Universal density and velocity distributions of dark matter around massive black holes

The distribution of dark matter at the galactic center, crucial for indirect searches, remains uncertain. In particular, in the vicinity of the massive black hole in the center of a galaxy where indirect signals may be stronger, the density of a dark matter spike may undergo redistribution. Here we calculate the density surrounding Schwarzschild black holes that originate from diverse initial dark halos and estimate the velocity distribution of dark matter particles. By employing a series of Hernquist and power-law initial dark halos, we obtain a fitting formula between dark matter spikes and black hole masses. The Maxwell-Boltzmann distribution is utilized to approximate the velocity distribution of dark matter particles. As an application, taking into account dark matter self-annihilation, we assess the relic densities of dark matter spikes around black holes. We find that the relic spikes for s-wave annihilation are higher than p-wave annihilation, and the relic densities obtained for p-wave annihilation depend on the velocity distribution, varying significantly with distance. The findings shall further provide useful insights for multi-messenger dark matter detections in the future.