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 (local Mixmaster dynamics (LMD)) [22Jump To The Next Citation Point In The Article]. For a review of homogeneous cosmologies, inhomogeneous cosmologies, and the relation between them, see [182]. The main difficulty with the acceptance of this conjecture has been the controversy over whether the required time slicing can be constructed globally [13, 122]. Montani [192], Belinskii [16], and Kirillov and Kochnev [175, 174] 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. BKL seem to imply [22] that LMD should only be expected to occur in completely general spacetimes with no spatial symmetries. However, LMD is actually possible in any spatially inhomogeneous cosmology with a local MSS with a ``closed'' potential (in the sense of the standard triangular potentials of Bianchi VIII and IX). This closure may be provided by spatial curvature, matter fields, or rotation. A class of cosmological models which appear to have local MD are vacuum universes on tex2html_wrap_inline2066 with a U (1) symmetry [190Jump To The Next Citation Point In The Article]. Even simpler plane symmetric Gowdy cosmologies [121Jump To The Next Citation Point In The Article, 26Jump To The Next Citation Point In The Article] have ``open'' local MSS potentials. However, these models are interesting in their own right since they have been conjectured to possess an AVTD singularity [123Jump To The Next Citation Point In The Article]. One way to obtain these Gowdy models is to allow spatial dependence in one direction in Bianchi I homogeneous cosmologies [26Jump To The Next Citation Point In The Article]. It is well-known that addition of matter terms to homogeneous Bianchi I, Bianchi VI tex2html_wrap_inline2006, and other AVTD models can yield Mixmaster behavior [168, 178, 177]. Allowing spatial dependence in one direction in such models might then yield a spacetime with LMD. Application of this procedure to magnetic Bianchi VI tex2html_wrap_inline2006 models yields magnetic Gowdy models [250Jump To The Next Citation Point In The Article, 247Jump To The Next Citation Point In The Article]. Of course, Gowdy cosmologies are not the most general tex2html_wrap_inline2040 symmetric vacuum spacetimes [121Jump To The Next Citation Point In The Article, 82, 40Jump To The Next Citation Point In The Article]. Restoring the ``twists'' introduces a centrifugal wall to close the MSS. Magnetic Gowdy and general tex2html_wrap_inline2040 symmetric models appear to admit LMD [247Jump To The Next Citation Point In The Article, 248, 41Jump To The Next Citation Point In The Article].

The past few years have seen the development of strong numerical evidence in support of the BKL claims [36Jump To The Next Citation Point In The Article]. The Method of Consistent Potentials (MCP) [123Jump To The Next Citation Point In The Article] has been used to organize the data obtained in simulations of spatially inhomogeneous cosmologies [42Jump To The Next Citation Point In The Article, 35Jump To The Next Citation Point In The Article, 250Jump To The Next Citation Point In The Article, 43Jump To The Next Citation Point In The Article, 36Jump To The Next Citation Point In The Article, 33Jump To The Next Citation Point In The Article, 41Jump To The Next Citation Point In The Article]. The main idea is to obtain a Kasner-like velocity term dominated (VTD) solution at every spatial point by solving Einstein's equations truncated by removing all terms containing spatial derivatives. If the spacetime really is AVTD, all the neglected terms will be subdominant (exponentially small in variables where the VTD solution is linear in the time tex2html_wrap_inline1866) when the VTD solution is substituted back into them. For the MCP to successfully predict whether or not the spacetime is AVTD, the dynamics of the full solution must be dominated at (almost) every spatial point by the VTD solution behavior. Surprisingly, MCP predictions have proved valid in numerical simulations of cosmological spacetimes with one [43Jump To The Next Citation Point In The Article] and two [42Jump To The Next Citation Point In The Article, 35Jump To The Next Citation Point In The Article, 41Jump To The Next Citation Point In The Article] spatial symmetries. In the case of U (1) symmetric models, a comparison between the observed behavior [43Jump To The Next Citation Point In The Article] and that in a vacuum, diagonal Bianchi IX model written in terms of U (1) variables provides strong support for LMD [45] since the phenomenology of the inhomogeneous cosmologies can be reproduced by this rewriting of the standard Bianchi IX MD.

Polarized plane symmetric cosmologies have been evolved numerically using standard techniques by Anninos, Centrella, and Matzner [5, 6]. The long-term project involving Berger, Garfinkle, and Moncrief and their students to study the generic cosmological singularity numerically has been reviewed in detail elsewhere [37Jump To The Next Citation Point In The Article, 36Jump To The Next Citation Point In The Article, 32] but will be discussed briefly here.

3.4.2 Gowdy cosmologies and their generalizations 

The Gowdy model on tex2html_wrap_inline2066 is described by gravitational wave amplitudes tex2html_wrap_inline2086 and tex2html_wrap_inline2088 which propagate in a spatially inhomogeneous background universe described by tex2html_wrap_inline2090 . (We note that the physical behavior of a Gowdy spacetime can be computed from the effect of the metric evolution on a test cylinder [39].) We impose tex2html_wrap_inline2092 and periodic boundary conditions. The time variable tex2html_wrap_inline1866 measures the area in the symmetry plane with tex2html_wrap_inline2002 being 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_inline2102) obtained from the variation of [188]

  eqnarray552

where tex2html_wrap_inline2104 and tex2html_wrap_inline2106 are canonically conjugate to P and Q respectively. 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_inline2112 -momentum constraints, respectively. These can be expressed as first order equations for tex2html_wrap_inline1950 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 [26]. For this case, it has been proven that the singularity is AVTD [165Jump To The Next Citation Point In The Article]. This has also been conjectured to be true for generic Gowdy models [123Jump To The Next Citation Point In The Article].

AVTD behavior is defined in [165] as follows: Solve the Gowdy wave equations neglecting all terms containing spatial derivatives. This yields the VTD solution [42Jump To The Next Citation Point In The Article]. If the approach to the singularity is AVTD, the full solution comes arbitrarily close to a VTD solution at each spatial point as tex2html_wrap_inline1996 . As tex2html_wrap_inline1996, the VTD solution becomes

equation438

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_inline2130  [123]. We have shown that, except at isolated spatial points, the nonlinear terms in the wave equation for P drive v into this range [35Jump To The Next Citation Point In The Article, 37Jump 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 [42Jump To The Next Citation Point In The Article, 35Jump To The Next Citation Point In The Article]. We conclude that generic Gowdy cosmologies have an AVTD singularity except at isolated spatial points [35Jump To The Next Citation Point In The Article, 37Jump To The Next Citation Point In The Article]. This has been confirmed by Hern and Stuart [143] and by van Putten [243]. After the nature of the solutions became clear through numerical experiments, it became possible to use Fuchsian asymptotic methods to prove that Gowdy solutions with 0 < v < 1 and AVTD behavior almost everywhere are generic [173]. These methods have recently been applied to Gowdy spacetimes with tex2html_wrap_inline1990 and tex2html_wrap_inline1964 topologies with similar conclusions [233].

One striking property of the Gowdy models are the development of ``spiky features'' at isolated spatial points where the coefficient of a local ``potential term'' vanishes [42, 35]. Recently, Rendall and Weaver have shown analytically how to generate such spikes from a Gowdy solution without spikes [220].

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_inline2006 models provides an additional potential which grows exponentially if 0 < v < 1 . Local Mixmaster behavior has recently been observed in these magnetic Gowdy models [250, 247].

Garfinkle has used a vacuum Gowdy model with tex2html_wrap_inline1990 spatial topology to test an algorithm for axis regularity [111]. Along the way, he has shown that these models are also AVTD with behavior at generic spatial points that is eventually identical to that in the tex2html_wrap_inline1988 case. Comparison of the two models illustrates that topology or other global or boundary conditions are important early in the simulation but become irrelevant as the singularity is approached.

Gowdy spacetimes are not the most general tex2html_wrap_inline2040 symmetric vacuum cosmologies. Certain off-diagonal metric components (the twists which are tex2html_wrap_inline2152, tex2html_wrap_inline2154 in the notation of (12Popup Equation)) have been set to zero [121]. Restoring these terms (see [83, 40]) yields spacetimes that are not AVTD but rather appear to exhibit a novel type of LMD [41Jump To The Next Citation Point In The Article, 249]. The LMD in these models is an inhomogeneous generalization of non-diagonal Bianchi models with ``centrifugal'' MSS potential walls [227, 169] in addition to the usual curvature walls. In [41], remarkable quantitative agreement is found between predictions of the MCP and numerical simulation of the full Einstein equations. A version of the code with AMR has been developed [15]. (Asymptotic methods have been used to prove that the polarized version of these spacetimes have AVTD solutions [163].)

3.4.3 tex2html_wrap_inline2156 symmetric cosmologies

Moncrief has shown [190] that cosmological models on tex2html_wrap_inline2066 with a spatial U (1) symmetry can be described by five degrees of freedom tex2html_wrap_inline2162 and their respective conjugate momenta tex2html_wrap_inline2164 . All variables are functions of spatial variables u, v and time tex2html_wrap_inline1866 . Einstein's equations can be obtained by variation of

  eqnarray554

Here tex2html_wrap_inline2172 and tex2html_wrap_inline2174 are analogous to P and Q while tex2html_wrap_inline2180 is a conformal factor for the metric tex2html_wrap_inline2182 in the u - v plane perpendicular to the symmetry direction. Symplectic methods are still easily applicable. Note particularly that tex2html_wrap_inline2028 contains two copies of the Gowdy tex2html_wrap_inline2028 plus a free particle term and is thus exactly solvable. The potential term tex2html_wrap_inline2030 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 [200] is used. A spectral evaluation of derivatives [100] which has been shown to work in Gowdy simulations [23] 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 [37Jump To The Next Citation Point In The Article].

 

Click on thumbnail to view image

Figure 6: Behavior of the gravitational wave amplitude at a typical spatial point in a collapsing U(1) symmetric cosmology. For details see [43Jump To The Next Citation Point In The Article, 36].

Current limitations of the U (1) code do not affect simulations for the polarized case since problematic spiky features do not develop. Polarized models have tex2html_wrap_inline2200 . Grubisic and Moncrief [124] have conjectured that these polarized models are AVTD. The numerical simulations provide strong support for this conjecture [37Jump To The Next Citation Point In The Article, 44]. Asymptotic methods have been used to prove that an open set of AVTD solutions exist for this case [164].

3.4.4 Going further

The MCP indicates that the term containing gradients of tex2html_wrap_inline2174 in (14Popup Equation) acts as a Mixmaster-like potential to drive the system away from AVTD behavior in generic U (1) models [24]. Numerical simulations provide support for this suggestion [37, 43Jump 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 of U (1) symmetric cosmologies [30, 31]. In fact, it has been found [43], that when the Hamiltonian constraint is enforced at every time step, the predicted local oscillatory behavior of the approach to the singularity is observed. (The momentum constraint is not enforced.) (Note that in a numerical study of vacuum Bianchi IX homogeneous cosmologies, Zardecki obtained a spurious enhancement of Mixmaster oscillations due to constraint violation [253, 147]. In this case, the constraint violation introduced negative energy.)

Mixmaster simulations with the new algorithm [38] can easily evolve more than 250 bounces reaching tex2html_wrap_inline2208 . This compares to earlier simulations yielding 30 or so bounces with tex2html_wrap_inline2210 . The larger number of bounces quickly reveals that it is necessary to enforce the Hamiltonian constraint. An explicitly constraint enforcing U (1) code was developed some years ago by Ove (see [207] and references therein).

It is well known [20] that a scalar field can suppress Mixmaster oscillations in homogeneous cosmologies. BKL argued that the suppression would also occur in spatially inhomogeneous models. This was demonstrated numerically for magnetic Gowdy and U (1) symmetric spacetimes [33]. Andersson and Rendall proved that completely general cosmological (spatially tex2html_wrap_inline1988) spacetimes (no symmetries) with sufficiently strong scalar fields have generic AVTD solutions [3]. Garfinkle [107] has constructed a 3D harmonic code which, so far, has found AVTD solutions with a scalar field present. Work on generic vacuum models is in progress.

Cosmological models inspired by string theory contain higher derivative curvature terms and exotic matter fields. Damour and Henneaux have applied the BKL approach to such models and conclude that their approach to the singularity exhibits LMD [88].

Finally, there has been a study of the relationship between the ``long wavelength approximation'' and the BKL analyses by Deruelle and Langlois [90].



4 Discussion3 Singularities in Cosmological Models3.3 Mixmaster dynamics

image Numerical Approaches to Spacetime Singularities
Beverly K. Berger
http://www.livingreviews.org/lrr-2002-1
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