Top-Hat Spherical collapse with clustering dark energy. I. Radius evolution and critical contrast density

Understanding the influence of dark energy on the formation of structures is currently a major challenge in Cosmology, since it can distinguish otherwise degenerated viable models. In this work we consider the Top-Hat Spherical-Collapse (SC) model with dark energy, which can partially (or totally) cluster, according to a free parameter $\gamma$. The {\it lack of} energy conservation has to be taken into account accordingly, as we will show. We determine characteristic quantities for the SC model, such as the critical contrast density and radius evolution, with particular emphasis on their dependence on the clustering parameter $\gamma$.