, 30]
and Misner [95] discovered
that the Einstein equations in the vacuum homogeneous Bianchi
type IX (or Mixmaster) cosmology exhibit
complex behavior and are sensitive
to initial conditions as the Big Bang singularity is approached.
In particular, the solutions near the singularity are
described qualitatively by a discrete
map [27, 29]
representing different sequences of Kasner spacetimes
with time changing exponents 
, but otherwise constrained by
.
Because this discrete mapping of Kasner epochs is chaotic,
the Mixmaster dynamics is presumed to be chaotic as well.
are the anisotropy canonical coordinates. Seven level
surfaces are shown at equally spaced decades ranging from 
to 
. For large isocontours (V>1), the potential is open and
exhibits a strong triangular symmetry with three narrow channels
extending to spatial infinity. For V<1, the potential closes and
is approximately circular for 
.
Mixmaster behavior can be studied in the context of Hamiltonian dynamics, with a Hamiltonian [96]
and a semi-bounded potential arising from the spatial scalar curvature (whose level curves are plotted in Figure 2)
where 
and are the scale factor and
anisotropies, and 
and 
are the corresponding
conjugate variables. A solution of this Hamiltonian system
is an infinite sequence of Kasner epochs with
parameters that change when the phase space trajectories
bounce off the potential walls, which become exponentially
steep as the system evolves towards the singularity.
Several numerical calculations of the dynamical equations
arising from (2
) and (3)
have indicated that the Liapunov exponents of the system vanish,
in apparent contradiction with the discrete maps [48, 77],
and putting into question the characterization of Mixmaster dynamics
as chaotic. However, it has since been
shown that the usual definition of the
Liapunov exponents is ambiguous in this case
as it is not invariant under time reparametrizations, and that
with a different time variable one obtains positive
exponents [33, 65]. Also,
coordinate independent methods using fractal basin boundaries
to map basins of attraction in the space of initial
conditions indicates Mixmaster spacetimes to be chaotic [58].
Although BLK conjectured that local Mixmaster oscillations
might be the generic behavior for singularities in more
general classes of spacetimes [30],
it is only recently that this conjecture has begun to
be supported by numerical evidence (see Section 3.1.2
and [31]).
),
are characterized by either
open or no potentials in the Hamiltonian framework [109],
and exhibit essentially monotonic or
Asymptotically Velocity Term Dominated (AVTD) behavior.
Considering inhomogeneous spacetimes, Isenberg and
Moncrief [81] proved that the singularity
in the polarized Gowdy model is AVTD type, as are more
general polarized 
symmetric cosmologies [34].
Early numerical studies using symplectic
methods have confirmed these conjectures
and found no evidence of BLK oscillations, even in
spacetimes with U(1) symmetry [32]
(although there were concerns about the solutions due to
difficulties in resolving steep spatial gradients
near the singularity [32]), which were addressed later by
Hern and Stewart [75] for the Gowdy 
models).
However, Weaver et al. [124] have established
the first evidence through numerical simulations that
Mixmaster dynamics can occur in (at least a restricted class of)
inhomogeneous spacetimes which generalize the Bianchi
type VI 
with a magnetic field and two-torus symmetry.
More recently, Berger and Moncrief [36, 37] have
shown U(1) symmetric vacuum cosmologies to exhibit local
Mixmaster dynamics, which tends to support the BLK conjecture.
Despite numerical difficulties in resolving steep
gradients (which they managed by enforcing the Hamiltonian
constraint and spatially averaging the solutions),
Berger and Moncrief have confirmed their findings
under increased spatial resolution and changes in initial
data.
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Computational Cosmology: from the Early Universe to the Large Scale Structure Peter Anninos http://www.livingreviews.org/lrr-2001-2 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |
