Gravitational cooling and density profile near caustics in collisionless dark matter haloes

Cold dark matter haloes are populated by high-density structures with sharply peaked profiles known as caustics, which have not yet been resolved by three-dimensional numerical simulations. Here, we derive semi-analytic expressions for the density profiles near caustics in haloes that form by self-similar accretions of dark matter with infinitesimal velocity dispersion. A simple rescaling shows that, similarly to the case of absolutely cold medium, these profiles are universal: they are valid for all caustics irrespective of the physical parameters of the halo. We derive the maximum density of the caustics and show that it depends on the velocity dispersion and the caustic location. We show that both the absolute and relative thickness of the caustic decrease monotonically towards the centre of the halo while the maximum density grows. This indicates that the radial component of the thermal velocities decreases in the inner streams, i.e. the collisionless medium cools down in the radial direction descending to the centre of the halo. Finally, we demonstrate that there can be a significant contribution to the emission measure from dark matter particle annihilation in the caustics.