Non-integrability of the axisymmetric Bianchi IX cosmological model via Differential Galois Theory

TL;DR

This paper proves that the anisotropic Bianchi IX cosmological model with matter and a cosmological constant is non-integrable in terms of meromorphic functions, using differential Galois theory to analyze the variational equations.

Contribution

It applies differential Galois theory to demonstrate the non-integrability of the Bianchi IX cosmological model with a cosmological constant.

Findings

The model is non-integrable by meromorphic functions.

The invariant isotropic plane is an integrable sub-space.

Differential Galois group analysis confirms non-integrability.

Abstract

We investigate the integrability of an anisotropic universe with matter and cosmological constant formulated as Bianchi IX models. The presence of the cosmological constant causes the existence of a critical point in the finite part of the phase space. The separatrix associated to this Einstein's static universe is entirely contained in an invariant isotropic plane forgetting the singularity at the origin. This invariant plane of isotropy is an integrable sub-space of the Taub type. In this paper we analyse the differential Galois group of the second order variational equations to this plane in order to apply the integrability theorem of the second author with Ramis and Sim\' o. The main result is that the model is non-integrable by meromorphic functions.