Post-Newtonian cosmological models

Abstract: 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.