Mapping luminosity-redshift relationship to Lemaitre-Tolman-Bondi cosmology

We derive a direct general map from the luminosity distance D-L(z) to the inhomogeneous matter distribution M(r) in the Lemaitre-Tolman-Bondi (LTB) cosmology and compute several examples. One of our examples explicitly demonstrates that it is possible to tune the LTB cosmological solution to approximately reproduce the luminosity distance curve of a flat Friedmann-Robertson-Walker universe with a cosmological constant. We also discuss how smooth matter distributions can evolve into naked singularities due to shell crossing when the inhomogeneous "curvature" E(r) is a function which changes sign.