Post-Newtonian cosmological models
  We construct a framework to probe the effect of non-linear structure
formation on the large-scale expansion of the universe. We take a bottom-up
approach to cosmological modelling by splitting our universe into cells. The
matter content within each cell is described by the post-Newtonian formalism.
We assume that most of the cell is in the vicinity of weak gravitational
fields, so that it can be described using a perturbed Minkowski metric. Our
cells are patched together using the Israel junction conditions. We impose
reflection symmetry across the boundary of these cells. This allows us to
calculate the equation of motion for the boundary of the cell and, hence, the
expansion rate of the universe. At Newtonian order, we recover the standard
Friedmann-like equations. At post-Newtonian orders, we obtain a correction to
the large-scale expansion of the universe. Our framework does not depend on the
process of averaging in cosmology. As an example, we use this framework to
investigate the cosmological evolution of a large number of regularly arranged
point-like masses. At Newtonian order, the Friedmann-like equations take the
form of dust and spatial curvature. At post-Newtonian orders, we get
corrections to the dust term and we get an additional term that takes the same
form as radiation. The radiation-like term is a result of the non-linearity of
Einstein's equations, and is due to the inhomogeneity present in our model.
