A dynamical systems approach to the tilted Bianchi models of solvable type

Abstract

We use a dynamical systems approach to analyse the tilting spatially homogeneous Bianchi models of solvable type (e.g., types VI${}_h$ and VII${}_h$) with a perfect fluid and a linear barotropic $\gamma$-law equation of state. In particular, we study the late-time behaviour of tilted Bianchi models, with an emphasis on the existence of equilibrium points and their stability properties. We briefly discuss the tilting Bianchi type V models and the late-time asymptotic behaviour of irrotational Bianchi VII${}_0$ models. We prove the important result that for non-inflationary Bianchi type VII${}_h$ models vacuum plane-wave solutions are the only future attracting equilibrium points in the Bianchi type VII${}_h$ invariant set. We then investigate the dynamics close to the plane-wave solutions in more detail, and discover some new features that arise in the dynamical behaviour of Bianchi cosmologies with the inclusion of tilt. We point out that in a tiny open set of parameter space in the type IV model (the loophole) there exists closed curves which act as attracting limit cycles. More interestingly, in the Bianchi type VII${}_h$ models there is a bifurcation in which a set of equilibrium points turn into closed orbits. There is a region in which both sets of closed curves coexist, and it appears that for the type VII${}_h$ models in this region the solution curves approach a compact surface which is topologically a torus.

29 pages

References in corpus (3)

• Future Asymptotic Behaviour of Tilted Bianchi models of type IV and VIIh
• Bianchi Cosmologies: A Tale of Two Tilted fluids
• The Dynamics of Inhomogeneous Cosmologies

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