Centrella and Matzner [52, 53] studied
a class of plane symmetric cosmologies representing
gravitational inhomogeneities in the form of shocks or
discontinuities separating two vacuum expanding
Kasner cosmologies (1
).
By a suitable choice of parameters,
the constraint equations can be
satisfied at the initial time with an Euclidean 3-surface and
an algebraic matching of parameters
across the different Kasner regions
that gives rise to a discontinuous extrinsic curvature tensor.
They performed both numerical calculations and analytical
estimates using a Green's function analysis to establish
and verify (despite the numerical
difficulties in evolving discontinuous
data) certain aspects of the solutions, including
gravitational wave interactions, the formation of tails,
and the singularity behavior of colliding waves in
expanding vacuum cosmologies.
Shortly thereafter, Centrella and Wilson [54
, 55]
developed a polarized plane symmetric code for cosmology,
adding also hydrodynamic sources with artificial viscosity
methods for shock capturing and Barton's method for
monotonic transport [126].
The evolutions are fully constrained (solving both the
momentum and Hamiltonian constraints at each time step) and use the
mean curvature slicing condition.
This work was subsequently extended by
Anninos et al. [8, 10, 6],
implementing more robust numerical methods, an improved
parametric treatment of the initial value problem,
and generic unpolarized metrics.
In applications of these codes,
Centrella [51] investigated nonlinear gravity waves
in Minkowski space and compared the full numerical
solutions against a first
order perturbation solution to benchmark certain numerical
issues such as numerical damping and dispersion. A second
order perturbation analysis was used to model the
transition into the nonlinear regime.
Anninos et al. [9] considered small and large
perturbations in the two degenerate Kasner models:
or 2/3, and 
or -1/3
respectively, where 
are parameters
in the Kasner metric (1).
Carrying out a second order perturbation expansion
and computing the Newman-Penrose (NP) scalars, Riemann
invariants and Bel-Robinson vector, they demonstrated,
for their particular class of spacetimes, that the
nonlinear behavior is in the Coulomb (or background) part
represented by the leading order term in the NP scalar 
,
and not in the gravitational wave component.
For standing-wave perturbations,
the dominant second order effects in their variables
are an enhanced monotonic increase in the background expansion rate,
and the generation of oscillatory behavior in the background
spacetime with frequencies equal to the harmonics
of the first order standing-wave solution.
Expanding their investigations of the
Coulomb nonlinearity, Anninos and McKinney [14]
used a gauge invariant perturbation formalism to
construct constrained initial data for
general relativistic cosmological sheets formed
from the gravitational collapse of an ideal gas in a
critically closed FLRW ``background'' model.
Results are compared to the Newtonian Zel'dovich [128] solution over a range of
field strengths and flows. Also, the growth rates
of nonlinear modes (in both the gas density and
Riemann curvature invariants), their effect in
the back-reaction to modify the cosmological
scale factor, and their role in generating
CMB anisotropies are discussed.
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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 |