Complete classification of Friedmann–Lemaître–Robertson–Walker solutions with linear equation of state: parallelly propagated curvature singularities for general geodesics

Abstract

We completely classify the Friedmann–Lemaître–Robertson–Walker solutions with spatial curvature K = 0, ±1 for perfect fluids with linear equation of state p = wρ, where ρ 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 (p.p.) 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.