Inhomogeneous cosmology in an anisotropic Universe
  With the era of precision cosmology upon us, and upcoming surveys expected to
further improve the precision of our observations below the percent level,
ensuring the accuracy of our theoretical cosmological model is of the utmost
importance. Current tensions between our observations and predictions from the
standard cosmological model have sparked curiosity in extending the model to
include new physics. Although, some suggestions include simply accounting for
aspects of our Universe that are ignored in the standard model. One example
acknowledges the fact that our Universe contains significant density contrasts
on small scales; in the form of galaxies, galaxy clusters, filaments, and
voids. This small-scale structure is smoothed out in the standard model, by
assuming large-scale homogeneity of the matter distribution, which could have a
measurable effect due to the nonlinearity of Einstein's equations. This
backreaction of small-scale structures on the large-scale dynamics has been
suggested to explain the measured accelerating expansion rate of the Universe.
Current standard cosmological simulations ignore the effects of General
Relativity by assuming purely Newtonian dynamics. In this thesis, we take the
first steps towards quantifying the backreaction of small-scale structures by
performing cosmological simulations that solve Einstein's equations directly.
Simulations like these will allow us to quantify potentially important effects
on our observations that could become measurable as the precision of these
observations increases into the future.
