Interaction of massive black hole binaries with their stellar environment. III. Scattering of bound stars

We develop a formalism for studying the dynamics of massive black hole binaries embedded in gravitationally bound stellar cusps, and study the binary orbital decay by three-body interactions, the impact of stellar slingshots on the density profile of the inner cusp, and the properties of the ejected hypervelocity stars (HVSs). We find that the scattering of bound stars shrinks the binary orbit and increases its eccentricity more effectively than that of unbound ambient stars. Binaries with initial eccentricities e ≳ 0.3 and/or unequal-mass companions (M2/M1 ≲ 0.1) can decay by three-body interactions to the gravitational wave emission regime in less than a Hubble time. The stellar cusp is significantly eroded, and cores as shallow as ρ ∝ r-07 may develop from a preexisting singular isothermal density profile. A population of HVSs is ejected in the host galaxy halo, with a total mass ∼M2. We scale our results to the scattering of stars bound to Sgr A*, the massive black hole in the Galactic center, by an inspiraling companion of intermediate mass. Depending on binary mass ratio, eccentricity, and initial slope of the stellar cusp, a core of radius ∼0.1 pc typically forms in 1-10 Myr. On this timescale about 500-2500 HVSs are expelled with speeds sufficiently large to escape the gravitational potential of the Milky Way.