Testing the Copernican and Cosmological Principles in the local universe
  with galaxy surveys
  Cosmological density fields are assumed to be translational and rotational
invariant, avoiding any special point or direction, thus satisfying the
Copernican Principle. A spatially inhomogeneous matter distribution can be
compatible with the Copernican Principle but not with the stronger version of
it, the Cosmological Principle which requires the additional hypothesis of
spatial homogeneity. We establish criteria for testing that a given density
field, in a finite sample at low redshifts, is statistically and/or spatially
homogeneous. The basic question to be considered is whether a distribution is,
at different spatial scales, self-averaging. This can be achieved by studying
the probability density function of conditional fluctuations. We find that
galaxy structures in the SDSS samples, the largest currently available, are
spatially inhomogeneous but statistically homogeneous and isotropic up to ~ 100
Mpc/h. Evidences for the breaking of self-averaging are found up to the largest
scales probed by the SDSS data. The comparison between the results obtained in
volumes of different size allows us to unambiguously conclude that the lack of
elf-averaging is induced by finite-size effects due to long-range correlated
fluctuations. We finally discuss the relevance of these results from the point
of view of cosmological modeling.
