Centrella and Matzner [20,
21] 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 [22, 23] developed a more general plane symmetric code for cosmology, adding also hydrodynamic sources. In order to simplify the resulting differential equations, they adopted a diagonal 3-metric of the form
which is maintained in time with a proper choice of shift
vector. The metric (6) allows an overall conformal factor
A
to simplify the initial value problem, and a dynamical
transverse wave component in the variable
h
. The hydrodynamic equations are solved using artificial
viscosity methods for shock capturing and Barton's method for
monotonic transport [57]. 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. [1,
3], implementing more robust numerical methods and an improved
parametric treatment of the initial value problem.
In applications of these codes, Centrella [19] 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 also used to model the transition into the nonlinear
regime. Anninos et al. [2] 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.
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Computational Cosmology: from the Early Universe to the
Large Scale Structure
Peter Anninos http://www.livingreviews.org/lrr-1998-9 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |