Detailed theoretical predictions for the outskirts of dark matter halos

In the present work we describe the formalism necessary to derive the properties of dark matter halos beyond 2 virial radii using the spherical collapse model ( without shell crossing) and provide the framework for the theoretical prediction presented by Prada et al. We show in detail how to obtain within this model the probability distribution for the spherically averaged enclosed density at any radius P(delta, r). Using this probability distribution, we compute the most probable and the mean density profiles, which turn out to differ considerably from each other. We also show how to obtain the typical profile, as well as the probability distribution and mean profile for the spherically averaged radial velocity. Three probability distributions are obtained: The first is derived using a simple assumption; that is, if Q is the virial radius in Lagrangian coordinates, then the enclosed linear contrast delta(l)(q) must satisfy the condition that delta(l)(q = Q) = delta(vir), where delta(vir) is the linear density contrast within the virial radius R-vir at the moment of virialization. Then we introduce an additional constraint to obtain a more accurate P(delta, r) that reproduces to a higher degree of precision the distribution of the spherically averaged enclosed density found in the simulations. This new constraint is that, for a given q > Q, delta(l)(q) < delta(vir). A third probability distribution, themost accurate, is obtained imposing the strongest constraint that delta(l)(q) < delta(vir) for all q > Q, which means that there are no radii larger than R-vir where the density contrast is larger than that used to define the virial radius. Finally, we compare our theoretical predictions for the mean density and the mean velocity profiles with the results found in the simulations.