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) {\em after} central relaxation. Our results can be
approximated by simple analytic scalings, with $M_{\rm IMBH} \propto v_{\rm
cl}^{3/2}$ where $v_{\rm cl}^{2} = G\,M_{\rm cl}/r_{\rm h}$ is the circular
velocity in terms of initial cluster mass $M_{\rm cl}$ and half-mass radius
$r_{\rm h}$. While this suggests IMBH formation is {\em possible} even in
typical clusters, we show that predicted IMBH masses for these systems are
small, $\sim 100-1000\,M_{\odot}$ or $\sim 0.0003\,M_{\rm cl}$, below even the
most conservative observational upper limits in all known cases. The IMBH mass
could reach $\gtrsim 10^{4}\,M_{\odot}$ 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.
