Probing violation of the Copernican principle via the integrated
  Sachs-Wolfe effect
  Recent observational data of supernovae indicate that we may live in an
underdense region, which challenges the Copernican principle. We show that the
integrated Sachs-Wolfe (ISW) effect is an excellent discriminator between
anti-Copernican inhomogeneous models and the standard Copernican models. As a
reference model, we consider an anti-Copernican inhomogeneous model that
consists of two inner negatively curved underdense regions and an outer flat
Einstein-de Sitter region. We assume that these regions are connected by two
thin-walls at redshifts z = 0.067 and z=0.45. In the inner two regions, the
first-order ISW effect is dominant and comparable to that in the concordant
flat-Lambda models. In the outer Einstein-de Sitter region, the first-order ISW
effect vanishes but the second-order ISW effect plays a dominant role, while
the first-order ISW effect is dominant in the flat-Lambda models at moderate
redshifts. This difference can discrimate the anti-Copernican models from the
concordant flat-Lambda model. At high redshits, the second-order ISW effect is
dominant both in our inhomogeneous model and the concordant model. In the outer
region, moreover, the ISW effect due to large-scale density perturbations with
a present matter density contrast much less than 0.37 is negligible, while the
effect due to small-scale density perturbations (such as clusters of galaxies,
superclusters and voids) with matter density contrast much larger than 0.37
would generate anisotropies which are larger than those generated by the ISW
effect in the concordant model.
