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 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.
Corrections to classical behavior are most drastic in the Bianchi IX model with symmetry group such that . The classical evolution can be described by a 3dimensional mechanics system with a potential obtained from Equation (37) such that the kinetic term is quadratic in derivatives of with respect to a time coordinate defined by . This potential
diverges at small , in particular (in a directiondependent manner) at the classical singularity where all . Figure 4 illustrates the walls of the potential, which with decreasing volume push the universe toward the classical singularity.
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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 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 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 and two anisotropy parameters defined such that
The classical potential then takes the form
which at fixed has three exponential walls rising from the isotropy point and enclosing a triangular region (see Figure 5).
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A cross section of a wall can be obtained by taking and , in which case the potential becomes . 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 [39], which suggests a complicated approach to classical singularities in general.

With the effectivedensity term, however, the potential for fixed does not diverge and the walls, as shown in Figure 6, break down at a small but nonzero volume [80]. The effective potential is illustrated in Figure 7 as a function of densitized triad components and as a function on the anisotropy plane in Figure 8. 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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Comparing Figure 5 with Figure 8 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.
If we use the potential for time coordinate rather than , it is replaced by , which in the isotropic reduction gives the curvature term . 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 approaches zero. Thus, there is a suppression of the divergence in 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 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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