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Spherical collapse has turned out to be a successful semi-analytic model tostudy structure formation in different DE models and theories of gravity.Nevertheless, the process of virialization is commonly studied on the basis ofthe virial theorem of classical mechanics. In the present paper, a fullygenerally-relativistic virial theorem based on the Tolman-Oppenheimer-Volkoff(TOV) solution for homogeneous, perfect-fluid spheres is constructed for theEinstein-de Sitter and \Lambda CDM cosmologies. We investigate the accuracy ofclassical virialization studies on cosmological scales and considervirialization from a more fundamental point of view. Throughout, we remainwithin general relativity and the class of FLRW models. The virializationequation is set up and solved numerically for the virial radius, y_{vir}, fromwhich the virial overdensity \Delta_{V} is directly obtained. Leading ordercorrections in the post-Newtonian framework are derived and quantified. Inaddition, problems in the application of this formalism to dynamical DE modelsare pointed out and discussed explicitly. We show that, in the weak fieldlimit, the relative contribution of the leading order terms of thepost-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 morefundamental point of view.

Relativistic virialization in the Spherical Collapse model for Einstein-de Sitter and \Lambda CDM cosmologies