The mass budget for intermediate-mass black holes in dense star clusters

Intermediate-mass black holes (IMBHs) could form via runaway merging of massive stars in a young massive star cluster (YMC). We combine a suite of numerical simulations of YMC formation with a semi-analytic model for dynamical friction and merging of massive stars and evolution of a central quasi-star, to predict how final quasi-star and relic IMBH masses scale with cluster properties (and compare with observations). The simulations argue that inner YMC density profiles at formation are steep (approaching isothermal), producing some efficient merging even in clusters with relatively low effective densities, unlike models which assume flat central profiles resembling those of globular clusters (GCs) after central relaxation. Our results can be approximated by simple analytic scalings, with M_(IMBH) ∝ v^(3/2)_(cl) where v²_(cl) = GM_(cl)/r_h is the circular velocity in terms of initial cluster mass M_(cl) and half-mass radius rh. While this suggests IMBH formation is possible even in typical clusters, we show that predicted IMBH masses for these systems are small, ∼100−1000 M_⊙ or ∼0.0003M_(cl), below even the most conservative observational upper limits in all known cases. The IMBH mass could reach ≳ 10⁴ M⊙ in the centers nuclear star clusters, ultra-compact dwarfs, or compact ellipticals, but in all these cases the prediction remains far below the present observed supermassive BH masses in these systems.