Some examples will now be presented. The following discussion makes use of the Bianchi classification of
homogenous cosmological models (see, e.g., [344]). If we take the Kasner solution and add a perfect fluid
with equation of state
,
, maintaining the symmetry (Bianchi type I), then
the eigenvalues
of the second fundamental satisfy
in the limit of infinite
expansion. The solution isotropizes. More generally this does not happen. If we look at models of
Bianchi type II with non-tilted perfect fluid, i.e. where the fluid velocity is orthogonal to the
homogeneous hypersurfaces, then the quantities
converge to limits that are positive
but differ from
(see [344
], Page 138.) There is partial but not complete isotropization.
The quantities
just introduced are called generalized Kasner exponents, since in the case
of the Kasner solution they reduce to the
in the metric form (3
). This kind of partial
isotropization, ensuring the definiteness of the second fundamental form at late times, seems to be
typical.
Intuitively, a sufficiently general vacuum spacetime should resemble gravitational waves propagating on
some metric describing the large-scale geometry. This could even apply to spatially homogeneous solutions,
provided they are sufficiently general. Hence, in that case also there should be partial isotropization. This
expectation is confirmed in the case of vacuum spacetimes of Bianchi type VIII [313]. In that case the
generalized Kasner exponents converge to non-negative limits different from . For a vacuum model this
can only happen if the quantity
, where
is the spatial scalar curvature, does not tend
to zero in the limit of large time. Detailed asymptotics for these spacetimes has been obtained
in [314].
The Bianchi models of type VIII are the most general indefinitely expanding models of class A. Note,
however, that models of class VI for all
together are just as general. The latter models with perfect
fluid and equation of state
sometimes tend to the Collins model for an open set of values of
for each fixed
(cf. [344], Page 160). These models do not in general exhibit partial
isotropization. It is interesting to ask whether this is connected to the issue of spatial boundary
conditions. General models of class B cannot be spatially compactified in such a way as to be
locally spatially homogeneous while models of Bianchi type VIII can. See also the discussion
in [33].
Another issue is what assumptions on matter are required in order that it have the effect of
(partial) isotropization. Consider the case of Bianchi I. The case of a perfect fluid has already been
mentioned. Collisionless matter described by kinetic theory also leads to isotropization (at
least under the assumption of reflection symmetry), as do fluids with almost any physically
reasonable equation of state [293]. There is, however, one exception. This is the stiff fluid, which
has a linear equation of state with . In that case the generalized Kasner exponents are
time-independent, and may take on negative values. In a model with two non-interacting fluids with
linear equation of state the one with the smaller value of
dominates the dynamics at late
times [120], and so the isotropization is restored. Consider now the case of a magnetic field and a
perfect fluid with linear equation of state. A variety of cases of Bianchi types I, II, and VI
have been studied in [221
, 222, 223], with a mixture of rigorous results and conjectures being
obtained. The general picture seems to be that, apart from very special cases, there is at least
partial isotropization. The asymptotic behaviour varies with the parameter
in the equation
of state and with the Bianchi type (only the case
will be considered here). At one
extreme, Bianchi type I models with
isotropize. At the other extreme, the long time
behaviour resembles that of a magnetovacuum model. This occurs for
in type I, for
in type II and for all
in type VI
. In all these cases there is partial
isotropization.
Under what circumstances can a spatially homogeneous spacetime have the property that the
generalized Kasner exponents are independent of time? The strong energy condition says that
for any causal vector
. It follows from the Hamiltonian constraint and the evolution
equation for
that if the generalized Kasner exponents are constant in time in a spacetime of
Bianchi type I, then the normal vector
to the homogeneous hypersurfaces gives equality in
the inequality of the strong energy condition. Hence the matter model is in a sense on the
verge of violating the strong energy condition and this is a major restriction on the matter
model.
A further question that can be posed concerning the dynamics of expanding cosmological models is
whether tends to zero. This is of cosmological interest since
is (up to a constant factor)
the density parameter
used in the cosmology literature. Note that it is not hard to show that
and
each tend to zero in the limit for any model with
which exists globally in the
future and where the matter satisfies the dominant and strong energy conditions. First, it can
be seen from the evolution equation for
that this quantity is monotone increasing and
tends to zero as
. Then it follows from the Hamiltonian constraint that
tends to
zero.
A reasonable condition to be demanded of an expanding cosmological model is that it be future
geodesically complete. This has been proved for many homogeneous models in [290].
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