A derivation of (half) the dark matter distribution function
  All dark matter structures appear to follow a set of universalities, such as
phase-space density or velocity anisotropy profiles, however, the origin of
these universalities remains a mystery. Any equilibrated dark matter structure
can be fully described by two functions, namely the radial and the tangential
velocity distribution functions (VDF), and when we will understand these two
then we will understand all the observed universalities. Here we demonstrate
that if we know the radial VDF, then we can derive and understand the
tangential VDF. This is based on simple dynamical arguments about properties of
collisionless systems. We use a range of controlled numerical simulations to
demonstrate the accuracy of this result. We therefore boil the question of the
dark matter structural properties down to understanding the radial VDF.
