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

Spherically symmetric dust universe models with a positive cosmological constant Lambda, known as Lambda-Lemaitre-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 (ohm(dis)(m0), ohm(dis)(Lambda 0))=(0.3,0.7), where ohm(dis)(m0) and ohm(dis)(Lambda 0) are the normalized matter density and the cosmological constant, respectively. Then, we construct the Lambda LTB model for several values of ohm(Lambda 0) :- Lambda/(3H(0)(2)), where H-0 is the present Hubble parameter observed at the symmetry center. We obtain void (over dense) structure around the symmetry center for ohm(Lambda 0) < ohm(dis)(Lambda 0)(ohm(Lambda 0) > ohm(dis)(Lambda 0)). We show the relation between the ratio ohm(Lambda 0)/ohm(dis)(Lambda 0) and the amplitude of the inhomogeneity