Asymptotic discrete self-similarity in cosmological models

In the previous post I discussed some ideas from the theory of dynamical systems and mentioned some applications of these ideas to models arising from chemistry and biology. Here I will explain an example where they can be applied to mathematical cosmology. In cosmology it is common to study spatially homogeneous solutions of the Einstein equations with matter modelled by a perfect fluid with linear equation of state p=(\gamma-1)\rho. In a spatially homogeneous solution there is a preferred foliation by spacelike hypersurfaces defined by the orbits of the symmetry group. An important simplifying assumption is that the four-velocity u^\alpha of the fluid is orthogonal to these hypersurfaces. If this assumption is not satisfied the fluid is said to be tilted and the dynamics of the models becomes significantly more complicated. Physically tilt means that in a frame of reference where the energy density is spatially constant there is a bulk motion of the fluid.

Starting with a paper of Coley and Hervik it has been observed that solutions of the Einstein equations with tilted fluid can show dynamical behaviour which is qualitatively different from anything which is possible in the absence of tilt. The approach used combines analytical and numerical techniques. I have been aware of this work and found it interesting for a long time but I now see that I had not gone into it deeply enough to get a real understanding of what is going on there. This dynamics involves periodic solutions and Hopf bifurcations, which is why it is related to the previous post. The interesting new phenomena occur for certain Bianchi types of class B, namely types {\rm IV}, {\rm VI} {}_h and {\rm VII} {}_h. Some background information on this terminology can be found here. The relevant dynamical regime is that where the spacetime expands forever. Many of the basic quantities tend to zero at late time and to get a less degenerate description it is useful to introduce a dynamical system for certain dimensionless quantities. When this is done a stationary solution of the dimensionless system corresponds to a continuously self-similar solution of the original Einstein-Euler system and a periodic solution of the dimensionless system corresponds to a discretely self-similar solution of the original system. The latter type of solution will be of particular interest in what follows.

In the classes of solutions considered here there are stationary solutions of the dimensionless systems which play the role of late time attractors in most cases. The solutions are asymptotically self-similar at late times. There is, however, a small exceptional region called the loophole where no stable self-similar solutions are available. In that case the asymptotic behaviour is more complicated. The dimensionless variable \Omega corresponding to the energy density of the fluid tends to zero at late times. Thus in a sense the solution is converging to a vacuum solution. However certain variables describing the fluid retain a non-trivial dynamics so that in a sense the fluid leaves a trace on the vacuum solution. More precisely, in the limit the spacetime geometry converges in a suitable sense to that of a solution of the vacuum Einstein equations while the fluid variables describing the tilt converge to a test fluid on that background. A test fluid means that while the fluid satisfies the Euler equations its energy-momentum tensor does not make a contribution to the Einstein equations. The limiting dynamical system is defined on the unit ball in R^3 with the unit sphere being an invariant submanifold. On the unit sphere the test fluid can be thought of as null dust, where the four-velocity is null. The parameter \gamma in the equation of state vanishes completely from the restrictions of the equations for the tilt variables to the boundary sphere. In terms of the dimensionless dynamical system the solutions with test fluid are not solutions of that system but solutions of a smooth extension of that system to part of the boundary of the physical region. The restriction of the extended system to that part of the boundary will be called the asymptotic system.

For some values of the parameters the asymptotic system has periodic solutions. These can be of different types. Probably the simplest case is that described in Theorem 4.3 of gr-qc/0409106. It concerns Bianchi type {\rm VII}{}_h. There the system on the boundary sphere has precisely two stationary points which are hyperbolic sources. Thus it follows immediately by Poincaré-Bendixson theory that there exists a periodic solution. The uniqueness of this periodic solution which is also part of the assertion of the theorem requires a more complicated proof. A Hopf bifurcation occurs explicitly in the treatment of Bianchi type {\rm VI}{}_{-\frac19} in arXiv:0706.3184 where the authors give the expression for the first Lyapunov number in this case.

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