Testing Cosmic Homogeneity Using Galaxy Clusters
  According to the cosmological principle, galaxy cluster sizes and cluster
densities, when averaged over sufficiently large volumes of space, are expected
to be constant everywhere, except for a slow variation with look-back time
(redshift). Thus, average cluster sizes or correlation lengths provide a means
of testing for homogeneity that is almost free of selection biases. Using ~10^6
galaxies from the SDSS DR7 survey, I show that regions of space separated by ~2
Gpc/h have the same average cluster size and density to 5 - 10 percent. I show
that the average cluster size, averaged over many galaxies, remains constant to
less than 10 percent from small redshifts out to redshifts of 0.25. The
evolution of the cluster sizes with increasing redshift gives fair agreement
when the same analysis is applied to the Millennium Simulation. However, the MS
does not replicate the increase in cluster amplitudes with redshift seen in the
SDSS data. This increase is shown to be caused by the changing composition of
the SDSS sample with increasing redshifts. There is no evidence to support a
model that attributes the SN Ia dimming to our happening to live in a large,
nearly spherical void.
