Handy Relation between Binary Black Hole Merger Times and Host Galaxy Properties

Abstract Over the past 15 yr, the evidence has clearly demonstrated that massive black hole (MBH) binary merger timescales depend strongly on the structural and kinematic properties of their host galaxy. Stellar density, gas content, shape, and kinematics all play a role, combining in nonlinear ways to affect the evolution of the binary. The binary properties themselves—such as the eccentricity, mass ratio, and orbital plane—all matter as well. This makes it nontrivial to estimate accurate cosmological MBH binary merger rates or to generate merger-rate ranges that reflect the distribution of galaxy hosts and orbits. We seek to accurately model the few-body stellar scattering phase, as this is the least well understood part of the journey from black holes in separate galaxies to a merged black hole. Using an extensive set of high-resolution, gas-free, direct N -body simulations, in which the shape, structure, and kinematics of each galaxy host are directly informed by observations, we map out MBH binary merger timescales over a range of galaxy hosts and MBH binary orbits. This yields a convenient set of scaling relations to determine MBH binary merger timescales—and the range of merger timescales—as functions of basic observables. Such scaling relations can be readily employed as a subgrid model in cosmological or semianalytic studies—for example, to model event rates for LISA or pulsar timing.