Complete classification of Friedmann-Lema\^{i}tre-Robertson-Walker solutions with linear equation of state: parallelly propagated curvature singularities for general geodesics

TL;DR

This paper provides a complete classification of FLRW solutions with linear equations of state, analyzing geodesics and curvature singularities, and presents Penrose diagrams for all cases without assuming energy conditions.

Contribution

It extends previous classifications by including all geodesics and curvature singularities, identifying null singularities and critical values of the equation of state parameter.

Findings

No non-null geodesic emanates from or terminates at null conformal infinity.

Initial singularity for certain parameters is a null non-scalar polynomial curvature singularity.

Penrose diagrams are constructed for all cases, highlighting the critical value w=-5/3.

Abstract

We completely classify the Friedmann-Lema\^{i}tre-Robertson-Walker solutions with spatial curvature $K=0,\pm 1$ for perfect fluids with linear equation of state $p=w\rho $, where $\rho$ and $p$ are the energy density and pressure, without assuming any energy conditions. We extend our previous work to include all geodesics and parallelly propagated curvature singularities, showing that no non-null geodesic emanates from or terminates at the null portion of conformal infinity and that the initial singularity for $K=0,-1$ and $-5/3<w<-1$ is a null non-scalar polynomial curvature singularity. We thus obtain the Penrose diagrams for all possible cases and identify $w=-5/3$ as a critical value for both the future big-rip singularity and the past null conformal boundary.