The generic singularity in spatially homogeneous cosmologies is
reasonably well understood. The approach to it asymptotically
falls into two classes. The first, called asymptotically velocity
term dominated (AVTD) [94,
165], refers to a cosmology that approaches the Kasner (vacuum,
Bianchi I) solution [171] as
. (Spatially homogeneous universes can be described as a sequence
of homogeneous spaces labeled by
. Here we shall choose
so that
coincides with the singularity.) An example of such a solution
is the vacuum Bianchi II model [236] which begins with a fixed set of Kasner-like anisotropic
expansion rates, and, possibly, makes one change of the rates in
a prescribed way (Mixmaster-like bounce) and then continues to
as a fixed Kasner solution. In contrast are the homogeneous
cosmologies which display Mixmaster dynamics such as vacuum
Bianchi VIII and IX [22,
187,
133] and Bianchi VI
and Bianchi I with a magnetic field [178,
29,
177]. Jantzen [168] has discussed other examples. Mixmaster dynamics describes an
approach to the singularity which is a sequence of Kasner epochs
with a prescription, originally due to Belinskii, Khalatnikov,
and Lifshitz (BKL) [22], for relating one Kasner epoch to the next. Some of the
Mixmaster bounces (era changes) display sensitivity to initial
conditions one usually associates with chaos, and in fact
Mixmaster dynamics is chaotic [86,
194]. Note that we consider an
epoch
to be a subunit of an
era
. In some of the literature, this usage is reversed. The vacuum
Bianchi I (Kasner) solution is distinguished from the other
Bianchi types in that the spatial scalar curvature
, (proportional to) the minisuperspace (MSS) potential [187,
227], vanishes identically. But
arises in other Bianchi types due to spatial dependence of the
metric in a coordinate basis. Thus an AVTD singularity is also
characterized as a regime in which terms containing or arising
from spatial derivatives no longer influence the dynamics. This
means that the Mixmaster models do not have an AVTD singularity
since the influence of the spatial derivatives (through the MSS
potential) never disappears - there is no last bounce. A more
general review of numerical cosmology has been given by
Anninos [4].