Inhomogeneous cosmologies

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)) [22 ]. 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 $T3 × R$ with a $U (1)$ symmetry [190 ]. Even simpler plane symmetric Gowdy cosmologies [121 , 23 ] have “open” local MSS potentials. However, these models are interesting in their own right since they have been conjectured to possess an AVTD singularity [123 ]. One way to obtain these Gowdy models is to allow spatial dependence in one direction in Bianchi I homogeneous cosmologies [23 ]. It is well-known that addition of matter terms to homogeneous Bianchi I, Bianchi VI $0$ , 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 $0$ models yields magnetic Gowdy models [250 , 247 ]. Of course, Gowdy cosmologies are not the most general $2 T$ symmetric vacuum spacetimes [121 , 82 , 34 ]. Restoring the “twists” introduces a centrifugal wall to close the MSS. Magnetic Gowdy and general $T2$ symmetric models appear to admit LMD [247 , 248 , 41 ].

The past few years have seen the development of strong numerical evidence in support of the BKL claims [37 ]. The Method of Consistent Potentials (MCP) [123 ] has been used to organize the data obtained in simulations of spatially inhomogeneous cosmologies [42 , 36 , 250 , 43 , 37 , 32 , 41 ]. 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 $τ$ ) 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 [43 ] and two [42 , 36 , 41 ] spatial symmetries. In the case of $U (1)$ symmetric models, a comparison between the observed behavior [43 ] 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 [38 , 37 , 31 ] but will be discussed briefly here.

3.4.2 Gowdy cosmologies and their generalizations

The Gowdy model on $T 3 × R$ is described by gravitational wave amplitudes $P (𝜃,τ)$ and $Q(𝜃,τ )$ which propagate in a spatially inhomogeneous background universe described by $λ(𝜃,τ)$ . (We note that the physical behavior of a Gowdy spacetime can be computed from the effect of the metric evolution on a test cylinder [40 ].) We impose $0 ≤ 𝜃 ≤ 2π$ and periodic boundary conditions. The time variable $τ$ measures the area in the symmetry plane with $τ = ∞$ 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 $,a = ∂∕ ∂a$ ) obtained from the variation of [188 ]

2∫π [ ] H = 1- d𝜃 π2P + e−2Pπ2Q 2 0 2π 1∫ [ −2τ ( 2 2P 2) ] + 2 d𝜃 e P,𝜃 +e Q,𝜃 = H1 + H2, (12 ) 0

where $πP$ and $πQ$ 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 $𝜃$ -momentum constraints, respectively. These can be expressed as first order equations for $λ$ 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 [23 ]. For this case, it has been proven that the singularity is AVTD [164 ]. This has also been conjectured to be true for generic Gowdy models [123 ].

AVTD behavior is defined in [164 ] as follows: Solve the Gowdy wave equations neglecting all terms containing spatial derivatives. This yields the VTD solution [42 ]. If the approach to the singularity is AVTD, the full solution comes arbitrarily close to a VTD solution at each spatial point as $τ → ∞$ . As $τ → ∞$ , the VTD solution becomes

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 $0 ≤ v < 1$ [123 ]. We have shown that, except at isolated spatial points, the nonlinear terms in the wave equation for $P$ drive $v$ into this range [36

, 38

]. 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 [42

, 36

]. We conclude that generic Gowdy cosmologies have an AVTD singularity except at isolated spatial points [36

, 38

]. 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 $S2 × S1$ and $S3$ 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 , 36 ]. 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 $0$ 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 $S2 × S1$ 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 $3 T$ 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 $2 T$ symmetric vacuum cosmologies. Certain off-diagonal metric components (the twists which are $g𝜃σ$ , $g𝜃δ$ in the notation of (12 )) have been set to zero [121 ]. Restoring these terms (see [83 , 34 ]) yields spacetimes that are not AVTD but rather appear to exhibit a novel type of LMD [41 , 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 $U (1 )$ symmetric cosmologies

Moncrief has shown [190 ] that cosmological models on $T 3 × R$ with a spatial $U(1)$ symmetry can be described by five degrees of freedom ${x,z,Λ, φ,ω }$ and their respective conjugate momenta ${px, pz,pΛ,p,r}$ . All variables are functions of spatial variables $u$ , $v$ and time $τ$ . Einstein’s equations can be obtained by variation of

∮ ∮ H = dudv ℋ ∮ ∮ ( 1 2 1 4z 2 1 2 1 4φ 2 1 2 ) = dudv 8pz + 2e px + 8p + 2e r − 2pΛ + 2pΛ −2τ ∮ ∮ {( Λ ab) ( Λ ab) +e dudv e e ,ab− e e ,a Λ,b Λ [( −2z) ( −2z) ] +e e ,u x,v − e ,v x,u Λ ab 1 Λ −4φ ab } +2e e φ,aφ,b+ 2e e e ω,aω,b = H1 + H2. (14 )

Here $φ$ and $ω$ are analogous to $P$ and $Q$ while $eΛ$ is a conformal factor for the metric $eab(x,z)$ in the $u$ – $v$ plane perpendicular to the symmetry direction. Symplectic methods are still easily applicable. Note particularly that $H1$ contains two copies of the Gowdy $H1$ plus a free particle term and is thus exactly solvable. The potential term $H2$ 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 [33 ] 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 [38

Figure 6: Behavior of the gravitational wave amplitude at a typical spatial point in a collapsing $U (1)$ symmetric cosmology. For details see [43

, 37 ].

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 $r = ω = 0$ . Grubisić and Moncrief [124 ] have conjectured that these polarized models are AVTD. The numerical simulations provide strong support for this conjecture [38 , 44 ]. Asymptotic methods have been used to prove that an open set of AVTD solutions exist for this case [165 ].

3.4.4 Going further

The MCP indicates that the term containing gradients of $ω$ in (14 ) acts as a Mixmaster-like potential to drive the system away from AVTD behavior in generic $U (1)$ models [30 ]. Numerical simulations provide support for this suggestion [38 , 43 ]. 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 [27 , 28 ]. 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 [39 ] can easily evolve more than 250 bounces reaching $|Ω | ≈ 1062$ . This compares to earlier simulations yielding 30 or so bounces with $|Ω| ≈ 108$ . 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 [32 ]. Andersson and Rendall proved that completely general cosmological (spatially $3 T$ ) spacetimes (no symmetries) with sufficiently strong scalar fields have generic AVTD solutions [3 ]. Garfinkle [113 ] 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 ].

3.4 Inhomogeneous cosmologies

3.4.1 Overview

3.4.2 Gowdy cosmologies and their generalizations

3.4.3 symmetric cosmologies

3.4.4 Going further

3.4.3 $U (1 )$ symmetric cosmologies