Equilibrium points of the tilted perfect fluid Bianchi VI$_h$ state space

TL;DR

This paper derives equilibrium points for tilted perfect fluid Bianchi VI_h cosmologies, revealing conditions for self-similar solutions, their stability, and explicit metrics, with implications for future attractors in these models.

Contribution

It provides a comprehensive set of equilibrium points for tilted perfect fluid Bianchi VI_h models, including new self-similar solutions and their explicit metrics, extending previous classifications.

Findings

Self-similar solutions exist only for h > -1.

For h > -1/9, solutions exist with specific gamma intervals.

Equilibrium points have a five-dimensional stable manifold.

Abstract

We present the full set of evolution equations for the spatially homogeneous cosmologies of type VI$_h$ filled with a tilted perfect fluid and we provide the corresponding equilibrium points of the resulting dynamical state space. It is found that only when the group parameter satisfies $h>-1$ a self-similar solution exists. In particular we show that for $h>-{\frac 19}$ there exists a self-similar equilibrium point provided that $\gamma \in ({\frac{2(3+\sqrt{-h})}{5+3\sqrt{-h}}},{\frac 32}) $ whereas for $h<-{\frac 19}$ the state parameter belongs to the interval $\gamma \in (1,{\frac{2(3+\sqrt{-h})}{5+3\sqrt{-h}}}) $. This family of new exact self-similar solutions belongs to the subclass $n_\alpha ^\alpha =0$ having non-zero vorticity. In both cases the equilibrium points have a five dimensional stable manifold and may act as future attractors at least for the models satisfying $n_\alpha ^\alpha =0$. Also we give the exact form of the self-similar metrics in terms of the state and group parameters. As an illustrative example we provide the explicit form of the corresponding self-similar radiation model ($\gamma={\frac 43}$), parametrised by the group parameter $h$. Finally we show that there are no tilted self-similar models of type III and irrotational models of type VI$_h$.