Geometry of the generical cosmological solution before the singularity limit

Abstract: The generic cosmological solution is analyzed both for the non-asymptotic limit to the cosmological singularity and in the asymptotic limit analytically. The Bianchi I solution and the Bianchi IX solution, described as a sequence of Bianchi I reparameterized solutions, are analyzed with respect to the asymptotic symmetry implied by the space part of the metric tensor. Numerical studies are explained. The semiclassical regime is proposed by using the degrees of freedom for the initial conditions to the Einstein field equations, i.e. those which are not necessarily characterizing for a Bianchi scheme. The appropriate homegeneization techniques and the de-homogenization techniques referred to a generic system of PDE's are discussed and applied to the affine (Misner) space containing the dynamics pertinent to the Hamiltonian problem associated to the solution of the Hamiltonian constraint. The asymptotic limit to the cosmological singularity for the generic cosmological solution is implemented within the asymptotic Kasner parametrization of the BKL approach and for the Misner-Chitre formalism. The EFE in the Misner-Chitre approach imply constraints on the non-asymptotical degrees of freedom, which allow one to define classes of solutions on the anisotropy plane.