138 Cosmological models
evolution patterns very difficult to follow. It is better to keep them separate,
but to learn to identify where multiple realizations of the same model occur
(which is just theequivalence problemfor cosmological models).3.7.3 Isotropization properties
An issue of importance is whether these models tend to isotropy at early or
late times. An important paper by Collins and Hawking [16] shows that for
ordinary matter, at late times, types I, V, VII, isotropize but other Bianchi models
become anisotropic at very late times, even if they are very nearly isotropic at
present. Thus isotropy is unstable in this case. However, a paper by Wald [130]
showed that Bianchi models will tend to isotropize at late times if there is a
positive cosmological constant present, implying that an inflationary era can
cause anisotropies to die away. The latter work, however, while applicable to
models with non-zero tilt angle, did not show this angle dies away, and indeed it
does not do so in general (Goliath and Ellis [62]). Inflation also only occurs in
Bianchi models if there is not too much anisotropy to begin with (Rothman and
Ellis [111]), and it is not clear that shear and spatial curvature are in fact removed
in all cases [109]. Hence, some Bianchi models isotropize due to inflation, but
not all.
An important idea that arises out of this study is that ofintermediate
isotropization: namely, models that become very like a FLRW model for a period
of their evolution but start and end quite unlike these models. It turns out that
many Bianchi types allow intermediate isotropization, because the FLRW models
are saddle points in the relevant phase planes. This leads to the following two
interesting results:
Bianchi evolution theorem 1.Consider a family of Bianchi models that allow
intermediate isotropization. Define an-neighbourhood of a FLRW model as a
region in state space where all geometrical and physical quantities are closer
thanto their values in a FLRW model. Choose a time scale L. Then no matter
how smalland how large L, there is an open set of Bianchi models in the state
space such that each model spends longer than L within the corresponding-
neighbourhood of the FLRW model.
This follows because the saddle point is a fixed point of the phase flow;
consequently the phase flow vector becomes arbitrarily close to zero at all
points in a small enough open region around the FLRW point in state space.
Consequently, although these models are quite unlike FLRW models at very
early and very late times, there is an open set of them that are observationally
indistinguishable from a FLRW model (chooseLlong enough to encompass
from today to last coupling or nucleosynthesis, andto correspond to current
observational bounds). Thus there exist many such models that are viable
as models of the real universe in terms of compatibility with astronomical
observations.