Complete solutions to the metric of spherically collapsing dust in an expanding spacetime with a cosmological constant

TL;DR

This paper provides semi-analytical, continuous solutions for the complete spacetime of a collapsing dust sphere within an expanding universe with a cosmological constant, avoiding numerical integration.

Contribution

It offers the first fully analytical solutions for the LTB metric with dust, curvature, and Lambda, covering both expansion and collapse phases without numerical methods.

Findings

Solutions describe complete spacetime of collapsing dust in expanding universe

No numerical integration needed for these solutions

Includes cases with dust, curvature, and cosmological constant

Abstract

We present semi-analytical solutions to the background equations describing the Lema\^itre-Tolman-Bondi (LTB) metric as well as the homogeneous Friedmann equations, in the presence of dust, curvature and a cosmological constant Lambda. For none of the presented solutions any numerical integration has to be performed. All presented solutions are given for expanding and collapsing phases, preserving continuity in time and radius. Hence, these solutions describe the complete space time of a collapsing spherical object in an expanding universe. In the appendix we present for completeness a solution of the Friedmann equations in the additional presence of radiation, only valid for the Robertson-Walker metric.