Scattering experiments meet N-body - I. A practical recipe for the evolution of massive black hole binaries in stellar environments

The apparent N-independence observed in the evolution of massive black hole binaries (MBHBs) in recent simulation of merging stellar bulges suggests a simple interpretation beyond complex time-dependent relaxation processes. We conjecture that the MBHB hardening rate is roughly equivalent to that of a binary immersed in a field of unbound stars with density ρ and typical velocity dispersion σ, provided that ρ and σ are the stellar density and the velocity dispersion at the influence radius of the MBHB. By comparing direct N-body simulations to a hybrid model based on three-body scattering experiments, we verify that the hardening rates matches reasonably well (within 30 per cent) in the two cases. This result is particularly practical because it provides an approximate estimate the lifetime of MBHBs forming in dry mergers based solely on the stellar density profile of the host galaxy. We briefly discuss some implications of our finding for the gravitational wave signal observable by pulsar timing arrays and for the expected population of MBHBs lurking in massive ellipticals.