Is the Radial Profile of the Phase-Space Density of Dark Matter Haloes a Power-Law?

The latest cosmological N -body simulations find two intriguing properties for dark matter haloes: (1) their radial density profile, ρ, is better fit by a form that flattens to a constant at the halo center (the Einasto profile) than the widely-used NFW form; (2) the radial profile of the pseudo-phase-space density, ρ/σ r , on the other hand, continues to be well fit by a power law, as seen in earlier lower-resolution simulations. In this paper we use the Jeans equation to argue that (1) and (2) cannot both be true at all radii. We examine the implied radial dependence of ρ/σ r over 12 orders of magnitude in radius by solving the Jeans equation for a broad range of input ρ and velocity anisotropy β. Independent of β, we find that ρ/σ r is approximately a power law only over the limited range of halo radius resolvable by current simulations (down to ∼ 0.1% of the virial radius), and ρ/σ r deviates significantly from a powerlaw below this scale for both the Einasto and NFW ρ. The same conclusion also applies to a more general density-velocity relation ρ/σ D. Conversely, when we enforce ρ/σ r ∝ r−η as an input, none of the physically allowed ρ (occurring for the narrow range 1.8 . η 6 1.9444) follows the Einasto form. We expect the next-generation simulations with better spatial resolution to settle the debate: either the Einasto profile will continue to hold and ρ/σ r will deviate from a power law, or ρ/σ 3 r will continue as a power law and ρ will deviate from its current parameterizations.