### 7.3 Homogeneous models with

One important aspect of the fragmentary picture of the dynamics of expanding cosmological models
presented in the last two Sections 7.1 and 7.2 is that it seems to be complicated. A situation
where we can hope for a simpler, more unified picture is that where there is sufficiently strong
acceleration of the cosmological expansion. This has the tendency to damp out irregularities. The
simplest way of achieving this is to introduce a positive cosmological constant. Recall first that
when the cosmological constant vanishes and the matter satisfies the usual energy conditions,
spacetimes of Bianchi type IX recollapse [229] and so never belong to the indefinitely expanding
models. When this is no longer true. Then Bianchi IX spacetimes show complicated
features, and it has been suggested in the literature that they exhibit chaotic behaviour (cf. [131]).
A more recent study [175] suggests that the claimed features of the solutions indicating the
presence of chaos may be artefacts of the numerical treatment of a dynamical system which is not
everywhere regular. The numerical work in [175] gives a different picture, parts of which the
authors confirm by mathematical proofs. As a consequence this system, while complicated,
may not be so intractable as previously feared and merits further analytical and numerical
investigation.
In discussing homogeneous models in the following we restrict to Bianchi types other than type IX.
Then a general theorem of Wald [346] states that any model whose matter content satisfies the strong and
dominant energy conditions and which expands for an infinite proper time is such that all generalized
Kasner exponents tend to as . A positive cosmological constant leads to isotropization. The
mean curvature tends to the constant value as , while the scale factors increase
exponentially.

Wald’s result is only dependent on energy conditions and uses no details of the matter field equations.
The question remains whether solutions corresponding to initial data for the Einstein equations with
positive cosmological constant, coupled to reasonable matter, exist globally in time under the sole
condition that the model is originally expanding. It can be shown that this is true for various
matter models using the techniques of [290, 287]. This has been worked out in detail for the
case of collisionless matter by Lee [224]. For the case of a perfect fluid with linear equation of
state see [303]. Once global existence is known and a specific matter model has been chosen,
details of the asymptotic behaviour of the matter fields can be determined and this was done
in [224, 303]. For instance, it was shown that the solutions of the Vlasov equation behave like dust
asymptotically.