  Cold dark matter haloes are populated by high-density structures with
sharply-peaked profiles known as caustics which have not yet been resolved by
3-dimensional numerical simulations. Here, we derive semi-analytic expressions
for the density profiles near caustics in haloes which 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 and irrespective of
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
monotonically decrease toward the center 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 center 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.
