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

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.