The Phase-Space Density Profiles of Cold Dark Matter Halos

We examine the coarse-grained phase-space density profiles of a set of recent, highresolution 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 ∝ r−α, with α ≈ 1.875. This behaviour closely matches the self-similar solution obtained by Bertschinger for secondary infall of gas onto a point-mass perturber in a uniformly expanding universe. On the other hand, the density profile corresponding to Bertschinger’s solution (a power-law of slope r2α−6) differs significantly from the density profiles of CDM halos. CDM halo density profiles are clearly not power laws, and have logarithmic slopes that gradually steepen with radius, roughly as described by Navarro, Frenk & White (NFW). We show that isotropic, spherically-symmetric equilibrium mass distributions with power-law phase-space density profiles form a one-parameter family of structures controlled by the ratio of the local velocity dispersion to the “natural” velocity dispersion at some fiducial radius, r0; κ = 4πGρ(r0)r 2 0 /σ(r0) . For κ = α = 1.875 one recovers the power-law solution ρ ∝ r2α−6. As κ increases, the density profiles become quite complex but still diverge like r2α−6 near the center. For κ larger than some critical value, κcrit(α), solutions become non-physical, leading to negative densities near the center. The critical solution, κ = κcrit, corresponds to the case where the phase-space density distribution is the narrowest compatible with the power-law phase-space density stratification constraint. Over three decades in radius the critical solution is indistinguishable from an NFW profile, although its logarithmic slope asymptotically approaches −2α/5 = −0.75 (rather than −1) at very small radii. 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 spherical infall model but which “mixes” the system maximally, perhaps as a result of repeated merging, leading to a relatively uniform phase-space density distribution across the system. This finding offers intriguing clues as to the origin of the similarity in the structure of dark matter halos formed in hierarchically clustering universes. Subject headings: cosmology: dark matter — cosmology: theory — galaxies: formation — galaxies: structure — methods: analytical — methods: numerical CIAR Scholar and Alfred P. Sloan Research Fellow