Inverse Construction of the $\Lambda$LTB Model from a Distance-redshift Relation

Abstract

Spherically symmetric dust universe models with a positive cosmological constant $\Lambda$, known as $\Lambda$-Lemaître-Tolman-Bondi($\Lambda$LTB) models, are considered. We report a method to construct the $\Lambda$LTB 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 $\Lambda$. As a demonstration, we fix the distance-redshift relation as that of the flat $\Lambda$CDM model with $({\Omega }_{m0}^{\mathrm{dis}},{\Omega }_{\Lambda 0}^{\mathrm{dis}})=(0.3,0.7)$, where ${\Omega }_{m0}^{\mathrm{dis}}$ and ${\Omega }_{\Lambda 0}^{\mathrm{dis}}$ are the normalized matter density and the cosmological constant, respectively. Then, we construct the $\Lambda$LTB model for several values of ${\Omega }_{\Lambda 0}:=\Lambda /(3H_0^2)$, where $H_0$ is the present Hubble parameter observed at the symmetry center. We obtain void structure around the symmetry center for ${\Omega }_{\Lambda 0}<{\Omega }_{\Lambda 0}^{\mathrm{dis}}$. We show the relation between the ratio ${\Omega }_{\Lambda 0}/{\Omega }_{\Lambda 0}^{\mathrm{dis}}$ and the amplitude of the inhomogeneity.

12 pages, 2 figures

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