Ray tracing and Hubble diagrams in post-Newtonian cosmology
  On small scales the observable Universe is highly inhomogeneous, with
galaxies and clusters forming a complex web of voids and filaments. The optical
properties of such configurations can be quite different from the perfectly
smooth Friedmann-Lema\^itre-Robertson-Walker (FLRW) solutions that are
frequently used in cosmology, and must be well understood if we are to make
precise inferences about fundamental physics from cosmological observations. We
investigate this problem by calculating redshifts and luminosity distances
within a class of cosmological models that are constructed explicitly in order
to allow for large density contrasts on small scales. Our study of optics is
then achieved by propagating one hundred thousand null geodesics through such
space-times, with matter arranged in either compact opaque objects or diffuse
transparent haloes. We find that in the absence of opaque objects, the mean of
our ray tracing results faithfully reproduces the expectations from FLRW
cosmology. When opaque objects with sizes similar to those of galactic bulges
are introduced, however, we find that the mean of distance measures can be
shifted up from FLRW predictions by as much as $10\%$. This bias is due to the
viable photon trajectories being restricted by the presence of the opaque
objects, which means that they cannot probe the regions of space-time with the
highest curvature. It corresponds to a positive bias of order $10\%$ in the
estimation of $\Omega_{\Lambda}$ and highlights the important consequences that
astronomical selection effects can have on cosmological observables.
