4 Discussion3 Singularities in Cosmological Models3.3 Mixmaster Dynamics

3.4 Inhomogeneous Cosmologies

3.4.1 Overview

BKL have conjectured that one should expect a generic singularity in spatially inhomogeneous cosmologies to be locally of the Mixmaster type [15]. For a review of homogeneous cosmologies, inhomogeneous cosmologies, and the relation between them, see [134]. The main difficulty with the acceptance of this conjecture has been the controversy over whether the required time slicing can be constructed globally [8, 89]. Montani [144], Belinskii {[10], and Kirillov and Kochnev [128, 127] have pointed out that if the BKL conjecture is correct, the spatial structure of the singularity could become extremely complicated as bounces occur at different locations at different times. A class of cosmological models which might have local MD are vacuum universes on tex2html_wrap_inline1488 with a U (1) symmetry [142Jump To The Next Citation Point In The Article]. The non-commuting Killing vectors of local MD can be constructed since only one Killing vector is already present. The two commuting Killing vectors of the even simpler plane symmetric Gowdy cosmologies [88, 18Jump To The Next Citation Point In The Article] preclude their use to test the conjecture. However, these models are interesting in their own right since they have been conjectured to possess an AVTD singularity [90Jump To The Next Citation Point In The Article].

Polarized plane symmetric cosmologies have been evolved numerically using standard techniques by Anninos, Centrella, and Matzner [2, 3]. A long-term project involving Berger, Garfinkle, and Moncrief and our students to study the generic cosmological singularity numerically has been reviewed in detail elsewhere [26Jump To The Next Citation Point In The Article] but will be discussed briefly here.

3.4.2 Gowdy Cosmologies

The Gowdy model on tex2html_wrap_inline1488 is described by gravitational wave amplitudes tex2html_wrap_inline1494 and tex2html_wrap_inline1496 which propagate in a spatially inhomogeneous background universe described by tex2html_wrap_inline1498 . (We note that the physical behavior of a Gowdy spacetime can be computed from the effect of the metric evolution on a test cylinder [29].) We impose tex2html_wrap_inline1500 and periodic boundary conditions. The time variable tex2html_wrap_inline1308 measures the area in the symmetry plane with tex2html_wrap_inline1432 a curvature singularity.

Einstein's equations split into two groups. The first is nonlinearly coupled wave equations for dynamical variables P and Q (where tex2html_wrap_inline1510) obtained from the variation of [140]

  eqnarray288

This Hamiltonian has the form required by the symplectic scheme. If the model is, in fact, AVTD, the approximation in the symplectic numerical scheme should become more accurate as the singularity is approached. The second group of Einstein equations contains the Hamiltonian and tex2html_wrap_inline1512 -momentum constraints respectively. These can be expressed as first order equations for tex2html_wrap_inline1394 in terms of P and Q . This break into dynamical and constraint equations removes two of the most problematical areas of numerical relativity from this model--the initial value problem and numerical preservation of the constraints.

For the special case of the polarized Gowdy model (Q =0), P satisfies a linear wave equation whose exact solution is well-known [18]. For this case, it has been proven that the singularity is AVTD [119Jump To The Next Citation Point In The Article]. This has also been conjectured to be true for generic Gowdy models [90Jump To The Next Citation Point In The Article].

AVTD behavior is defined in [119] as follows: Solve the Gowdy wave equations neglecting all terms containing spatial derivatives. This yields the AVTD solution [32Jump To The Next Citation Point In The Article]. If the approach to the singularity is AVTD, the full solution comes arbitrarily close to an AVTD solution at each spatial point as tex2html_wrap_inline1426 . As tex2html_wrap_inline1426, the AVTD solution becomes

equation303

where v > 0. Substitution in the wave equations shows that this behavior is consistent with asymptotic exponential decay of all terms containing spatial derivatives only if tex2html_wrap_inline1530 [90]. We have shown that, except at isolated spatial points, the nonlinear terms in the wave equation for P drive v into this range [25Jump To The Next Citation Point In The Article, 26Jump To The Next Citation Point In The Article]. The exceptional points occur when coefficients of the nonlinear terms vanish and are responsible for the growth of spiky features seen in the wave forms [32, 25Jump To The Next Citation Point In The Article]. We conclude that generic Gowdy cosmologies have an AVTD singularity except at isolated spatial points [25, 26Jump To The Next Citation Point In The Article]. A claim to the contrary [107] is incorrect [27]. Recently, it has been proven analytically that Gowdy solutions with 0 < v < 1 and AVTD behavior almost everywhere are generic [126].

Addition of a magnetic field to the vacuum Gowdy models (plus a topology change) which yields the inhomogeneous generalization of magnetic Bianchi VI tex2html_wrap_inline1436 models provides an additional potential which grows exponentially if 0 < v < 1. Local Mixmaster behavior has recently been observed in these magnetic Gowdy models [181].

3.4.3 U (1) Symmetric Cosmologies

Given the success of the symplectic method in studying the singularity behavior of the Gowdy model, we can consider its extension to the case of U (1) symmetric cosmologies. Moncrief has shown [142] that cosmological models on tex2html_wrap_inline1488 with a spatial U (1) symmetry   can be described by five degrees of freedom tex2html_wrap_inline1550 and their respective conjugate momenta tex2html_wrap_inline1552 . All variables are functions of spatial variables u, v and time, tex2html_wrap_inline1308 . Einstein's equations can be obtained by variation of

  eqnarray317

Here tex2html_wrap_inline1560 and tex2html_wrap_inline1562 are analogous to P and Q while tex2html_wrap_inline1568 is a conformal factor for the metric tex2html_wrap_inline1570 in the u - v plane perpendicular to the symmetry direction. Note particularly that tex2html_wrap_inline1456 contains two copies of the Gowdy tex2html_wrap_inline1456 plus a free particle term and is thus exactly solvable. The potential term tex2html_wrap_inline1458 is very complicated. However, it still contains no momenta, so its equations are trivially exactly solvable. However, issues of spatial differencing are problematic. (Currently, a scheme due to Norton [150] is used. A spectral evaluation of derivatives [74] which has been shown to work in Gowdy simulations [17] does not appear to be helpful in the U (1) case.) A particular solution to the initial value problem is used, since the general solution is not available [26Jump To The Next Citation Point In The Article]. Currently, except as discussed below, the constraints are monitored but not explicitly enforced.

Despite the current limitations of the U (1) code, conclusions can be confidently drawn for polarized U (1) models in our restricted class of initial data. No numerical difficulties arise in this case. Polarized models have tex2html_wrap_inline1588 . Grubisic and Moncrief [91Jump To The Next Citation Point In The Article] have conjectured that these polarized models are AVTD. The numerical simulations provide strong support for this conjecture [26Jump To The Next Citation Point In The Article, 31].

3.4.4 Going Further

Methods similar to those used in [91], indicate that the term containing gradients of tex2html_wrap_inline1562 in (14Popup Equation) acts as a Mixmaster-like potential to drive the system away from AVTD behavior in generic U (1) models [16]. Numerical simulations provide support for this suggestion [26, 30Jump To The Next Citation Point In The Article]. Whether this potential term grows or decays depends on a function of the field momenta. This in turn is restricted by the Hamiltonian constraint. However, failure to enforce the constraints can cause an erroneous relationship among the momenta to yield qualitatively wrong behavior. There is numerical evidence that this error tends to suppress Mixmaster-like behavior leading to apparent AVTD behavior in extended spatial regions [22, 23]. In fact, it has been found very recently [30] that when the Hamiltonian constraint is enforced at every time step, the predicted local oscillatory behavior of the approach to the singularity is observed.

Mixmaster simulations with the new algorithm [28] can easily evolve more than 250 bounces reaching tex2html_wrap_inline1595 . This compares to earlier simulations yielding 30 or so bounces with tex2html_wrap_inline1597 . The larger number of bounces quickly reveals that it is necessary to enforce the Hamiltonian constraint. A similar conclusion for gravitational wave equations was obtained by Gundlach and Pullin [99]. Although this particular analysis may be incorrect [47], it is still likely that constraint enforcement will be essential for sufficiently long simulations. An explicitly constraint enforcing U (1) code was developed some years ago by Ove (see [153] and references therein).



4 Discussion3 Singularities in Cosmological Models3.3 Mixmaster Dynamics

image Numerical Approaches to Spacetime Singularities
Beverly K. Berger
http://www.livingreviews.org/lrr-1998-7
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