Role of shell crossing on the existence and stability of trapped matter shells in spherical inhomogeneous Λ -CDM models

We analyze the dynamics of trapped matter shells in spherically symmetric inhomogeneous Λ-CDM models. The investigation uses a generalized Lema\^{}tre-Tolman-Bondi description with initial conditions subject to the constraints of having spatially asymptotic cosmological expansion, initial Hubble-type flow, and a regular initial density distribution. We discuss the effects of shell crossing and use a qualitative description of the local trapped matter shells to explore global properties of the models. Once shell crossing occurs, we find a splitting of the global shells separating expansion from collapse into, at most, two global shells: an inner and an outer limit trapped matter shell. In the case of expanding models, the outer limit trapped matter shell necessarily exists. We also study the role of shear in this process, compare our analysis with the Newtonian framework, and give concrete examples using density profile models of structure formation in cosmology.