3+2 Cosmology: unifying FRW metrics in the bulk

TL;DR

This paper introduces a five-dimensional bulk model with two time coordinates to unify all spatially flat FRW metrics, revealing new solutions and limitations in embedding these cosmologies within higher-dimensional manifolds.

Contribution

It presents a novel 5D bulk framework with two time dimensions that unifies all flat FRW metrics and explores conditions for their embedding, extending the understanding of higher-dimensional cosmological models.

Findings

Derived a general 'mother' metric from which all flat FRW metrics can be obtained.

Identified solutions evolving from an infinite past, including emergent universe scenarios.

Showed that not all flat FRW metrics can be embedded in a 3+2 bulk, indicating limitations of the Campbell theorem extension.

Abstract

The Cosmological Problem is considered in a five-dimensional (bulk) manifold with two time coordinates, obeying vacuum Einstein field equations. The evolution formalism is used there, in order to get a simple form of the resulting constraints. In the spatially flat case, this approach allows to find out the general solution, which happens to consist in a single metric. All the embedded Friedmann-Robertson-Walker (FRW) metrics can be obtained from this 'mother' metric ('M-metric') in the bulk, by projecting onto different four-dimensional hypersurfaces (branes). Having a time plane in the bulk allows to devise the specific curve which will be kept as the physical time coordinate in the brane. This method is applied for identifying FRW regular solutions, evolving from the infinite past (no Big Bang), even with an asymptotic initial state with non-zero radius (emergent universes). Explicit counter-examples are provided, showing that not every spatially-flat FRW metric can actually be embedded in a 3+2 bulk manifold. This implies that the extension of the Campbell theorem to the General Relativity case works only in its weaker form in this case, requiring as an extra assumption that the constraint equations hold at least in a single 4D hypersurface.