run one of the paper's dense, low-metallicity cluster setups (e.g., M_cl ~1.5e5 Msun, Z~0.1 Zsun, Sigma_h~1e5 Msun/pc2) with a wind-loss rate capped at 1.5e-4 Msun/yr: the paper itself reports that this produces monotonic growth past 20,000 Msun. If a single such run with a different but physically standard wind model reproduces the plateau near 7,000 Msun, the reported IMBH masses are robust; if it grows into the supermassive-star regime, the headline masses are an artifact of the extrapolated wind law. Observationally, detecting a >300 Msun black hole in a solar-metallicity massive cluster w

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

If this is right

  • The fitted M_bullet(Sigma_h,Z) can seed black holes in semi-analytic galaxy-formation models and high-resolution cosmological simulations without running star-by-star collision calculations.
  • The IMBH formation efficiency is strongly peaked below Z~0.1-0.2 Zsun; local young massive clusters with solar-like metallicity and low density should rarely produce IMBHs, explaining their absence.
  • The cosmic IMBH formation rate density peaks at z~2-4 with values up to ~1e-7 per year per cMpc^3; roughly half of IMBHs form below z~1.5-3, so a local IMBH need not be a leftover high-z SMBH seed.
  • The densest hierarchical models reach sustained tidal-disruption rates above 1e-5 per year per cluster, providing a growth channel that can push IMBHs beyond 1e4 Msun.
  • The sensitivity study shows that if a maximum wind-loss rate (a luminous-blue-variable-like cap) is used instead, collisionally grown stars pass 20,000-25,000 Msun, making the channel a viable supermassive-star seed route.
  • If the paper's picture holds, targeted searches for IMBHs should prioritize dense, low-metallicity old clusters rather than solar-metallicity clusters, where the suppression boundary predicts near-zero occupation.
  • If the formula is read as a predictive map, the same dense clusters that form IMBHs also produce 5-10% of their mass in wind and collisional ejecta, connecting IMBH formation to the chemical peculiarities seen in globular cluster multiple populations.
  • The wind-rate sensitivity suggests the dominant uncertainty is the physics of extremely massive stellar winds at low metallicity; if the cap is real, the channel may produce supermassive stars and heavier seeds than the paper's headline masses.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper presents a large suite of more than 1,440 direct N-body simulations of young massive star clusters with the BIFROST code, varying cluster mass, half-mass surface density, and metallicity, to study the formation of intermediate-mass black holes (IMBHs) via runaway stellar collisions. The central quantitative claim is that the maximum IMBH mass is determined almost entirely by cluster half-mass surface density and metallicity through a fitting formula M_bullet(Sigma_h, Z) (Eq. 9, with Eq. 10 for extrapolation), with IMBHs of roughly 300–6000 Msun forming at Z ≲ 0.2–0.3 Zsun and formation suppressed at higher metallicity and below Sigma_h ≲ 3e4 Msun/pc^2. The paper also presents 30 hierarchical cluster assembly simulations, estimates wind and collisional ejecta budgets, and constructs a cosmic IMBH formation rate model that peaks at z~2–4, with ~50% of IMBHs forming below z~1.5–3, thereby challenging the picture that all local IMBHs are failed high-redshift SMBH seeds. The authors explicitly test the sensitivity of their results to the adopted massive-star wind prescription in Section 4.2 and Appendices B and C.

Significance. If the central scaling holds, the fitting formulas provide a practical seeding prescription for galaxy-formation simulations and semi-analytic models, and the derived redshift-dependent IMBH formation rates would be an important constraint on black-hole seeding scenarios. The paper's main strengths are the exceptionally large and systematic simulation sample, the inclusion of hierarchical assembly alongside isolated clusters, and the unusually transparent treatment of the dominant stellar-wind uncertainty: the authors show in their own post-processing and dedicated test runs that an alternative widely used wind prescription (Vink 2001 with an LBV-like cap of 1.5e-4 Msun/yr) changes the maximum IMBH mass by a factor of about four and pushes collisionally grown stars into the supermassive-star regime. This honesty is commendable, but it also directly exposes the fragility of the headline numerical predictions and of the metallicity threshold that drives the cosmic rate model.

major comments (3)
  1. [§2.2.3, Eq. (5); §4.2, Fig. 11; Appendix C] The load-bearing premise of the paper is the Vink (2018) wind prescription, Eq. (5), extrapolated beyond its stated 900 Msun calibration range with no upper rate limit. The paper's own analysis shows that imposing the commonly used LBV-like ceiling of 1.5e-4 Msun/yr, as in NBODY6++GPU/MOCCA, changes the mass growth from a plateau near ~7000 Msun to monotonic growth beyond 25000 Msun (Fig. 11 and Fig. C1), a factor of ~4 in the final IMBH mass, and removes much of the metallicity dependence that produces the Z~0.2 Zsun threshold. Since Eq. (9), the threshold quoted in the abstract, and the cosmic rate model in Section 6 all inherit this choice, the central quantitative claims are conditional on an extrapolated and uncapped wind law. The authors should either justify this extrapolation quantitatively or present the main fitting formulas and cosmic rates for both wind prescriptions, with a
  2. [§3.4, Eq. (9); §6.1–6.6] The fitting formula Eq. (9) is a LOESS-plus-piecewise fit to the authors' own simulation data, and it is then used as input to the cosmic rate model in Section 6. The 'consistency' with JWST little-red-dot number densities reported in Section 6.6 is therefore a self-consistency check of the model chain, not an independent test of the fitting formula or of the IMBH channel. This is especially important because the rate model also depends on the cluster formation efficiency and the normalization of the cluster mass–radius relation (f_h), which are varied but not constrained by the N-body results. The text should be revised to state explicitly that the JWST comparison validates the assumed seeding efficiency only under the combined model assumptions, and should avoid presenting it as independent support for M_bullet(Sigma_h, Z).
  3. [§2.2.1, Eq. (2); Appendix A] The collisional mass-loss model, Eq. (2) with C=0.3 and mass loss taken from both the primary and secondary, is another ad-hoc choice with a strong cumulative effect: Appendix A shows that for a star doubling its mass the cumulative fractional loss is ~0.214, about 2.85 times the single-collision maximum and 1–2 orders of magnitude larger than secondary-only loss prescriptions. Recent hydrodynamical work on extended, marginally bound extremely massive stars suggests that mass loss can be even more catastrophic in some collisions. Although the authors acknowledge this uncertainty, it is not varied in the main sample, so the error bars on the final IMBH masses and on the metallicity threshold reflect only a subset of plausible prescriptions. At minimum, the paper should state how Eq. (2) affects the peak IMBH masses relative to the wind uncertainty, and ideally provide a small number of te
minor comments (5)
  1. [§3.2.1] Typo in the heading: 'The faction of clusters' should read 'The fraction of clusters'.
  2. [Fig. 2 caption] In the caption, 'IMBHs cannot from at Z=1.0 Zsun' should read 'cannot form'.
  3. [§4.6] The sentence 'The maximum collision and TDE rates by IMBHs in models HD9Z2 is are' has a grammatical error; 'is are' should be 'are'.
  4. [§3.4, Tables 4–5] The piecewise definitions of A(Z), B(Z), C(Z), D(Z), E(Z) are terse; an explicit worked example of how to assemble, say, A(Z) and Sigma_crit(Z) from Table 4 would help users implement Eq. (9) without error.
  5. [§6.1] The cluster formation efficiency Gamma is set to 0.3 and treated as constant; the text notes the linear scaling, but a brief discussion of the observed environmental variation of Gamma and its likely redshift evolution would improve the robustness of the rate estimates.

Circularity Check

0 steps flagged

No significant circularity: the fitting formula summarizes the simulations; the cosmic-rate model is a transparent application, not a validation loop.

full rationale

The derivation chain is: adopted stellar-physics inputs (Vink 2018 winds, Eq. 5; collisional mass loss, Eq. 2; EMS radii, Eq. 3) -> direct N-body simulations -> empirical fitting formulae M_bullet(Sigma_h,Z), Eqs. 9-10, fitted to the LOESS-smoothed simulation data of Fig. 7 -> cosmic formation rate model (Section 6) using external SFRD(Z) models and assumed cluster mass-size relations. No step reduces to its own output by construction. The fitting formulae are explicitly fits to the authors' own simulation results, and the paper labels them as fitting/empirical models rather than as independent predictions. The use of Eqs. 9-10 in Section 6 to populate clusters with IMBHs and to estimate R_IMBH is an application of the fitted relation, not a test of it; the JWST little-red-dot consistency check is explicitly framed as an upper-limit estimate relying on several simplified assumptions, not as evidence for the fitting formula itself. The strong sensitivity to the extrapolated Vink (2018) wind law beyond 900 Msun with no cap is a physical-modeling uncertainty, openly acknowledged in the abstract, Section 2.2.3, Section 4.2, and Appendices B and C; it is a robustness concern, not circularity. The self-citations (e.g., f_h=1/8 from earlier FROST papers, IMBH-IMBH merger rates) are used for initial-condition choices and context, not to justify the target result, and no uniqueness theorem or ansatz is imported from the authors' prior work. The paper is self-contained against external comparisons (Vergara et al. 2025a; Mapelli 2016; Gieles et al. 2018) and conducts additional direct simulations (Appendix C) to test the wind-rate-cap scenario. Thus no circular step can be substantiated with the paper's own equations.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 0 invented entities

The central claim depends on four main families of inputs: collision physics (C=0.3), massive-star radii and winds (PARSEC extrapolation plus Vink 2018), initial conditions (Plummer, Kroupa IMF, binary fraction, f_h), and the cosmic assembly model (SFRD(Z), Gamma=0.3, mass-radius relation). The first two are the most fragile, since the paper itself shows that changing the wind ceiling shifts final masses by roughly a factor of four.

free parameters (6)
  • Collisional mass-loss constant C = 0.3
    Eq. (2), f_loss = C q/(q+1)^2; chosen from Glebbeek et al. (2013) and directly sets cumulative mass loss in collision cascades (Appendix A).
  • ZAMS radius power-law coefficients a(Z), delta(Z) = Table 1, e.g. a=9.588, delta=0.562 at Z=0.01 Zsun
    Eq. (3) is fit to PARSEC tracks in the 400-600 Msun range and extrapolated above 600 Msun; it sets the collision cross-sections of extremely and supermassive stars.
  • Massive-star wind normalization and power-law coefficients = log(dm/dt) = -9.13 + 2.1 log(m*/Msun) + 0.74 log(Z/Zsun), no upper cap
    Vink (2018) recipe, Eq. (5), extrapolated above 900 Msun; the paper's own Section 4.2 and Appendix C show that capping the wind rate at 1.5e-4 Msun/yr raises the maximum SMS mass from about 7000 to above 25000 Msun.
  • Parametrized IMBH-mass fit coefficients = Tables 4-5: A1,A2,B1,B2,C1,C2,Sigma1,Sigma2,D1,D2,E1,E2
    Eqs. (9)-(10) are fitted to the LOESS-smoothed maximum IMBH masses from 1350 isolated simulations; the cosmic rate model in Section 6 inherits these fits.
  • Cluster mass-radius normalization f_h = Isolated: 0.125-0.580; cosmic model: 1, 1/2, 1/4, 1/8 plus evolving variants
    Parameterizes initial cluster density via Eq. (7) from Brown & Gnedin (2021); it is the dominant driver of IMBH formation efficiency in the cosmic model (Fig. 18).
  • TDE accretion fraction = 0.5
    Section 2.1.4 and 3.2.2: the BH accretes 50% of a tidally disrupted star; the authors note this is high and that BH masses grown by micro-TDEs should be treated as upper limits.
axioms (6)
  • standard math Newtonian gravity with PN corrections up to 3.5PN for BH-BH interactions
    Section 2.1.1; standard gravitational dynamics, no new physics assumed.
  • domain assumption PARSEC/SEVN stellar tracks with overshooting lambda=0.4; binary stars evolve as single stars in this code branch
    Section 2.1.3; the SEVN binary stellar evolution coupling is absent, which affects binary merger histories and mass transfer.
  • domain assumption Young clusters are initially gas-free Plummer spheres with Kroupa IMF, coeval ZAMS stars, and primordial binaries; no gas accretion onto collision products
    Section 2.3; gas is known to affect collision rates and SMS radii (Reinoso et al. 2023; Fujii et al. 2024) and is omitted here.
  • ad hoc to paper Collisional mass loss follows Eq. (2) with C=0.3 and removes mass from both primary and secondary stars
    Section 2.2.1; chosen from Glebbeek et al. (2013); the paper itself cites newer results suggesting possible catastrophic mass loss for extended envelopes.
  • ad hoc to paper Vink (2018) wind-loss rates are extrapolated above 900 Msun with no upper limit
    Section 2.2.3; this is the central sensitivity: Appendix C shows an LBV-like cap changes final masses from about 7000 to above 25000 Msun.
  • domain assumption Cosmic rate model assumes Chruslinska et al. (2025) SFRD(Z), cluster formation efficiency Gamma=0.3, -2 power-law cluster mass function, and f_h-based birth mass-size relation
    Section 6.1; these external assumptions, especially birth densities, dominate the volumetric IMBH formation rate estimates.

pith-pipeline@v1.3.0-alltime-deepseek · 56340 in / 14861 out tokens · 138599 ms · 2026-08-03T10:58:37.960362+00:00 · methodology

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read the original abstract

We explore the formation of intermediate mass black holes (IMBHs), potential seeds for supermassive black holes (SMBHs), via runaway stellar collisions for a wide range of star cluster (surface) densities ($4\times10^3 M_\odot$ pc$^{-2} \lesssim \Sigma_\mathrm{h} \lesssim 4\times10^6 M_\odot$ pc$^{-2}$) and metallicities $(0.01 Z_\odot \lesssim Z \lesssim 1.0 Z_\odot)$. Our sample of isolated (>1400) and hierarchical (30) simulations of young, massive star clusters with up to $N=1.8\times10^6$ stars includes collisional stellar dynamics, stellar evolution, and post-Newtonian equations of motion for black holes using the BIFROST code. High stellar wind rates suppress IMBH formation at high metallicities ($Z\gtrsim0.2 Z_\odot$) and low collision rates prevent their formation at low densities ($\Sigma_\mathrm{h}\lesssim 3\times10^4 M_\odot$ pc$^{-2}$). The assumptions about stellar wind loss rates strongly affect the maximum final IMBH masses ($M_\bullet\sim 6000 M_\odot$ vs. $25000 M_\odot$). The total stellar mass loss from collisions and collisionally boosted winds before $t=3$ Myr can together reach up to $5$--$10\%$ of the final cluster mass. We present fitting formulae for IMBH masses as a function of host star cluster $\Sigma_\mathrm{h}$ and $Z$ which can be used to seed SMBHs in high resolution cosmological hydrodynamical simulations and in semi-analytic models for galaxy formation. Our results favour IMBH formation in dense low metallicity environments similar to $z\sim10$ James Webb Space Telescope (\textit{JWST}) proto globular clusters. IMBH formation is suppressed in the high metallicity and low density conditions of the local Universe.

Figures

Figures reproduced from arXiv: 2601.07917 by Antti Rantala, Basti\'an Reinoso, Klaus Reuter, Markus Rampp, Martyna Chru\'sli\'nska, Natalia Lah\'en, Thorsten Naab.

Figure 1
Figure 1. Figure 1: A schematic illustration of the initial conditions for the hierarchi￾cally assembling cluster setups. 2.3 Initial conditions 2.3.1 Isolated models We construct idealised, isolated, spherically symmetric star clus￾ter models following the Plummer (1911) density profile. For the isolated models we use three different cluster masses of 𝑀cl = 2.3 × 104 M⊙, 𝑀cl = 5.9 × 104 M⊙ and 𝑀cl = 1.5 × 105 M⊙ cor￾respondi… view at source ↗
Figure 2
Figure 2. Figure 2: The mass growth histories of selected massive stars and IMBHs in the isolated dense star cluster models IM3D5Z1–9 with 𝑀cl = 1.5 × 105 and Σh = 2.9 × 105 M⊙ pc−2 . Each line shows the growth history of a star that reached the highest mass in its host cluster during the simulation before collapsing into an IMBH. In a number of models the massive star is disrupted by a stellar BH or an IMBH, or the IMBH is e… view at source ↗
Figure 3
Figure 3. Figure 3: The fraction 𝑓IMBH of star clusters forming an IMBH with 𝑀• > 300 M⊙ via runaway collisions in the isolated setups and retain￾ing it until 𝑡 = 7.5 Myr with different densities and metallicities. In the models 𝑀cl = 1.5 × 105 M⊙. The IMBH fraction sensitively depends on both cluster density and metallicity. creasingly high cluster densities, the fraction of clusters with the runaway IMBH formation rapidly i… view at source ↗
Figure 4
Figure 4. Figure 4: Furthermore, we display the IMBH masses from models with different metallicities and surface densities in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The maximum IMBH masses 𝑀• in the simulations with 𝑀cl = 1.5 × 105 M⊙ from [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The most massive stellar BHs and IMBHs in the isolated simulations at 𝑡 = 7.5 Myr at different metallicities as a function of the cluster half mass surface density Σh. The cluster masses 2.3 × 104 M⊙ ≲ 𝑀• ≲ 1.5 × 105 M⊙ are indicated by the increasingly dark shaded symbols. The maximum stellar BH masses typically lie close to the lower edge of the (P)PISN mass gap at each different metallicity. At high sur… view at source ↗
Figure 7
Figure 7. Figure 7: The LOESS smoothed map presenting the maximum IMBH mass as a function of their host star cluster half mass surface density Σh and metallicity 𝑍. The contours for 𝑀• = 100 M⊙ and 𝑀• = 1000 M⊙ show very regular angular shapes. Stellar BHs occupy the parameter space towards lower surface densities and higher metallicities from the critical contour of 𝑀• = 100 M⊙ while increasingly massive IMBHs are found towa… view at source ↗
Figure 8
Figure 8. Figure 8: The parametrized model for IMBH masses as a function of their host cluster half mass surface densities Σh and metallicities 𝑍. For the illustration we use Eq. (9) below log10 (Σh/ M⊙ pc−2 ) < 5.45, and Eq. (10) above it. population synthesis model (Hurley et al. 2000, 2002; Iorio et al. 2023) instead. Our aim is to find a relatively simple parametrized functional form for 𝑀• (Σh, 𝑍) which remains well beha… view at source ↗
Figure 9
Figure 9. Figure 9: The mass growth and loss histories of six most massive stars formed in our densest hierarchical models HD9Z1 (𝑍 = 0.01 Z⊙) and HD9Z2 (𝑍 = 0.10 Z⊙). First panel: the masses of the stars as function of time. Second panel: the cumulative gained mass by the stars i.e. the sum of secondary impacting stars 𝑚2. Third and fourth panels: the cumulative mass lost in collisions and stellar winds. While the gained mas… view at source ↗
Figure 10
Figure 10. Figure 10: The rates for gained (dots) and total lost mass (solid lines) from the most massive stars in our densest hierarchical models with 𝑍 = 0.10 Z⊙ (HD9Z2) and 𝑍 = 0.01 Z⊙ (HD9Z1). While the collisional mass gain rates at different metallicities are comparable, the total mass loss rates peak and exceed the mass gain rates earlier, and are in general higher in the models with 𝑍 = 0.10 Z⊙. This is mainly due to t… view at source ↗
Figure 11
Figure 11. Figure 11: , we further include the primary mass loss in stellar collisions in the simple post-processed model. The inclusion of collisional mass loss from the primary in the models only somewhat decreases the final stellar masses at 𝑡 = 4 Myr: without the collisional mass loss the final masses are 15-16% higher. Finally, the radii of extremely massive stars in the models included in NBODY6++GPU, MOCCA and BIFROST c… view at source ↗
Figure 12
Figure 12. Figure 12: The 3D stellar density profiles (solid lines) of the assembling massive central star cluster at 𝑡 = 7.5 Myr. We show for comparison the medium density low metallicity model HB-150 at 𝑡 = 10 Myr from our previous study (Rantala et al. 2025) as the dotted line. Initially denser models result in higher final central densities, as expected, however the central densities decrease in our densest models during t… view at source ↗
Figure 13
Figure 13. Figure 13: The fraction of destroyed binary systems (square symbols) and potential supernova progenitors (plus symbols) as a function of the host star cluster surface density. Up to ∼ 40% of all supernova progenitors (cross symbols) can experience a stellar collision in the densest clusters. At 𝑡 = 7.5 Myr, the centres of the low density 𝑍 = 0.10 Z⊙ setups HD1Z2–HD3Z2 do not contain any IMBHs. For the interme￾diate … view at source ↗
Figure 14
Figure 14. Figure 14: Top panel: the IMBH TDE rates ΓTDE in the hierarchical models HD1Z2–HD9Z2. In the densest models the TDE rates ΓTDE > 10−5 yr−1 can be sustained from the IMBH formation until the end of the simulations at 𝑡 = 7.5 Myr. Bottom panel: the rates of VMS and EMS collisions in the hierarchical models. In the densest models the maximum collision rates reach Γcoll ∼ 2 × 10−3 yr−1 . and denser) the collision rates … view at source ↗
Figure 15
Figure 15. Figure 15: The cumulative mass loss rates from the stars through winds (left panel) and by collisional ejecta (right panel) in the hierarchical setups of different initial densities. The nine models with 𝑍 = 0.10 Z⊙ are displayed using the solid line while the densest comparison setup with 𝑍 = 0.01 Z⊙ is shown as the dotted line. At 0.10 Z⊙ the total wind losses in the models are very similar at different densities … view at source ↗
Figure 16
Figure 16. Figure 16: The total mass losses in winds and stellar collisions per initial stellar mass 𝑚loss/𝑀cl,0 as a function of the central cluster surface density Σh in the models with 𝑍 = 0.10 Z⊙. In low density models up to Σh ∼ 1– 2 × 105 M⊙ pc−2 the wind losses are relatively constant (𝑚loss/𝑀cl,0 ∼ 7 × 10−3 ) after which the total losses rise due to extremely massive star formation at high cluster densities. The collis… view at source ↗
Figure 18
Figure 18. Figure 18: The IMBH formation efficiency 𝜒IMBH per unit of star formation as a function of metallicity 𝑍 calculated using the model from Section 6. The IMBH formation efficiency, but not its dependence on metallicity, sensitively depends on the normalization of the cluster mass radius relation 𝑓h. For a fixed 𝑓h, 𝜒IMBH remains at low metallicities constant up to 𝑍 = 0.1 Z⊙ above which it steadily declines by a facto… view at source ↗
Figure 19
Figure 19. Figure 19: The cosmic formation rate densities of IMBHs RIMBH. Left panel: the case of the fixed non-evolving normalization 𝑓h of the star cluster birth mass radius relation. The RIMBH sensitively depends on the initial cluster densities through 𝑓h. With 𝑓h = 1/8, max RIMBH ∼ 10−7 yr−1 cMpc−3 , and the maximum RIMBH declines over by and order of magnitude per a factor of 2 in 𝑓h. The overall RIMBH history follows th… view at source ↗
Figure 20
Figure 20. Figure 20: The total volumetric integrated number density of IMBHs 𝑛IMBH (< 𝑧) formed below redshift 𝑧. In all the shown models 𝑓h (𝑧 = 0) = 1/8 while we display model variants with increasingly dense clusters forming at high redshifts. For a fixed cluster mass size relation (thin solid line), 𝑛IMBH ∼ 804 Mpc−3 IMBHs form below 𝑧 < 15, while for the model with the densest early clusters we have 𝑛IMBH ∼ 1465 Mpc−3 . … view at source ↗
Figure 21
Figure 21. Figure 21: The cumulative IMBH mass function formed in an average volume of space. The IMBH mass function (solid lines) sensitively depends on the cluster mass radius relation normalization 𝑓h. Without IMBH-BH GW kicks, the model variants follow a Schechter like function with a power law shape at 𝑀• < 1000 M⊙ followed by a sharp exponential decline. Including GW kicks in the models (dot-dashed lines) efficiently sup… view at source ↗

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Reference graph

Works this paper leans on

3 extracted references · 1 linked inside Pith · cited by 5 Pith papers

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    2024

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    fast"("slow

    The low (high) model variation adopts (i) the galaxy main sequence from Popesso et al. (2023) with a loga- rithmic slope aSFR = 1 (aSFR = 0.8) at low masses, (ii) the galaxy mass - gas-phase (oxygen) metallicity relation with a high (low) normalization (as given in Table C.1 in Chruślińska et al. 2025), (iii) the oxygen-to-iron abundance ratio - specific ...

    2023