Dynamical evolution of stellar-mass black holes in dense stellar clusters: estimate for merger rate of binary black holes originating from globular clusters

We have performed N-body simulations of globular clusters (GCs) in order to estimate a detection rate of mergers of Binary stellar-mass Black Holes (BBHs) by means of gravitational wave (GW) observatories. For our estimate, we have only considered mergers of BBHs which escape from GCs (BBH escapers). BBH escapers merge more quickly than BBHs inside GCs because of their small semi-major axes. N-body simulation can not deal with a GC with the number of stars N ~ 10^6 due to its high calculation cost. We have simulated dynamical evolution of small-N clusters (10^4 <~ N <~ 10^5), and have extrapolated our simulation results to large-N clusters. From our simulation results, we have found the following dependence of BBH properties on N. BBHs escape from a cluster at each two-body relaxation time at a rate proportional to N. Semi-major axes of BBH escapers are inversely proportional to N, if initial mass densities of clusters are fixed. Eccentricities, primary masses, and mass ratios of BBH escapers are independent of N. Using this dependence of BBH properties, we have artificially generated a population of BBH escapers from a GC with N ~ 10^6, and have estimated a detection rate of mergers of BBH escapers by next-generation GW observatories. We have assumed that all the GCs are formed 10 or 12Gyrs ago with their initial numbers of stars N_i=5 x 10^5 -- 2 x 10^6 and their initial stellar mass densities inside their half-mass radii \rho_h,i=6 x 10^3 -- 10^6M_sun pc^-3. Then, the detection rate of BBH escapers is 0.5 -- 20 yr^-1 for a BH retention fraction R_BH=0.5. A few BBH escapers are components of hierarchical triple systems, although we do not consider secular perturbation on such BBH escapers for our estimate. Our simulations have shown that BHs are still inside some of GCs at the present day. These BHs may marginally contribute to BBH detection.