Detailed theoretical predictions of the outskirts of dark matter halos
  In the present work we describe the formalism necessary to derive the properties of dark matter halos beyond two virial radius using the spherical collapse model (without shell crossing), and provide the framework for the theoretical prediction presented in Prada et al. (2005). We show in detail how to obtain within this model the probability distribution for the spherically-averaged enclosed density at any radii P(delta,r). Using this probability distribution, we compute the most probable and mean density profiles, which turns 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. Two probability distributions are obtained: a first one 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,Q) must satisfy the condition that delta_l(q=Q) = delta_vir, where delta_vir is the linear density contrast within the virial radius Rvir at the moment of virialization. Then we introduce an additional constraint to obtain a more accurate P(delta,r) which reproduces to a higher degree of precision the distribution of the spherically averaged enclosed density found in the simulations. This new constraint is delta_l(q,Q) < delta_vir for all q > Q, which means that there are no radii larger than Rvir where the density contrast is larger than that used to define the virial radius. Finally, we compare in detail our theoretical predictions for the probability distributions with the results found in the simulations. 
Juan E. Betancort-Rijo (1), Miguel A. Sanchez-Conde (2), Francisco   Prada (2) and Santiago G. Patiri (1) ((1) Instituto de ...