Solving Einstein Field Equations in Observational Coordinates with Cosmological Data Functions: Spherically Symmetric Universes with Cosmological Constant

TL;DR

This paper develops a method to construct and analyze spherically symmetric cosmological models with a cosmological constant using observational data from our past light cone, demonstrating how such data fully determines the models and their parameters.

Contribution

It extends previous methods to include non-flat universes and shows how observational data can determine key cosmological parameters, including the vacuum-energy density.

Findings

Successfully integrates Einstein field equations with observational data in spherically symmetric models.

Shows how to determine the cosmological constant from observational data.

Demonstrates the full determination of cosmological models from light cone data.

Abstract

Extending the approach developed by Ara\'ujo and Stoeger [1] and improved in Ara\'ujo {\it et al} [2], we have shown how to construct dust-filled $\Lambda \neq 0$ Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) cosmological models from FLRW cosmological data on our past light cone. Apart from being of interest in its own right -- demonstrating how such data fully determines the models -- it is also illustrated in the flat case how the more general spherically symmetric (SS) Einstein field equations can be integrated in observational coordinates with data fit to FLRW forms arrayed on our past light cone, thus showing how such data determines a FLRW universe -- which is not {\it a priori} obvious. It is also shown how to integrate these exact SS equations, in cases where the data are not FLRW, and the space-time is not known to be flat. It is essential for both flat and non-flat cases to have data giving the maximum of the observer area (angular-diameter) distance, and the redshift $z_{max}$ at which that occurs. This enables the determination of the vacuum-energy density $\mu_{\Lambda}$, which would otherwise remain undetermined.