  In this paper we study the evolution of a spherical matter overdensity in the
context of the recently introduced Galileon field theory. Our analysis
considers the complete covariant Lagrangian in four dimensions. This theory is
composed by a potential and a standard kinetic term, a cubic kinetic term and
two additional terms that include the coupling between the Galileon and the
metric, to preserve the original properties of Galileons also in curved
space-times. Here we extend previous studies, which considered both the
quintessence and the cubic terms, by focussing on the role of the last two
terms. The background evolution we consider is driven by a tracker solution.
Studying scalar perturbations in the non-linear regime, we find constraints on
the parameter of the model. We will show how the new terms contribute to the
collapse phase and how they modify physical parameters, such as the linearized
density contrast and the virial overdensity. The results show that the Galileon
modifies substantially the dynamics of the collapse, thus making it possible to
observationally constrain the parameters of this theory.
