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

Spherically symmetric dust universe models with a positive cosmological constant $Λ$, known as $Λ$-Lemaître-Tolman-Bondi($Λ$LTB) models, are considered. We report a method to construct the $Λ$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 $Λ$. As a demonstration, we fix the distance-redshift relation as that of the flat $Λ$CDM model with $(Ω^{\rm dis}_{\rm m0}, Ω^{\rm dis}_{\rm Λ0})=(0.3,0.7)$, where $Ω^{\rm dis}_{\rm m0}$ and $Ω^{\rm dis}_{\rm Λ0}$ are the normalized matter density and the cosmological constant, respectively. Then, we construct the $Λ$LTB model for several values of $Ω_{\rm Λ0}:=Λ/(3H_0^2)$, where $H_0$ is the present Hubble parameter observed at the symmetry center. We obtain void structure around the symmetry center for $Ω_{Λ0}<Ω^{\rm dis}_{Λ0}$. We show the relation between the ratio $Ω_{\Lambda0}/Ω^{\rm dis}_{Λ0}$ and the amplitude of the inhomogeneity.

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