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

TL;DR

This paper advances methods for empirically determining the Universe's geometry using inhomogeneous models, improving numerical schemes to handle observational uncertainties and special regions like the origin and maximum areal radius.

Contribution

It introduces an improved numerical scheme for LTB models that accounts for observational uncertainties and special regions, enabling more accurate empirical cosmological measurements.

Findings

Validated data reduction with perfect test data

Enhanced numerical scheme for real data application

Demonstrated correction method for systematic errors

Abstract

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 an LTB 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 an LTB 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.