3.7 Regge calculus model 3 Relativistic Cosmology 3.5 Nucleosynthesis

3.6 Plane symmetric gravitational waves 

Gravitational waves are an inevitable product of the Einstein equations, and one can expect a wide spectrum of wave signals propagating throughout our Universe due to shear anisotropies, primordial metric and matter fluctuations, collapsing matter structures, ringing black holes, and colliding neutron stars, for example. The discussion here is restricted to the pure vacuum field dynamics and the fundamental nonlinear behavior of gravitational waves in numerically generated cosmological spacetimes.

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 (1Popup Equation). 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 [54Jump To The Next Citation Point
 In The Article, 55Jump To The Next Citation Point In The Article] 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 [126Jump To The Next Citation Point In The 
Article]. 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, 6Jump To The Next Citation Point In The Article], 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: tex2html_wrap_inline2705 or 2/3, and tex2html_wrap_inline2709 or -1/3 respectively, where tex2html_wrap_inline2573 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 tex2html_wrap_inline2715 , 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 [14Jump To The Next Citation Poi
nt In The Article] 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 [128J
ump To The Next Citation Point In The Article] 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.

3.7 Regge calculus model 3 Relativistic Cosmology 3.5 Nucleosynthesis

Computational Cosmology: from the Early Universe to the Large Scale Structure
Peter Anninos
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de