Formation of Intermediate-Mass Black Holes through Runaway Collisions in
  the First Star Clusters
  We study the formation of massive black holes in the first star clusters. We
first locate star-forming gas clouds in proto-galactic haloes of $\gtrsim
\!10^7\,{\rm M}_{\odot}$ in cosmological hydrodynamics simulations and use them
to generate the initial conditions for star clusters with masses of $\sim
\!10^5\,{\rm M}_{\odot}$. We then perform a series of direct-tree hybrid
$N$-body simulations to follow runaway stellar collisions in the dense star
clusters. In all the cluster models except one, runaway collisions occur within
a few million years, and the mass of the central, most massive star reaches
$\sim \!400-1900\,{\rm M}_{\odot}$. Such very massive stars collapse to leave
intermediate-mass black holes (IMBHs). The diversity of the final masses may be
attributed to the differences in a few basic properties of the host haloes such
as mass, central gas velocity dispersion, and mean gas density of the central
core. Finally, we derive the IMBH mass to cluster mass ratios, and compare them
with the observed black hole to bulge mass ratios in the present-day Universe.
