Is a Topology Change After a Big Rip Possible?

TL;DR

This paper investigates whether the universe's topology can change during a Big Rip singularity, analyzing conditions under which such a change is theoretically possible within modified gravity frameworks.

Contribution

It critically examines the possibility of topology change at a Big Rip singularity, extending classical arguments and considering modified gravity scenarios.

Findings

Topology change is possible at Big Rip if spacetime is non-compact and boundary hypersurfaces are topologically distinct.

Type II and IV singularities do not allow topology change due to geodesic completeness.

Additional conditions, including the weak energy condition, are necessary for topology change in the Big Rip scenario.

Abstract

Motivated by condensed matter physical systems, in which a finite-time singularity indicates that the topology of the system changes, we critically examine the possibility of the Universe's topology change at a finite-time cosmological singularity. We emphasize on Big Rip and Type II and IV cosmological singularities, which we classify to future spacelike and timelike singularities. For the Type IV and Type II singularities, since no geodesics incompleteness occurs, no topological change is allowed, by using Geroch's theorem arguments. However, for the Big Rip case, Tipler's arguments allow a topology change, if the spacetime in which the topology change occurs is non-compact and the boundary of this region are two topologically distinct three dimensional spacelike partial Cauchy hypersurfaces. Also, some additional requirements must hold true, among which the weak energy condition, which can be satisfied in a geometric way in the context of a modified gravity. We critically examine Tipler's arguments for the Big Rip case, and we discuss the mathematical implications of such a topological change, with regard to the final hypersurface on which geodesics incompleteness occurs.