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

TL;DR

This paper provides a comprehensive classification of FLRW cosmological solutions with a linear equation of state, exploring their conformal structures, singularities, and horizons without energy condition restrictions, including negative densities.

Contribution

It offers the first complete classification of FLRW solutions with arbitrary curvature and negative energy densities, extending beyond traditional assumptions.

Findings

Identifies conditions for big-bang and big-rip singularities based on parameters.

Describes the nature of trapping horizons and conformal boundaries for various cases.

Shows negative energy densities are only possible for negative spatial curvature.

Abstract

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.