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 σ∼v^-a, the gas again forms a power-law cusp, but now the SIDM density scales as r^-β, where β=(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.