The phase-space density distribution of dark matter halos

High resolution N-body simulations have all but converged on a common empirical form for the shape of the density profiles of halos, but the full understanding of the underlying physics of halo formation has eluded them so far. We investigate the formation and structure of dark matter halos using analytical and semi-analytical techniques. Our halos are formed via an extended secondary infall model (ESIM); they contain secondary perturbations and hence random tangential and radial motions which affect the halo’s evolution at it undergoes shell-crossing and virialization. Even though the density profiles of NFW and ESIM halos are different their phase-space density distributions are the same: \rho/\sigma^3 ~ r^{-\alpha}, with \alpha=1.875 over ~3 decades in radius. We use two approaches to try to explain this universal'' slope: (1) The Jeans equation analysis yields many insights, however, does not answer why \alpha=1.875. (2) The secondary infall model of the 1960's and 1970's, augmented by thermal motions’’ of particles does predict that halos should have \alpha=1.875. However, this relies on assumptions of spherical symmetry and slow accretion. While for ESIM halos these assumptions are justified, they most certainly break down for simulated halos which forms hierarchically. We speculate that our argument may apply to an ``on-average’’ formation scenario of halos within merger-driven numerical simulations, and thereby explain why \alpha=1.875 for NFW halos. Thus, \rho/\sigma^3 ~ r^{-1.875} may be a generic feature of violent relaxation.