Inhomogeneity of the $\Lambda$LTB models

The Lema\'itre-Toman-Bondi (LTB) models have reported to suffer from incompatibility with cosmological observations and fine-tuning of the observer's location. Further analysis of these issues indicates that they could be resolved by models that are compatible with the supernova Ia data, but less inhomogeneous than those that have been presented in the literature so far. We study if such models exist by employing the degrees of freedom of the LTB models in a novel manner. We discovered two scenarios which may meet the expectations, but extensive numerical and analytical investigation showed them inviable. We extended our studies to the $\Lambda$LTB models, which generalizes the LTB models by including a non-zero cosmological constant $\Lambda$ in Einsteins equations. This adds an additional degree of freedom for the earlier scenarios and introduces a new scenario capable of meeting the expectations. However, extensive numerical and analytical investigation reveals that inclusion of $\Lambda$ does not enhance the viability of the models. We identify the lack of degrees of freedom to be the reason for the unviability. However, the method presented here can be generalized to models including more degrees of freedom, like the Szekeres models, which have more promise to overcome the issues in the LTB models.