Part I: Is our Universe spatially homogeneous on the largest observable scales? To investigate this question, we develop a flexible method based on spherically symmetric, but radially inhomogeneous Lemaître-Tolman-Bondi models that allows us to study a wide range of non-Copernican cosmological models. We employ a Monte Carlo sampler to systematically vary the shape of the (local) matter density profile and determine the likelihood of the sampled models given a selected set of observational data. After analysing non-Copernican models with and without cosmological constant, we arrive at the final conclusion that the observational data considered provide no statistical evidence for deviations from spatial homogeneity on large scales. However, more accurate constraints are required to ultimately confirm the validity of the cosmological principle.
Part II: Are the strongest observed gravitational lenses in conflict with the predictions of the standard cosmological model? To address this question, we apply extreme value and order statistics to the cosmological distribution of the largest Einstein radii. We show that cluster mergers can substantially increase the Einstein radii of the strongest gravitational lenses. A comparison with current observational data reveals that, presently, there is no reliable statistical evidence for observed Einstein radii to exceed the theoretical expectations of the standard cosmological model
