POST-NEWTONIAN COSMOLOGY IN THE EULERIAN AND LAGRANGIAN FRAMES

This thesis is devoted to the post-Newtonian (PN) approximation of General Relativity in Cosmology. We analysed the procedure for passing from GR to Newtonian theory in both the Eulerian and the Lagrangian approaches to gravitational dynamics and also the connection between the two. In the GR framework we provided the transformation equations from the Poisson gauge to the synchronous and comoving gauge for the metric in the Newtonian approximation, being completely non-linear in the standard perturbation theory. We fully recovered equations and solution for the metric both in the relativistic perturbation theory up to second order and in the Newtonian limit at any order. We then extend our transformation to the PN approximation and our results are consistent with GR up to second order in perturbation theory.  We specialise the PN Lagrangian dynamics to globally plane-parallel configurations. We obtained the analytical solution of the Einstein field equations. The solution is non-perturbative in the standard sense, exact up to PN order and extends the Zel’dovich approximation which, in turn, is exact for non-linear plane-parallel dynamics in Newtonian gravity. An application of our solution in the context of the back-reaction proposal is eventually given, providing a PN estimation of kinematical back-reaction, mean spatial curvature, average scale-factor and expansion rate.