Determining the metric of the Cosmos: Stability, accuracy, and consistency

The ultimate application of Einstein's field equations is to empirically determine the geometry of the Universe from its matter content, rather than simply assuming the Universe can be represented by a homogeneous model on all scales. Choosing, a Lemaitre-Tolman-Bondi model as the most convenient inhomogeneous model for the early stages of development, a data reduction procedure was recently validated using perfect test data. Here, we simulate observational uncertainties and improve the previous numerical scheme to ensure that it will be usable with real data as soon as observational Surveys are sufficiently deep and complete. Two regions require special treatment - the origin and the maximum in the areal radius. To minimize numerical errors near the origin, we use a Lemaitre-Tolman-Bondi series expansion to provide the initial values for integrating the differential equations. We also use an improved method to match the numerical integration to the series expansion that bridges the region near the maximum in the areal radius. Because the mass enclosed within the maximum obeys a specific relationship, we show that it is possible to correct for a fixed systematic error in either the distance scale or the redshift-space mass density, such that the integrated values are consistent with the data at the maximum.