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

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 (Ω{sup dis}{sub m0}, Ω{sup dis}{sub Λ0})=(0.3,0.7), where Ω{sup dis}{sub m0} and Ω{sup dis}{sub Λ0} are the normalized matter density and the cosmological constant, respectively. Then, we construct the ΛLTB model for several values of Ω{sub Λ0}:=Λ/(3 H {sub 0}{sup 2}), where H {sub 0} is the present Hubble parameter observed at the symmetry center. We obtain void (over dense) structure around the symmetry center for Ω{sub Λ0} < Ω{sup dis}{sub Λ0}(Ω{sub Λ0} > Ω{sup dis}{sub Λ0}). We show the relation between the ratio Ω{sub Λ0}/Ω{sup dis}{sub Λ0} and the amplitude of the inhomogeneity.