Evolution of massive black hole binaries

We present the results of large-scale N-body simulations of the stellar-dynamical evolution of massive black hole binaries at the center of spherical galaxies. We focus on the dependence of the hardening rate on the relaxation timescale of the parent galaxy. A simple theoretical argument predicts that a binary black hole creates a "loss cone" around it. Once the stars in the loss cone are depleted, the hardening rate is determined by the rate at which field stars diffuse into the loss cone. Therefore, the hardening timescale becomes proportional to the relaxation timescale. Recent N-body simulations, however, have failed to confirm this theory, and various explanations have been proposed. By performing simulations with sufficiently large N (up to 10(6)) for sufficiently long time, we found that the hardening rate does indeed depend on N. Our result is consistent with the simple theoretical prediction that the hardening timescale is proportional to the relaxation timescale. This dependence implies that massive black hole binaries are unlikely to merge within a Hubble time through interaction with field stars and gravitational wave radiation alone.