Relaxing and virializing a dark matter halo

Navarro, Frenk & White have suggested that the density profiles of simulated dark matter haloes have a 'universal' shape, so that a given halo can be characterized by a single free parameter which fixes its mass. In this paper we revisit the spherical infall model in the hope of recognizing in detail the existence and origin of any such universality. A system of particles is followed from linear perturbation, through first shell crossing, then through an accretion or infall phase, and finally to virialization. During the accretion phase, the system relaxes through a combination of phase mixing, phase-space instability, and moderate violent relaxation. It is driven quickly, by the flow of mass through its surface, toward self-similar evolution. The selfsimilar solution plays its usual role of intermediate attractor, and can be recognized from a virial-type theorem in scaled variables and from our numerical simulations. The transition to final equilibrium state once infall has ceased is relatively gentle, an observation which leads to an approximate form for the distribution function of the final object. The infall phase fixes the density profile in intermediate regions of the halo to be close to r(-2). We make contact with the standard hierarchical clustering scenario and explain how modifications of the self-similar infall model might lead to density profiles in agreement with those found in cosmological simulations.