  We construct high-precision models of the Universe that contain radiation, a
cosmological constant, and periodically distributed inhomogeneous matter. The
density contrasts in these models are allowed to be highly non-linear, and the
cosmological expansion is treated as an emergent phenomenon. This is achieved
by employing a generalised version of the post-Newtonian formalism, and by
joining together inhomogeneous regions of space-time at reflection symmetric
junctions. Using these models, we find general expressions that precisely and
unambiguously quantify the effect of small-scale inhomogeneity on the
large-scale expansion of space (an effect referred to as "back-reaction", in
the literature). We then proceed to specialize our models to the case where the
matter fields are given by a regular array of point-like particles. This allows
us to derive extremely simple expressions for the emergent Friedmann-like
equations that govern the large-scale expansion of space. It is found that the
presence of radiation tends to reduce the magnitude of back-reaction effects,
while the existence of a cosmological constant has only a negligible effect.
