Bianchi universes (s= 3 ) 137In this case the parameterH 0 is no longer essential, since it can be set to unity by
a translation ofτ,τ→τ+constant; then (3.90) implies that
Ht=1
1 +q∗, (3.95)
so that the commutation functions are of the form(constant)×t−^1. It follows
that the resulting cosmological model isself-similar. It thus turns out thatto
each equilibrium point of the DE (3.92) there corresponds a unique self-similar
cosmological model. In such a model the physical states at different times differ
only by an overall change in the length scale. Such models are expanding, but
in such a way that their dimensionless state does not change. They include the
flat FLRW model (=1) and the Milne model (=0). All vacuum and
non-tilted perfect fluid self-similar Bianchi solutions have been given by Hsu and
Wainwright [73]. The equilibrium points determine the asymptotic behaviour of
other more general models.
3.7.2.4 Phase planes
Many phase planes can be constructed explicitly. The reader is referred to
Wainright and Ellis [128] for a comprehensive presentation and survey of results.
Several interesting points emerge
(1) Variety of singularities. Various types of singularity can occur in Bianchi
universes: cigar, pancake and oscillatory in the orthogonal case. In the case
of tilted models, one can, in addition get non-scalar singularities, associated
with a change in the nature of the spacetime symmetries—a horizon occurs
where the surfaces of homogeneity change from being timelike to being
spacelike, so the model changes from being spatially homogeneous to
spatially inhomogeneous [15, 42]. The fluid can then run into timelike
singularities, quite unlike the spacelike singularities in FL models. Thus the
singularity structure can be quite unlike that in a FL model, even in models
that are arbitrarily similar to a FL model today and indeed since the time of
decoupling.
(2) Relation to lower dimensional spaces. It seems that the lower dimensional
spaces, delineating higher symmetry models, may be skeletons guiding the
development of the higher dimensional spaces (the more generic models).
This is one reason why study of the exact higher symmetry models is of
significance.
(3) Identification of models in state space. The analysis of the phase planes for
Bianchi models shows that the procedure sometimes adopted of identifying
all points in state space corresponding to the same model, is not a good idea.
For example the Kasner ring that serves as a framework for evolution of
many other Bianchi models contains multiple realizations of the same Kasner
model. To identify them as the same point in state space would make the