Dark Matter Velocity Distributions: Comparing Numerical Simulations to Analytic Results

We test the consistency of dark matter velocity distributions obtained from dark matter-only numerical simulations with analytic predictions, using the publicly available Via Lactea 2 dataset as an example. We find that, well inside the scale radius, the velocity distribution obtained from numerical simulation is consistent with a function of a single integral of motion -- the energy -- and moreover is consistent with the result obtained from Eddington inversion. This indicates that the assumptions underlying the analytic result, namely, spherical symmetry, isotropy, and a static potential, are sufficiently accurate to govern the coarse properties of the velocity distribution in the inner regions of the halo. We discuss implications for the behavior of the high-velocity tail of the distribution, which can dominate dark matter annihilation from a $p$- or $d$-wave state.