Asymptotic behavior of a matter filled universe with exotic topology

TL;DR

This paper investigates the long-term behavior of matter-filled universes with exotic topologies using the ADM formalism and a Lyapunov function, revealing conditions under which the universe approaches a state where matter becomes negligible.

Contribution

It introduces a weak Lyapunov function for Einstein equations in expanding universes with matter and positive cosmological constant, linking its infimum to topological invariants.

Findings

Lyapunov function decays monotonically in expanding universes.

Infimum of the Lyapunov function relates to the Sigma constant of the manifold.

Construction of a true Hamiltonian for matter-coupled gravity is challenging.

Abstract

The ADM formalism together with a constant mean curvature (CMC) temporal gauge is used to derive the monotonic decay of a weak Lyapunov function of the Einstein dynamical equations in an expanding universe with a positive cosmological constant and matter sources satisfying suitable energy conditions. While such a Lyapunov function does not, in general, represent a true Hamiltonian of the matter-coupled gravity dynamics (unlike in the vacuum case when it does), it can nevertheless be used to study the asymptotic behavior of the spacetimes. The Lyapunov function attains its infimum only in the limit that the matter sources be `turned off` or, at least, become asymptotically negligible provided that the universe model does not re-collapse and form singularities. Later we specialize our result to the case of a perfect fluid which satisfies the desired energy conditions. However, even in this special case, we show using Shutz's velocity potential formalism cast into Hamiltonian form that unlike the vacuum spacetimes (with or without a positive cosmological constant), construction of a true Hamiltonian for the dynamics in constant mean curvature temporal gauge is difficult and therefore the Lyapunov function does not have a straightforward physical interpretation. Nevertheless, we show, for the fluid with equation of state $P=(\gamma-1)\rho$ ($1\leq\gamma\leq2$), that the general results obtained hold true and the infimum of the weak Lyapunov function can be related to the Sigma constant, a topological invariant of the manifold. Utilizing these results, some general conclusions are drawn regarding the asymptotic state of the universe and the dynamical control of the allowed spatial topologies in the cosmological models.