Future Asymptotic Behaviour of Tilted Bianchi models of type IV and VIIh

Using dynamical systems theory and a detailed numerical analysis, the late-time behaviour of tilting perfect fluid Bianchi models of types IV and VII${}_h$ are investigated. In particular, vacuum plane-wave spacetimes are studied and the important result that the only future attracting equilibrium points for non-inflationary fluids are the plane-wave solutions in Bianchi type VII${}_h$ models is discussed. A tiny region of parameter space (the loophole) in the Bianchi type IV model is shown to contain a closed orbit which is found to act as an attractor (the Mussel attractor). From an extensive numerical analysis it is found that at late times the normalised energy-density tends to zero and the normalised variables ‘freeze’ into their asymptotic values. A detailed numerical analysis of the type VII${}_h$ models then shows that there is an open set of parameter space in which solution curves approach a compact surface that is topologically a torus.