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

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

2004-07-12

arxiv.org/abs/gr-qc/0407040v2

Gen.Rel.Grav. 37 (2005) 937-952

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

Latex, 17 pages, no figures; (v2) minor corrections, additional comments and 2 new references; revised version to appear in Ge

Scientific paper

10.1007/s10714-005-0076-1

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$_h$.

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