Relativistic virialization in the Spherical Collapse model for
  Einstein-de Sitter and \Lambda CDM cosmologies
  Spherical collapse has turned out to be a successful semi-analytic model to
study structure formation in different DE models and theories of gravity.
Nevertheless, the process of virialization is commonly studied on the basis of
the virial theorem of classical mechanics. In the present paper, a fully
generally-relativistic virial theorem based on the Tolman-Oppenheimer-Volkoff
(TOV) solution for homogeneous, perfect-fluid spheres is constructed for the
Einstein-de Sitter and \Lambda CDM cosmologies. We investigate the accuracy of
classical virialization studies on cosmological scales and consider
virialization from a more fundamental point of view. Throughout, we remain
within general relativity and the class of FLRW models. The virialization
equation is set up and solved numerically for the virial radius, y_{vir}, from
which the virial overdensity \Delta_{V} is directly obtained. Leading order
corrections in the post-Newtonian framework are derived and quantified. In
addition, problems in the application of this formalism to dynamical DE models
are pointed out and discussed explicitly. We show that, in the weak field
limit, the relative contribution of the leading order terms of the
post-Newtonian expansion are of the order of 10^{-3}% and the solution of Wang
& Steinhardt (1998) is precisely reproduced. Apart from the small corrections,
the method could provide insight into the process of virialization from a more
fundamental point of view.
