The apparent accelerating expansion of the Universe is forcing us to examine
the foundational aspects of the standard model of cosmology -- in particular,
the fact that dark energy is a direct consequence of the homogeneity
assumption. We discuss the foundations of the assumption of spatial
homogeneity, in the case when the Copernican Principle is adopted. We present
results that show how (almost-) homogeneity follows from (almost-) isotropy of
various observables. The analysis requires the fully nonlinear field equations
-- i.e., it is not possible to use second- or higher-order perturbation theory,
since one cannot assume a homogeneous and isotropic background. Then we
consider what happens if the Copernican Principle is abandoned in our Hubble
volume. The simplest models are inhomogeneous but spherically symmetric
universes which do not require dark energy to fit the distance modulus. Key
problems in these models are to compute the CMB anisotropies and the features
of large-scale structure. We review how to construct perturbation theory on a
non-homogeneous cosmological background, and discuss the complexities that
arise in using this to determine the growth of large-scale structure.Comment: 26 pages and 1 figure. Invited review article for the CQG special
issue on nonlinear cosmological perturbations. v2 has additional refs and
comments, minor errors corrected, version in CQ
