Oscillatory spacelike singularities: The Bianchi type $\mathrm{VI}_{-1/9}$ vacuum models

TL;DR

This paper explicitly solves the equations governing oscillatory spacelike singularities in Bianchi type VI_{-1/9} vacuum models, introduces a new graph representation of heteroclinic chains, and identifies which chains are relevant for the singularity behavior.

Contribution

It provides the first explicit solutions on the boundary sets of type VI_{-1/9} models and introduces a novel graph-based framework for analyzing heteroclinic networks.

Findings

Explicit solutions on boundary sets of type VI_{-1/9} models

Introduction of a new graph representation for heteroclinic chains

Identification of asymptotically relevant heteroclinic chains

Abstract

The Bianchi type $\mathrm{VI}_{-1/9}$, $\mathrm{VIII}$ and $\mathrm{IX}$ vacuum models all have 4-dimensional Hubble-normalized state spaces and are expected to have a generic initial oscillatory singularity, but the invariant boundary sets responsible for the oscillations are much more complicated for type $\mathrm{VI}_{-1/9}$ than those of type $\mathrm{VIII}$ and $\mathrm{IX}$. For the first time, we explicitly solve the equations on these type $\mathrm{VI}_{-1/9}$ boundary sets and also introduce a new graph representation of the associated network of heteroclinic chains (i.e. sequences of solutions describing the oscillations). In particular, we give examples of networks of entangled cyclic heteroclinic chains and show that only some of these cyclic heteroclinic chains are asymptotically relevant.