Self-interacting dark matter cusps around massive black holes
  We adopt the conduction fluid approximation to model the steady-state
distribution of matter around a massive black hole at the center of a weakly
collisional cluster of particles. By "`weakly collisional" we mean a cluster in
which the mean free time between particle collisions is much longer than the
characteristic particle crossing (dynamical) time scale, but shorter than the
cluster lifetime. When applied to a star cluster, we reproduce the familiar
Bahcall-Wolf power-law cusp solution for the stars bound to the black hole.
Here the star density scales with radius as $r^{-7/4}$ and the velocity
dispersion as $r^{-1/2}$ throughout most of the gravitational well of the black
hole. When applied to a relaxed, self-interacting dark matter (SIDM) halo with
a velocity-dependent cross section $\sigma \sim v^{-a}$, the gas again forms a
power-law cusp, but now the SIDM density scales as $r^{-\beta}$, where $\beta =
(a+3)/4$, while its velocity dispersion again varies as $r^{-1/2}$. Results are
obtained first in Newtonian theory and then in full general relativity.
Although the conduction fluid model is a simplification, it provides a
reasonable first approximation to the matter profiles and is much easier to
implement than a full Fokker-Planck treatment or an $N$-body simulation of the
Boltzmann equation with collisional perturbations.
