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

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 $(Z \gtrsim 0.2 Z_\odot)$ 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, 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.