On strong mass segregation around a massive black hole: Implications for lower-frequency gravitational-wave astrophysics

We present, for the first time, a clear $N$-body realization of the {\it strong mass segregation} solution for the stellar distribution around a massive black hole. We compare our $N$-body results with those obtained by solving the orbit-averaged Fokker-Planck (FP) equation in energy space. The $N$-body segregation is slightly stronger than in the FP solution, but both confirm the {\it robustness} of the regime of strong segregation when the number fraction of heavy stars is a (realistically) small fraction of the total population. In view of recent observations revealing a dearth of giant stars in the sub-parsec region of the Milky Way, we show that the time scales associated with cusp re-growth are not longer than $(0.1-0.25) \times T_{rlx}(r_h)$. These time scales are shorter than a Hubble time for black holes masses $\mbul \lesssim 4 \times 10^6 M_\odot$ and we conclude that quasi-steady, mass segregated, stellar cusps may be common around MBHs in this mass range. Since EMRI rates scale as $\mbul^{-\alpha}$, with $\alpha \in [1\4,1]$, a good fraction of these events should originate from strongly segregated stellar cusps.