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4.12 Anisotropy: Applications

4.12.1 Isotropization

Matter fields are not the only contributions to the Hamiltonian in cosmology. The effect of anisotropies can also be included in an isotropic model in this way. The late time behavior of this contribution can be shown to behave as −6 a in the shear energy density [228], which falls off faster than any other matter component. Thus, toward later times the universe becomes more and more isotropic.

In backward evolution, on the other hand, this means that the shear term diverges most strongly, which suggests that this term should be most relevant for the singularity issue. Even if matter densities are cut off as discussed before, the presence of bounces would depend on the fate of the anisotropy term. This simple reasoning is not true, however, since the behavior of shear is only effective and uses assumptions about the behavior of matter. Thus, it cannot simply be extrapolated to early times. Anisotropies are independent degrees of freedom which affect the evolution of the scale factor. Only in certain regimes can this contribution be modeled simply by a function of the scale factor alone; in general, one has to use the coupled system of equations for the scale factor, anisotropies and possible matter fields.

4.12.2 Bianchi IX

Corrections to classical behavior are most drastic in the Bianchi IX model with symmetry group ∼ S = SU (2) such that I n = 1. The classical evolution can be described by a 3-dimensional mechanics system with a potential obtained from Equation (37View Equation) such that the kinetic term is quadratic in derivatives of aI with respect to a time coordinate τ defined by dt = a1a2a3dτ. This potential

I 1 2 3 2 1 3 3 1 2 W (p ) = (Γ(2Γ 3 − n Γ 1)p p + (Γ 1Γ 3 − n Γ 2)p p + (Γ 1Γ 2 − n Γ 3)p p ) (40 ) 1 ( p2p3)2 ( p1p3)2 ( p1p2)2 = -- --1-- + --2-- + --3-- − 2(p1)2 − 2 (p2)2 − 2(p3)2 4 p p p
diverges at small I p, in particular (in a direction-dependent manner) at the classical singularity where all I p = 0. Figure 4Watch/download Movie illustrates the walls of the potential, which with decreasing volume push the universe toward the classical singularity.

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Figure 4: mpg-Movie (1700 KB) An illustration of the Bianchi IX potential (40View Equation) and the movement of its walls, rising toward zero p1 and p2 and along the diagonal direction, toward the classical singularity with decreasing volume V = ∘ |p1p2p3|-. The contours are plotted for the function 1 2 2 1 2 W (p ,p ,V ∕(p p )).

As before in isotropic models, phenomenological equations in which the behavior of eigenvalues of the spin connection components is used, do not have this divergent potential. Instead, if two pI are held fixed and the third approaches zero, the effective potential is cut off and goes back to zero at small values, which changes the approach to the classical singularity. Yet, the effective potential is unbounded if one pI diverges while another goes to zero and the situation is qualitatively different from the isotropic case. Since the effective potential corresponds to spatial intrinsic curvature, curvature is not bounded in anisotropic effective models. However, this is a statement only about (kinematical) curvature expressions on minisuperspace, and the more relevant question is what happens to curvature along dynamical trajectories obtained by solving equations of motion. This demonstrates that dynamical equations must always be considered to draw conclusions for the singularity issue.

The approach to the classical singularity is best analyzed in Misner variables [229] consisting of the scale factor Ω := − 1 log V 3 and two anisotropy parameters β± defined such that

√- √- a1 = e −Ω+β++ 3β− , a2 = e−Ω+ β+ − 3β− , a3 = e− Ω−2β+ .

The classical potential then takes the form

( ) W (Ω,β ) = 1e− 4Ω e− 8β+ − 4e −2β+ cosh(2√3-β ) + 2e4β+(cosh(4√3-β ) − 1) , ± 2 − −

which at fixed Ω has three exponential walls rising from the isotropy point β± = 0 and enclosing a triangular region (see Figure 5Watch/download Movie).

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Figure 5: mpg-Movie (1705 KB) An illustration of the Bianchi IX potential in the anisotropy plane and its exponentially rising walls. Positive values of the potential are drawn logarithmically with solid contour lines and negative values with dashed contour lines.

A cross section of a wall can be obtained by taking β− = 0 and β+ < 0, in which case the potential becomes W (Ω, β+, 0) ≈ 12e−4Ω− 8β+. One obtains the picture of a point moving almost freely until it is reflected at a wall. In between reflections, the behavior is approximately given by the Kasner solution described previously. This behavior, with infinitely many reflections before the classical singularity is reached, can be shown to be chaotic [39Jump To The Next Citation Point], which suggests a complicated approach to classical singularities in general.

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Figure 6: Approximate effective wall of finite height [80Jump To The Next Citation Point] as a function of x = − β+, compared to the classical exponential wall (upper dashed curve). Also shown is the exact wall 1 1 1 2 W (p ,p ,(V ∕p ) ) (lower dashed curve), which for x smaller than the peak value coincides well with the approximation up to a small, nearly constant shift.

With the effective-density term, however, the potential for fixed Ω does not diverge and the walls, as shown in Figure 6View Image, break down at a small but non-zero volume [80Jump To The Next Citation Point]. The effective potential is illustrated in Figure 7Watch/download Movie as a function of densitized triad components and as a function on the anisotropy plane in Figure 8Watch/download Movie. In this scenario, there are only a finite number of reflections, which do not lead to chaotic behavior but instead result in asymptotic Kasner behavior [81].

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Figure 7: mpg-Movie (1639 KB) An illustration of the effective Bianchi IX potential and the movement and breakdown of its walls. The contours are plotted as in Figure 4Watch/download Movie.

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Figure 8: mpg-Movie (1699 KB) An illustration of the effective Bianchi IX potential in the anisotropy plane and its walls of finite height, which disappear at finite volume. Positive values of the potential are drawn logarithmically with solid contour lines and negative values with dashed contour lines.

Comparing Figure 5Watch/download Movie with Figure 8Watch/download Movie shows that at their centers they are very close to each other, while strong deviations occur for large anisotropies. This demonstrates that most of the classical evolution, which happens almost entirely in the inner triangular region, is not strongly modified by the effective potential. Quantum effects are important only when anisotropies become too large, for instance when the system moves deep into one of the three valleys, or the total volume becomes small. In those regimes the quantum evolution will take over and describe the further behavior of the system.

4.12.3 Isotropic curvature suppression

If we use the potential for time coordinate t rather than τ, it is replaced by W ∕(p1p2p3), which in the isotropic reduction p1 = p2 = p3 = 1a2 4 gives the curvature term ka −2. Although the anisotropic effective curvature potential is not bounded, it is, unlike the classical curvature, bounded from above at any fixed volume. Moreover, it is bounded along the isotropy line and decays when a approaches zero. Thus, there is a suppression of the divergence in ka−2 when the closed isotropic model is viewed as embedded in a Bianchi IX model. Similar to matter Hamiltonians, intrinsic curvature then approaches zero at zero scale factor.

This is a further illustration of the special nature of isotropic models compared to anisotropic ones. In the classical reduction, the I p in the anisotropic spin connection cancel such that the spin connection is a constant and no special steps are needed for its quantization. By viewing isotropic models within anisotropic ones, one can consistently realize the model and see a suppression of intrinsic curvature terms. Anisotropic models, on the other hand, do not have, and do not need, complete suppression since curvature functions can still be unbounded.


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