Inverse Construction of the $\Lambda$LTB Model from a Distance-redshift
  Relation
  Spherically symmetric dust universe models with a positive cosmological
constant $\Lambda$, known as $\Lambda$-Lema\^itre-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 $(\Omega^{\rm dis}_{\rm m0}, \Omega^{\rm
dis}_{\rm \Lambda 0})=(0.3,0.7)$, where $\Omega^{\rm dis}_{\rm m0}$ and
$\Omega^{\rm dis}_{\rm \Lambda 0}$ are the normalized matter density and the
cosmological constant, respectively. Then, we construct the $\Lambda$LTB model
for several values of $\Omega_{\rm \Lambda 0}:=\Lambda/(3H_0^2)$, where $H_0$
is the present Hubble parameter observed at the symmetry center. We obtain void
structure around the symmetry center for $\Omega_{\Lambda 0}<\Omega^{\rm
dis}_{\Lambda 0}$. We show the relation between the ratio
$\Omega_{\Lambda0}/\Omega^{\rm dis}_{\Lambda 0}$ and the amplitude of the
inhomogeneity.
