Relativistic virialization in the spherical collapse model for Einstein-de Sitter and ΛcDM cosmologies

Spherical collapse has turned out to be a successful semianalytic model to study structure formation in different dark energy 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 general-relativistic virial theorem based on the Tolman-Oppenheimer- Volkoff solution for homogeneous, perfect-fluid spheres is constructed for the Einstein-de Sitter and ΛCDM 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 Friedmann-Lemaître-Robertson- Walker models. The virialization equation is set up and solved numerically for the virial radius, y vir, from which the virial overdensity Δ 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 dark energy 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 and Steinhardt 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. © 2012 American Physical Society.