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.