Complete conformal classification of the Friedmann-Lemaitre-Robertson-Walker solutions with a linear equation of state

Abstract

We completely classify Friedmann-Lemaître-Robertson-Walker solutions with spatial curvature $K=0,\pm 1$ and equation of state $p=wρ$, according to their conformal structure, singularities and trapping horizons. We do not assume any energy conditions and allow $ρ< 0$, thereby going beyond the usual well-known solutions. For each spatial curvature, there is an initial spacelike big-bang singularity for $w>-1/3$ and $ρ>0$, while no big-bang singularity for $w<-1$ and $ρ>0$. For $K=0$ or $-1$, $-1 0$, there is an initial null big-bang singularity. For each spatial curvature, there is a final spacelike future big-rip singularity for $w<-1$ and $ρ>0$, with null geodesics being future complete for $-5/3\le w<-1$ but incomplete for $w<-5/3$. For $w=-1/3$, the expansion speed is constant. For $-1 -1/3$, the universe contracts from infinity, then bounces and expands to infinity; for $-1<w<-1/3$, it starts from a big-bang singularity and contracts to a big-crunch singularity; for $w<-1$, it expands from a regular null hypersurface and contracts to another regular null hypersurface.