We derive an observational constraint on a spherical inhomogeneity of the void centered at our position from the angular power spectrum of the cosmic microwave background(CMB) and local measurements of the Hubble parameter. The late time behaviour of the void is assumed to be well described by the so-called Λ-Lemaître-Tolman-Bondi (ΛLTB) solution. Then, we restrict the models to the asymptotically homogeneous models each of which is approximated by a flat Friedmann-Lemaître-Robertson-Walker model. The late time ΛLTB models are parametrized by four parameters including the value of the cosmological constant and the local Hubble parameter. The other two parameters are used to parametrize the observed distance-redshift relation. Then, the ΛLTB models are constructed so that they are compatible with the given distance-redshift relation. Including conventional parameters for the CMB analysis, we characterize our models by seven parameters in total. The local Hubble measurements are reflected in the prior distribution of the local Hubble parameter. As a result of a Markov-Chains-Monte-Carlo analysis for the CMB temperature and polarization anisotropies, we found that the inhomogeneous universe models with vanishing cosmological constant are ruled out as is expected. However, a significant under-density around us is still compatible with the angular power spectrum of CMB and the local Hubble parameter.

Spherically symmetric dust universe models with a positive cosmological constant Λ, known as Λ-Lemaître-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 distanceredshift 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 dis m0 , Ω dis Λ0 ) = (0.3, 0.7), where Ω dis m0 and Ω dis Λ0 are the normalized matter density and the cosmological constant, respectively. Then, we construct the ΛLTB model for several values of Ω Λ0 := Λ/(3H 2 0 ), 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 dis Λ0 . We show the relation between the ratio Ω Λ0 /Ω dis Λ0 and the amplitude of the inhomogeneity. * Electronic address: tokutake@gravity.phys.nagoya-u.ac.jp † Electronic address: yoo@gravity.phys.nagoya-u.ac.jp 2