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.