Phase Space Distribution Functions and Energy Distributions of Dark Matter Particles in Haloes

We calculate the phase space distribution function (DF) and the energy distribution of dark matter particles for a spherical halo in dynamical equilibrium assuming the Navarro-Frenk-White (NFW) density profile. Comparing the results with simulations of a wide range of haloes, we find that with appropriate matching, the energy distribution for a simulated halo can be well described by that derived from the best-fit NFW profile. Deviations occur at low energy when the NFW profile provides a poor fit for r<0.05R_vir, where R_vir is the virial radius. The comparisons of DFs are similar to those of energy distributions, but the DF derived from the best-fit NFW profile has somewhat less accuracy because additional deviations are introduced through the density of energy states. We also compare the NFW fits to the simulated DFs and energy distributions with the DarkEXP fits of Hjorth Williams (arXiv:1010.0265). We find that these fits have comparable accuracy in the region where both fit well, and that there is an approximate relation between the energy scale of the DarkEXP fits and the parameters of the NFW profile. The DarkEXP fits are better at low energy because they require the central gravitational potential as an input.