Self-interacting dark matter cusps around massive black holes

Abstract

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.

11 pages, 2 figures

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• A Theory of Dark Matter
• Secluded WIMP Dark Matter
• Cores in Dwarf Galaxies from Dark Matter with a Yukawa Potential
• Dark matter distributions around massive black holes: A general relativistic analysis
• The loss cone problem in axisymmetric nuclei

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