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

The 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 equivalent to that of a binary immersed in a field of unbound stars with density $\rho$ and typical velocity $\sigma$, provided that $\rho$ and $\sigma$ are the stellar density and the velocity dispersion at the influence radius of the MBHB. By comparing direct N-body simulations to an hybrid model based on 3-body scattering experiments, we verify this hypothesis: when normalized to the stellar density and velocity dispersion at the binary influence radius, the N-body MBHB hardening rate approximately matches that predicted by 3-body scatterings in the investigated cases. The eccentricity evolution obtained with the two techniques is also in reasonable agreement. This result is particularly practical because it allows to 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.