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Power Laws and Non-Power Laws in Dark Matter Halos

Simulated dark matter profiles are often modeled as an NFW density profile rather than a single power law. Recently, attention has turned to the rather rigorous power-law behavior exhibited by the "pseudo-phase-space density" of the dark matter halo, which is defined dimensionally in terms of the local density and velocity dispersion of the dark matter particles. The non-power-law behavior of the density profile is generally taken to exclude simple scale-free infall models; however, the power-law behavior of the "pseudodensity" is a counter indication. We argue in this paper that both behaviors may be at least qualitatively understood in terms of a dynamically evolving self-similarity, rather than as the form for self-similar infall that is fixed by cosmological initial conditions. The evolution is likely due to collective relaxation such as that provided by the radial-orbit instability on large scales. We deduce, from a distribution function given by first-order coarse-graining, both the NFW-type density profile and the power-law pseudodensity profile. The results are not greatly sensitive to variations of about 3 in the power of the velocity dispersion used in the definition of the phase-space pseudodensity. We suggest that the power 2 may create the more physical quantity, whose deviations from a power law are a diagnostic of incomplete relaxation.