Black hole growth through hierarchical black hole mergers in dense star
  clusters: implications for gravitational wave detections
  In a star cluster with a sufficiently large escape velocity, black holes
(BHs) that are produced by BH mergers can be retained, dynamically form new BH
binaries, and merge again. This process can repeat several times and lead to
significant mass growth. In this paper, we calculate the mass of the largest BH
that can be formed through repeated mergers of stellar seed BHs and determine
how its value depends on the physical properties of the host cluster. We adopt
an analytical model in which the energy generated by the black hole binaries in
the cluster core is assumed to be regulated by the process of two-body
relaxation in the bulk of the system. This principle is used to compute the
hardening rate of the binaries and to relate this to the time-dependent global
properties of the parent cluster. We demonstrate that in clusters with initial
escape velocity $\gtrsim 300\rm km\ s^{-1}$ in the core and density $\gtrsim
10^5\ M_\odot\rm pc^{-3}$, repeated mergers lead to the formation of BHs in the
mass range $100-10^5 \,M_\odot$, populating any upper mass gap created by
pair-instability supernovae. This result is independent of cluster metallicity
and the initial BH spin distribution. We show that about $10\%$ of the
present-day nuclear star clusters meet these extreme conditions, and estimate
that BH binary mergers with total mass $\gtrsim 100\,M _\odot$ should be
produced in these systems at a maximum rate $\approx 0.05 \,\rm Gpc^{-3}
yr^{-1}$, corresponding to one detectable event every few years with Advanced
LIGO/VIRGO at design sensitivity. The contribution of globular clusters is
likely to be negligible instead because the first BH merger remnant escapes
following the relativistic kick. A possible connection of our results to the
formation of massive BH seeds in galaxy nuclei and globular clusters is
discussed.
