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

We completely classify Friedmann-Lema\^{i}tre-Robertson-Walker solutions with spatial curvature $K=0,\pm 1$ and equation of state $p=w\rho$, according to their conformal structure, singularities and trapping horizons. We do not assume any energy conditions and allow $\rho < 0$, 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 $\rho>0$, while no big-bang singularity for $w<-1$ and $\rho>0$. For $K=0$ or $-1$, $-1<w<-1/3$ and $\rho>0$, 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 $\rho>0$, 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 $-1<w<-1/3$ and $K=1$, the universe contracts from infinity, then bounces and expands back to infinity. For $K=0$, the past boundary consists of timelike infinity and a regular null hypersurface for $-5/3<w<-1$, while it consists of past timelike and past null infinities for $w\le -5/3$. For $w<-1$ and $K=1$, the spacetime contracts from an initial spacelike past big-rip singularity, then bounces and blows up at a final spacelike future big-rip singularity. For $w<-1$ and $K=-1$, the past boundary consists of a regular null hypersurface. The trapping horizons are timelike, null and spacelike for $w\in (-1,1/3)$, $w\in \{1/3, -1\}$ and $w\in (-\infty,-1)\cup (1/3,\infty)$, respectively. A negative energy density ($\rho <0$) is possible only for $K=-1$. In this case, for $w>-1/3$, 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.