Is the Radial Profile of the Phase-Space Density of Dark Matter Halos a
  Power-Law?
  The latest cosmological N-body simulations find two intriguing properties for
dark matter haloes: (1) their radial density profile, rho, 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, rho/sigma3, 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 rho/sigma3 over 12 orders of magnitude
in radius by solving the Jeans equation for a broad range of input rho and
velocity anisotropy beta. Independent of beta, we find that rho/sigma3 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 rho/sigma3
deviates significantly from a power-law below this scale for both the Einasto
and NFW rho. The same conclusion also applies to a more general
density-velocity relation rho/sigma_D^epsilon. Conversely, when we enforce
rho/sigma^3 r^{-eta} as an input, none of the physically allowed rho (occurring
for the narrow range 1.8<eta<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 rho/sigma3 will
deviate from a power law, or rho/sigma3 will continue as a power law and rho
will deviate from its current parameterizations.
