Obtaining the Time Evolution for Spherically Symmetric Lema\^{i}tre-Tolman-Bondi Models Given Data on Our Past Light Cone

TL;DR

This paper presents a rigorous method to determine and evolve spherically symmetric Lemaître-Tolman-Bondi cosmological models from observational data on our past light cone, including the maximum of the angular-diameter distance.

Contribution

It provides a new, mathematically rigorous procedure to reconstruct the metric functions and their derivatives from light cone data, incorporating the maximum distance point to determine vacuum energy density.

Findings

Method allows evolution of models off the light cone.

Includes Taylor series expansion for solutions.

Determines vacuum-energy density from observational data.

Abstract

A rigorous demonstration that given appropriate data on our past light cone leads to the determination of the metric functions and all their time derivatives on our past light cone is presented, thus showing how to evolve the solution we obtain from data on the light cone off it in a well-defined and straightforward way. It also automatically gives a procedure for constructing the solution for all spherically symmetric, inhomogeneous cosmological Lema\^itre-Tolman-Bondi models in observational coordinates as a Taylor series in time of however many terms we need. Our procedure takes into account the essential 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.