Global integrability of cosmological scalar fields

TL;DR

This paper analyzes the mathematical integrability of cosmological models with scalar fields using differential Galois theory, finding most are non-integrable and linking this to chaos in the models.

Contribution

It applies differential Galois theory to determine integrability of scalar field cosmologies, identifying specific cases of integrability and non-integrability.

Findings

Most minimal coupling models are non-integrable.

Conformally coupled models are integrable in four specific cases.

Connections between integrability and chaos are discussed.

Abstract

We investigate the Liouvillian integrability of Hamiltonian systems describing a universe filled with a scalar field (possibly complex). The tool used is the differential Galois group approach, as introduced by Morales-Ruiz and Ramis. The main result is that the generic systems with minimal coupling are non-integrable, although there still exist some values of parameters for which integrability remains undecided; the conformally coupled systems are only integrable in four known cases. We also draw a connection with chaos present in such cosmological models, and the issues of integrability restricted to the real domain.