Dynamical Equilibrium States of a Class of Irrotational Non-Orthogonally Transitive $G_{2}$ Cosmologies I: The Conjecture of Chaotic Cosmological Inhomogeneity

TL;DR

This paper formulates Einstein's equations for a specific class of irrotational, non-orthogonally transitive $G_{2}$ cosmologies, analyzing their equilibrium states and proposing a conjecture about chaotic inhomogeneity in cosmological models.

Contribution

It derives a system of PDEs for these cosmologies, reduces it to an ODE for equilibrium points, and explores their mathematical and cosmological implications, including connections to Bianchi models.

Findings

Equilibrium points are self-similar and described by a one-parameter ODE.

The models exhibit evolution with one-dimensional inhomogeneity.

A conjecture about chaotic cosmological inhomogeneity is proposed.

Abstract

The Einstein field equations for a class of irrotational non-orthogonally transitive $G_{2}$ cosmologies are written down as a system of partial differential equations. The equilibrium points are self-similar and can be written as a one-parameter, five-dimensional, ordinary differential equation. The corresponding cosmological models both evolve and have one-dimension of inhomogeneity. The major mathematical features of this ordinary differential equation are derived, and a cosmological interpretation is given. The relationship to the exceptional Bianchi models is explained and exploited to provide a conjecture about future generalizations.