Dark matter haloes and self-similarity

This paper explores the self-similar solutions of the Vlasov-Poisson system and their relation to the gravitational collapse of dynamically cold systems. Analytic solutions are derived for power-law potentials in one dimension, and extensions of these solutions in three dimensions are proposed. Next, the self-similarity of the collapse of cold dynamical systems is investigated numerically. The fold system in phase space is consistent with analytic self-similar solutions, which present all the proper self-similar scaling. An additional point is the appearance of an x(-1/2) law at the centre of the system for initial conditions with power-law index larger than -1/2 (the Binney conjecture). It is found that the first appearance of the x(-1/2) law corresponds to the formation of a singularity very close to the centre. Finally, the general properties of self-similar multidimensional solutions near equilibrium are investigated. Smooth and continuous self-similar solutions have power-law behaviour at equilibrium. However, cold initial conditions result in discontinuous phase-space solutions, and the smoothed phase-space density loses its auto-similar properties. This problem is easily solved by observing that the probability distribution of the phase-space density P is identical except for scaling parameters to the probability distribution of the smoothed phase-space density P-S. As a consequence, P-S inherits the self-similar properties of P. This particular property is at the origin of the universal power law observed in numerical simulation for rho/sigma(3). The self-similar properties of P-S imply that other quantities should also have a universal power-law behaviour with predictable exponents. This hypothesis is tested using a numerical model of the phase-space density of cold dark matter haloes, and excellent agreement is obtained.