The Effect of Stars on the Dark Matter Spike Around a Black Hole: A Tale of Two Treatments

TL;DR

This paper compares two dynamical models, Fokker-Planck and two-fluid, to understand how stars influence dark matter distribution around black holes, with implications for observable gamma-ray signals.

Contribution

It extends existing models to include flux into the black hole and relativistic effects, providing a detailed comparison of two prevalent dynamical approaches.

Findings

Both methods produce similar density profiles away from boundaries.

Differences are observed in the calculated dark matter accretion rates.

The two-fluid approach is recast as a heated Bondi accretion problem.

Abstract

We revisit the role that gravitational scattering off stars plays in establishing the steady-state distribution of collisionless dark matter (DM) around a massive black hole (BH). This is a physically interesting problem that has potentially observable signatures, such as $\gamma-$rays from DM annihilation in a density spike. The system serves as a laboratory for comparing two different dynamical approaches, both of which have been widely used: a Fokker-Planck treatment and a two-component conduction fluid treatment. In our Fokker-Planck analysis we extend a previous analytic model to account for a nonzero flux of DM particles into the BH, as well as a cut-off in the distribution function near the BH due to relativistic effects or, further out, possible DM annihilation. In our two-fluid analysis, following an approximate analytic treatment, we recast the equations as a "heated Bondi accretion" problem and solve the equations numerically without approximation. While both the Fokker-Planck and two-fluid methods yield basically the same DM density and velocity dispersion profiles away from the boundaries in the spike interior, there are other differences, especially the determination of the DM accretion rate. We discuss limitations of the two treatments, including the assumption of an isotropic velocity dispersion.