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

Abstract

We present elliptic solutions to the background equations describing the Lemaître-Tolman-Bondi metric as well as the homogeneous Friedmann equation, in the presence of dust, curvature and a cosmological constant Λ. 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; both radial and angular scale functions are given. Hence, these solutions describe the complete spacetime of a collapsing spherical object in an expanding universe, as well as those of ever expanding objects. In the appendix we present for completeness a solution of the Friedmann equation in the additional presence of radiation, only valid for the Robertson-Walker metric. © 2012 The Author(s).