Coarse-graining the distribution function of cold dark matter II
We study analytically the coarse and fine-grained distribution function established by the self-similar infall of collisionless matter. We find this function explicitly for isotropic and spherically symmetric systems in terms of cosmological initial conditions. The coarse-grained function is structureless and steady but the familiar phase space sheet sub-structure is recovered in the fine-grained limit. By breaking the self-similarity of the halo infall we are able to argue for a central density flattening. In addition there will be an edge steepening. The best fitting analytic density function is likely to be provided by a high order polytrope fit smoothly to an outer power law of index -3 for isolated systems. There may be a transition to a -4 power law in the outer regions of tidally truncated systems. We find a progressive central flattening that is expected to end either in the non-singular isothermal sphere, or in non-singular metastable polytropic cores. Therefore a collisionless system may pass through a family of polytropes of increasing order, finally approaching the limit of the non-singular isothermal sphere, if the `violent' collective relaxation is frequently re-excited by `merger' events. Our results suggest that no physics beyond that of the moderate collective relaxation often known as `violent relaxation' (due we think to wave-particle scattering) is necessary to explain the nature of dark matter density profiles.