Complete conformal classification of the Friedmann–Lemaître–Robertson–Walker solutions with a linear equation of state

Abstract

We completely classify Friedmann–Lemaître–Robertson–Walker solutions with spatial curvature and equation of state , according to their conformal structure, singularities and trapping horizons. We do not assume any energy conditions and allow , 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 , while there is no big-bang singularity for w < −1 and . For K = 0 or −1, −1 < w < −1/3 and , 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 , with null geodesics being future complete for 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 . 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 , and , respectively. A negative energy density () 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.