Kovalevski exponents and integrability properties in class A homogeneous cosmological models

TL;DR

This paper investigates the integrability of class A homogeneous cosmological models using Kovalevski exponents, revealing nonintegrability in certain models and effects of matter terms on integrability.

Contribution

It applies Kovalevski exponents to analyze algebraic integrability in Bianchi class A models, highlighting nonintegrability inheritance and matter effects.

Findings

Vacuum Bianchi VII_0 model is nonintegrable.

Nonintegrability is inherited by Bianchi VIII and IX models.

Matter terms induce nonintegrability in otherwise integrable models.

Abstract

Qualitative approach to homogeneous anisotropic Bianchi class A models in terms of dynamical systems reveals a hierarchy of invariant manifolds. By calculating the Kovalevski Exponents according to Adler - van Moerbecke method we discuss how algebraic integrability property is distributed in this class of models. In particular we find that algebraic nonintegrability of vacuum Bianchi VII_0 model is inherited by more general Bianchi VIII and Bianchi IX vacuum types. Matter terms (cosmological constant, dust and radiation) in the Einstein equations typically generate irrational or complex Kovalevski exponents in class A homogeneous models thus introducing an element of nonintegrability even though the respective vacuum models are integrable.