Dark-matter distributions around massive black holes: A general relativistic analysis

The cold dark matter at the center of a galaxy will be redistributed by the presence of a massive black hole. The redistribution may be determined using an approach pioneered by Gondolo and Silk: begin with a model distribution function for the dark matter, and ``grow'' the black hole adiabatically, holding the adiabatic invariants of the motion constant. Unlike the approach of Gondolo and Silk, which adopted Newtonian theory together with ad hoc correction factors to mimic general relativistic effects, we carry out the calculation fully relativistically, using the exact Schwarzschild geometry of the black hole. We find that the density of dark matter generically vanishes at $r=2{R}_{\mathrm{S}}$, not $4{R}_{\mathrm{S}}$ as found by Gondolo and Silk, where ${R}_{\mathrm{S}}$ is the Schwarzschild radius, and that the spike very close to the black hole reaches significantly higher densities. We apply the relativistic adiabatic growth framework to obtain the final dark-matter density for both cored and cusped initial distributions. Besides the implications of these results for indirect detection estimates, we show that the gravitational effects of such a dark-matter spike are significantly smaller than the relativistic effects of the black hole, including frame dragging and quadrupolar effects, for stars orbiting close to the black hole that might be candidates for testing the black-hole no-hair theorems.