Inverse construction of the ΛLTB model from a distance-redshift relation

Abstract

Spherically symmetric dust universe models with a positive cosmological constant Λ, known as Λ-Lemaȋtre-Tolman-Bondi (ΛLTB) models, are considered. We report a method to construct the ΛLTB model from a given distance-redshift relation observed at the symmetry center. The spherical inhomogeneity is assumed to be composed of growing modes. We derive a set of ordinary differential equations for three functions of the redshift, which specify the spherical inhomogeneity. Once a distance-redshift relation is given, with careful treatment of possible singular points, we can uniquely determine the model by solving the differential equations for each value of Λ. As a demonstration, we fix the distance-redshift relation as that of the flat ΛCDM model with (Ωdism0, ΩdisΛ0)=(0.3,0.7), where Ωdism0 and ΩdisΛ0 are the normalized matter density and the cosmological constant, respectively. Then, we construct the ΛLTB model for several values of ΩΛ0:=Λ/(3H02), where H0 is the present Hubble parameter observed at the symmetry center. We obtain void (over dense) structure around the symmetry center for ΩΛ0 < ΩdisΛ0(ΩΛ0 > ΩdisΛ0). We show the relation between the ratio ΩΛ0/ΩdisΛ0 and the amplitude of the inhomogeneity.