Recoiling supermassive black hole escape velocities from dark matter haloes

We simulate recoiling black hole trajectories from z = 20 to z = 0 in dark matter haloes, quantifying how parameter choices affect escape velocities. These choices include the strength of dynamical friction, the presence of stars and gas, the accelerating expansion of the Universe (Hubble acceleration), host halo accretion and motion, and seed black hole mass. Lambda cold dark matter halo accretion increases escape velocities by up to 0.6 dex and significantly shortens return time-scales compared to non-accreting cases. Other parameters change orbit damping rates but have subdominant effects on escape velocities; dynamical friction is weak at halo escape velocities, even for extreme parameter values. We present formulae for black hole escape velocities as a function of host halo mass and redshift. Finally, we discuss how these findings affect black hole mass assembly as well as minimum stellar and halo masses necessary to retain supermassive black holes.