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 } \propto r ^ { - \alpha } , with \alpha \approx 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 r ^ { 2 \alpha - 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 , r _ { 0 } ; \kappa = 4 \pi G \rho ( r _ { 0 } ) r _ { 0 } ^ { 2 } / \sigma ( r _ { 0 } ) ^ { 2 } .
For \kappa = \alpha = 1.875 one recovers the power-law solution \rho \propto r ^ { 2 \alpha - 6 } .
As \kappa increases , the density profiles become quite complex but still diverge like r ^ { 2 \alpha - 6 } near the center .
For \kappa larger than some critical value , \kappa _ { crit } ( \alpha ) , solutions become non-physical , leading to negative densities near the center .
The critical solution , \kappa = \kappa _ { 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 is indistinguishable from 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 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 .