Singular behavior in Lema\^{i}tre-Tolman-Bondi cosmological models and the difficulties in using them as realistic models of the Universe

There has been much debate over whether or not one could explain the observed acceleration of the Universe with inhomogeneous cosmological models, such as the spherically-symmetric Lema\^{i}tre-Tolman-Bondi (LTB) models. It has been claimed that the central observer in these models can observe a local acceleration, which would contradict general theorems. We resolve the contradiction by noting that many of the models that have been explored contain a weak singularity at the location of the observer, which makes them unphysical. In the absence of this singularity, we show that LTB models must have a positive central deceleration parameter $q_{0}$, in agreement with the general theorems. We also show that at nonzero redshifts the apparent deceleration parameter measured by the central observer can be negative. However, we find other singularities that tend to arise in LTB models when attempting to match luminosity distance data, and these generally limit the range of redshifts for which these models can mimic observations of an accelerating Universe.