Solving the Inverse Problem with Inhomogeneous Universes
  We construct the Lema\^itre-Tolman-Bondi (LTB) dust universe whose distance-redshift relation is equivalent to that in the concordance $\Lambda$ cold dark matter ($\Lambda$CDM) cosmological model. In our model, the density distribution and velocity field are not homogeneous, whereas the big-bang time is uniform, which implies that the universe is homogeneous at its beginning. We also study the effects of local clumpiness in the density distribution as well as the effects of large-scale inhomogeneities on the distance-redshift relation, and show that these effects may reduce the amplitude of large-scale inhomogeneities necessary for having a distance-redshift relation that is the same as that of the concordance $\Lambda$CDM universe. We also study the temporal variation of the cosmological redshift and show that, by the observation of this quantity, we can distinguish our LTB universe model from the concordance $\Lambda$CDM model, even if their redshift-distance relations are equivalent to each other. 
Chul-Moon Yoo, Tomohiro Kai and Ken-ichi Nakao