Post-Newtonian cosmological dynamics of plane-parallel perturbations and
  back-reaction
  We study the general relativistic non-linear dynamics of self-gravitating
irrotational dust in a cosmological setting, adopting the comoving and
synchronous gauge, where all the equations can be written in terms of the
metric tensor of spatial hyper-surfaces orthogonal to the fluid flow.
Performing an expansion in inverse powers of the speed of light, we obtain the
post-Newtonian equations, which yield the lowest-order relativistic effects
arising during the non-linear evolution. We then specialize our analysis to
globally plane-parallel configurations, i.e. to the case where the initial
perturbation field depends on a single coordinate. The leading order of our
expansion, corresponding to the "Newtonian background", is the Zel'dovich
approximation, which, for plane-parallel perturbations in the Newtonian limit,
represents an exact solution. This allows us to find the exact analytical form
for the post-Newtonian metric, thereby providing the post-Newtonian extension
of the Zel'dovich solution: this accounts for some relativistic effects, such
as the non-Gaussianity of primordial perturbations. An application of our
solution in the context of the back-reaction proposal is eventually given,
providing a post-Newtonian estimation of kinematical back-reaction, mean
spatial curvature and average scale-factor.
