Is the Universe homogeneous?
  The standard model of cosmology is based on the existence of homogeneous
surfaces as the background arena for structure formation. Homogeneity underpins
both general relativistic and modified gravity models and is central to the way
in which we interpret observations of the CMB and the galaxy distribution.
However, homogeneity cannot be directly observed in the galaxy distribution or
CMB, even with perfect observations, since we observe on the past lightcone and
not on spatial surfaces. We can directly observe and test for isotropy, but to
link this to homogeneity, we need to assume the Copernican Principle. First, we
discuss the link between isotropic observations on the past lightcone and
isotropic spacetime geometry: what observations do we need to be isotropic in
order to deduce spacetime isotropy? Second, we discuss what we can say with the
Copernican assumption. The most powerful result is based on the CMB: the
vanishing of the dipole, quadrupole and octupole of the CMB is sufficient to
impose homogeneity. Real observations lead to near-isotropy on large scales -
does this lead to near-homogeneity? There are important partial results, and we
discuss why this remains a difficult open question. Thus we are currently
unable to prove homogeneity of the Universe on large-scales, even with the
Copernican Principle. However we can use observations of the CMB, galaxies and
clusters to test homogeneity itself.
