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

TL;DR

This study investigates the metallicity- and density-dependent formation of intermediate-mass black holes (IMBHs) via runaway stellar collisions in dense star clusters, spanning surface densities from Σh ≈ 4×10^3 to 4×10^6 M⊙ pc^-2 and metallicities from 0.01 to 1 Z⊙. By extending the FROST-CLUSTERS suite with an updated BIFROST code, the authors model collisional growth, wind losses, and post-Newtonian BH dynamics across isolated and hierarchically assembling clusters, and they derive practical IMBH-mass fits M_bh(Σh,Z) plus a LOESS-smoothed map to capture trends. Key findings show IMBHs with M_bh ≈ 300–6000 M⊙ can form in dense, low-metallicity environments (Z ≲ 0.2–0.3 Z⊙); wind losses at higher Z quench growth, shifting the IMBH mass function and reducing formation fractions. The work further develops a cosmic formation-rate model for IMBHs, showing a peak around redshift z ≈ 2–4 and highlighting that about half of IMBHs may form at z ≲ 1.5–3, depending on cluster birth densities; this challenges the view that local IMBHs are predominantly failed high-z SMBH seeds. Overall, the study provides actionable IMBH-mass predictions and a framework to seed IMBHs in cosmological models and semi-analytic galaxy formation codes, while emphasizing the crucial role of EMS/SMS winds and cluster-density evolution in shaping IMBH demographics and their gravitational-wave signatures.

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$^{-2} \lesssim Σ_\mathrm{h}$ $\lesssim 4\times10^6 M_\odot$ 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 $(Z \gtrsim 0.2 Z_\odot)$ and low collision rates prevent their formation at low densities ($Σ_\mathrm{h} \lesssim 3\times10^4 M_\odot$ 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 $Σ_\mathrm{h}$ and Z, and formulate a model for the cosmic IMBH formation rate density. Depending on the cluster birth densities, the IMBH formation rates peak at $z\sim2$-$4$ at up to $\sim10^{-7}$ yr$^{-1}$cMpc$^{-3}$. As more than 50% form below $z\lesssim1.5$-$3$, the model challenges a view in which all local IMBHs are failed early Universe SMBH seeds.

Figures (26)

• Figure 1: A schematic illustration of the initial conditions for the hierarchically assembling cluster setups.
• Figure 2: The mass growth histories of selected massive stars and IMBHs in the isolated dense star cluster models IM3D5Z1--9 with $M_\mathrm{cl} = 1.5\times10^5$ and $\Sigma_\mathrm{h} = {2.9\times10^5}\:\mathrm{M_\odot\:pc^{-2}}$. 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 ejected from its host cluster after merging with a stellar BH due to a GW recoil kick before the end of the simulations at $t=7.5$ Myr. The metallicity increases in each panel starting from $Z={0.01}\:\mathrm{Z_\odot}$ on the top left. When increasing the metallicity from $Z={0.01}\:\mathrm{Z_\odot}$ to $Z={0.10}\:\mathrm{Z_\odot}$, the overall picture of the stellar mass growth remains similar while the maximum IMBH mass decreases from $M_\bullet \sim {1900}\:\mathrm{M_\odot}$ to ${1500}\:\mathrm{M_\odot}$ in the models due to increased wind losses. Above $Z\gtrsim{0.2}\:\mathrm{Z_\odot}$ the picture qualitatively changes as the stronger wind losses can quench the collision cascades, and the massive stars lose substantial amounts of mass before the ends of their lives. At solar metallicity, only few IMBHs can form.
• Figure 3: The fraction $f_\mathrm{IMBH}$ of star clusters forming an IMBH with $M_\bullet > {300}\:\mathrm{M_\odot}$ via runaway collisions in the isolated setups and retaining it until $t=7.5$ Myr with different densities and metallicities. In the models $M_\mathrm{cl}={1.5\times10^5}\:\mathrm{M_\odot}$. The IMBH fraction sensitively depends on both cluster density and metallicity.
• Figure 4: The masses $M_\bullet$ of the most massive stellar BHs and IMBHs in models with $M_\mathrm{cl}={1.5\times10^5}\:\mathrm{M_\odot}$. From the top panel down the host cluster surface density increases from $\Sigma_\mathrm{h}={1.4\times10^4}\:\mathrm{M_\odot}$ to $\Sigma_\mathrm{h}={2.9\times10^5}\:\mathrm{M_\odot}$. In the two models with the lowest densities, no runaway collisions occur and IMBHs rarely form via a single massive binary merger. At $\Sigma_\mathrm{h} \gtrsim {6.3\times10^4}\:\mathrm{M_\odot\:pc^{-2}}$ IMBHs form through stellar collision cascades with $M_\bullet$ increasing with increasing $\Sigma_\mathrm{h}$ especially at low metallicities $Z\lesssim{0.10}\:\mathrm{Z_\odot}$. At higher metallicities the IMBH formation and their masses are suppressed due to the stellar wind mass losses. In the models with highest density, individual stellar BHs may grow into the (P)PISN mass gap via micro-TDEs.
• Figure 5: The maximum IMBH masses $M_\bullet$ in the simulations with $M_\mathrm{cl}={1.5\times10^5}\:\mathrm{M_\odot}$ from Fig. \ref{['fig: Z-Mbh1']}. Above the critical surface density $\Sigma_\mathrm{h} \gtrsim {6.3\times10^4}\:\mathrm{M_\odot\:pc^{-2}}$ IMBHs can form via runaway collision cascades, exceeding $M_\bullet = {10^3}\:\mathrm{M_\odot}$ in the models with $\Sigma_\mathrm{h} = {2.9\times10^5}\:\mathrm{M_\odot\:pc^{-2}}$. At low metallicities $Z \lesssim {0.10}\:\mathrm{Z_\odot}$ the stellar wind losses in the models are relatively weak and do not affect the final IMBH masses. However, above a critical metallicity threshold (e.g. $Z \sim {0.06}\:\mathrm{Z_\odot}$ for ${6.3\times10^4}\:\mathrm{M_\odot\:pc^{-2}} \lesssim \Sigma_\mathrm{h} \lesssim {1.4\times10^5}\:\mathrm{M_\odot\:pc^{-2}}$) the masses of the formed IMBH decline due to the wind mass losses of their progenitor stars before the ends of their lives.