Evolution of the DM distribution function in the density spikes around PBHs

Abstract: At the cosmological stage of radiation dominance, dark matter density spikes should form around primordial black holes. In the case when dark matter particles are able to annihilate, the density in the central regions of the spikes decreases due to the elimination of particles, which gives an upper bound on the central density. In this paper, the modification of the central density profile is investigated, taking into account the distribution of the particle orbits. The orbits in spike around a primordial black hole are very elongated, almost radial, and the angular momentum distribution has an exponential form. For such an initial distribution function, it is obtained that a cusp with the exponent $\approx-0.7$ is formed in the central region, instead of an annihilation plateau. The presence of the cusp provides some correction to the rate of dark matter annihilation around primordial black holes.