Power Laws and Non–Power Laws in Dark Matter Halos

Simulated dark matter profiles are often modelled as a `NFW' density profilerather than a single power law. Recently, attention has turned to the ratherrigorous power-law behaviour exhibited by the `pseudo phase-space density' ofthe dark matter halo, which is defined dimensionally in terms of the localdensity and velocity dispersion of the dark matter particles. The non-power-lawbehaviour of the density profile is generally taken to exclude simplescale-free, in-fall models; however the power-law behaviour of the`pseudo-density' is a counter indication. We argue in this paper that bothbehaviours may be at least qualitatively understood in terms of a dynamicallyevolving self-similarity, rather than the form for self-similar infall that isfixed by cosmological initial conditions. The evolution is likely due tocollective relaxation such as that provided by the radial-orbit instability onlarge scales. We deduce, from a distribution function given by first ordercoarse-graining, both the NFW-type density profile and the power-lawpseudo-density profile. The results are not greatly sensitive to variationabout 3 in the power of the velocity dispersion used in the definition of thephase space pseudo-density. We suggest that the power 2 may create the morephysical quantity, whose deviations from a power-law are a diagnostic ofincomplete relaxation.Comment: 31 pages, 4 figure