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The N-independence observed in the evolution of massive black hole binaries(MBHBs) in recent simulation of merging stellar bulges suggests a simpleinterpretation beyond complex time-dependent relaxation processes. Weconjecture that the MBHB hardening rate is equivalent to that of a binaryimmersed in a field of unbound stars with density $\rho$ and typical velocity$\sigma$, provided that $\rho$ and $\sigma$ are the stellar density and thevelocity dispersion at the influence radius of the MBHB. By comparing directN-body simulations to an hybrid model based on 3-body scattering experiments,we verify this hypothesis: when normalized to the stellar density and velocitydispersion at the binary influence radius, the N-body MBHB hardening rateapproximately matches that predicted by 3-body scatterings in the investigatedcases. The eccentricity evolution obtained with the two techniques is also inreasonable agreement. This result is particularly practical because it allowsto estimate the lifetime of MBHBs forming in dry mergers based solely on thestellar density profile of the host galaxy. We briefly discuss someimplications of our finding for the gravitational wave signal observable bypulsar timing arrays and for the expected population of MBHBs lurking inmassive ellipticals.

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