Matter fields are not the only contributions to the Hamiltonian in cosmology, but also the effect of anisotropies can be included in this way to an isotropic model. The late time behavior of this contribution can be shown to behave as in the shear energy density [156], which falls off faster than any other matter component. Thus, toward later times the universe becomes more and more isotropic.
In the backward direction, 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. It can thus not simply be extrapolated to early times. Anisotropies are independent degrees of freedom which affect the evolution of the scale factor. But 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.
Modifications 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 (34) 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 direction dependent 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.

The approach to the classical singularity is best analyzed in Misner variables [157] 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 (Figure 5).
A cross section of a wall can be obtained by taking and to be negative, in which case the potential becomes . One thus 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 before. This behavior with infinitely many reflections before the classical singularity is reached, can be shown to be chaotic [32], which suggests a complicated approach to classical singularities in general.


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. Similarly to matter Hamiltonians, intrinsic curvature then approaches zero at zero scale factor.
This is a further illustration for 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.
http://www.livingreviews.org/lrr200511 
© Max Planck Society and the author(s)
Problems/comments to 