3.2. Physical motivation

in the next section we shall analyse the type IX model as a dynamical system using the ideas introduced in section 2. It is worth remarking why the Bianchi type IX or 'Mixmaster' Universe [32] has been much studied by cosmologists:

It is an anisotropically expanding Universe with closed space sections (although it is not known whether it will generally recollapse to a second singularity). It begins expanding in a strongly irregular fashion but can, in the course of time, isotropize and provide a good description of the large scale Universe today. When the anisotropy level is small it resembles an anisotropic perturbation of the closed Friedmann Universe.

It has been shown by York [33] and by Fischer and Marsden [34] that for general (in- homogeneous) vacuum closed Universes away from initial data with symmetries (Killing vectors) the initial data for the Einstein equations can be completely defined by four independent functions of three variables. That is, there is ocally a diffeomorphism from the space of pairs of three-tensors describing the induced metric on a space-like hypersurface and its extrinsic curvature onto the Hilbert space consisting of four arbitrary functions of three variables (see ref. [11] for a fu ler discussion). The general Bianchi IX (as well as VIh, VIIfc, VIII), Cauchy data is specified by four arbitrary constants in vacuum and so in some sense, that is yet to be made precise [11], might be near the general vacuum solution to the Einstein equations. Investigation of aspects of the Mixmaster evolution should provide clues to what can be expected in the general solution. This approach has been particularly stressed by Belinskii, Khalatnikov and Lifshitz [31]. Lastly, it is worth remarking that the type VIII and IX models have no Newtonian analogues, unlike types I—VII. Their unusual dynamics are a consequence of the non- Newtonian, or 'magnetic' portion of the Weyl conformal curvature. They are intrinsically general relativistic phenomena.

Misner [32] originally discovered that the Mixmaster Universe has the unusual property ihat periodically, close to the initial singularity, light can circumnavigate the Universe. This is not possible in other simpler homogeneous models and probably also not in inhomogeneous models either. Misner hoped that this discovery might go some way towards providing an explanation for the remarkable degree of regularity displayed by the present-day Universe; regularity that extends over regions, which in the more conventional cosmological models like Friedmann's, could never have been causally connected during the expansion history of the Universe. How then did they manage to coordinate their structure to within one part in ten thousand today? Misner's suggestion was that if the Universe began in a manner resembling the Bianchi type IX model then causal communication could, in principle, be established over the entire Universe in its earliest stages. Irregularities associated with the 'big bang' would be efficiently ironed-out by viscous transport processes and diffusive mixing.t Unfortunately, subsequent studies revealed that the Mixmaster model very rarely visited configurations conducive to 'mixing' over very large regions [31,35]. The possibility of explaining the present large scale regularity of the Universe by 'mixing' could not be sustained unless the initial conditions of the model were very specially chosen. The Mixmaster model could not guarantee the present structure of the Universe independently of the initial data-the goal of the so-called chaotic cosmology' programme,

In the course of investigating the Mixmaster model and formulating the chaotic cosmology programme Misner [36] introduced a complete Hamiltonian formalism for establishing the qualitative llK- Mm uisier' owes lis name to such physical processes. This has nothing to do with the technical property of strong or weak mixing of !.n.,mu ,il wMcms introduced in section 2. although, as we shall see in section 4 the Mixmaster possesses these trajectory mixing properties also.Ш Barrow, Chaotk behaviour in general relativity

behaviour of the homogeneous Bianchi Universes. The Mixmaster model can be formulated as a conservative dynamic system with a two-dimensional, time-dependent potential in which the type and degree of anisotropy in the universal expansion is described by the motion of a point within a closed potential well.

With this motivation and background we are now in a position to connect the Einstein equations with the dynamical notions of chaos introduced in section 2.