The Phase-Space Density Profiles of Cold Dark Matter Halos
We examine the coarse-grained phase-space density profiles of a set of recent, high-resolution simulations of galaxy-sized Cold Dark Matter (CDM) halos. Over two and a half decades in radius the phase-space density closely follows a power-law, $ρ/σ^3 \propto r^{-α}$, with $α= 1.875$. This behaviour matches the self-similar solution obtained by Bertschinger for secondary infall in a uniformly expanding universe. On the other hand, the density profile corresponding to Bertschinger's solution (a power-law of slope $r^{2α-6}$) differs significantly from the density profiles of CDM halos. We show that isotropic mass distributions with power-law phase-space density profiles form a one-parameter family of structures controlled by $κ$, the ratio of the velocity dispersion to the peak circular velocity. For $κ=α=1.875$ one recovers the power-law solution $ρ\propto r^{2α-6}$. For $κ$ larger than some critical value, $κ_{cr}$, solutions become non-physical, leading to negative densities near the center. The critical solution, $κ=κ_{cr}$, has the narrowest phase-space density distribution compatible with the power-law phase-space density stratification constraint. Over three decades in radius the critical solution is indistinguishable from an NFW profile. Our results thus suggest that the NFW profile is the result of a hierarchical assembly process that preserves the phase-space stratification of Bertschinger's infall model but which ``mixes'' the system maximally, perhaps as a result of repeated merging.