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, rho/sigma (3) proportional to r(-alpha), with alpha approximate to 1.875. This behavior 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 r(2 alpha -6)) differs significantly from the density profiles of CDM halos. CDM halo density profiles are clearly not power laws, and they have logarithmic slopes that steepen gradually 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 r(o); k = 4 piG rho (r(o))r(o)(2)/sigma (2)(r(o)). For k = alpha = 1.875, one recovers the power-law solution rho proportional to r(2 alpha -6). As k increases, the density profiles become quite complex but still diverge as r(2 alpha -6) near the center. For k larger than some critical value k(crit)(alpha), solutions become nonphysical, leading to negative densities near the center. The critical solution, k = k(crit), 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 follows closely an NFW profile, although its logarithmic slope asymptotically approaches -2 alpha /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 "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.