Integrability and explicit solutions in some Bianchi cosmological dynamical systems

TL;DR

This paper investigates specific Bianchi cosmological models by applying Darboux's theory to solve polynomial differential equations, providing explicit solutions for Bianchi V and partial results for Bianchi VI models.

Contribution

It introduces a novel application of Darboux's theory to Bianchi cosmologies, achieving explicit solutions for Bianchi V and identifying first integrals in Bianchi VI models.

Findings

Explicit solutions for Bianchi V models.

First integrals identified for Bianchi VI models.

Application of Darboux's theory to cosmological differential equations.

Abstract

The Einstein field equations for several cosmological models reduce to polynomial systems of ordinary differential equations. In this paper we shall concentrate our attention to the spatially homogeneous diagonal G_2 cosmologies. By using Darboux's theory in order to study ordinary differential equations in the complex projective plane CP^2 we solve the Bianchi V models totally. Moreover, we carry out a study of Bianchi VI models and first integrals are given in particular cases.