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4.14 Anisotropy: Implications for inhomogeneities

Even without implementing inhomogeneous models the previous discussion allows some tentative conclusions as to the structure of general singularities. This is based on the BKL picture [38], whose basic idea is to study Einstein’s field equations close to a singularity. One can then argue that spatial derivatives become subdominant compared to time-like derivatives such that the approach should locally be described by homogeneous models, in particular the Bianchi IX model, since it has the most freedom in its general solution.

Since spatial derivatives are present, though, they lead to small corrections and couple the geometries in different spatial points. One can visualize this by starting with an initial slice, which is approximated by a collection of homogeneous patches. For some time, each patch evolves independently of the others, but this is not precisely true since coupling effects have been ignored. Moreover, each patch geometry evolves in a chaotic manner, which means that two initially nearby geometries depart rapidly from each other. The approximation can thus be maintained only if the patches are subdivided during the evolution, which goes on without limits in the approach to the singularity. There is, thus, more and more inhomogeneous structure being generated on arbitrarily small scales, which leads to a complicated picture of a general singularity.

This picture can be used to describe the behavior of the Bianchi IX model with effective density corrections (higher powers of the connection have not yet been included). Here, the patches do not evolve chaotically, even though at larger volume they follow the classical behavior. The subdivision thus has to also be done for the initial effective evolution. At some point, however, when reflections on the potential walls stop, the evolution simplifies and subdivisions are no longer necessary. There is thus a lower bound to the scale of structure, whose precise value depends on the initial geometries. Nevertheless, from the scale at which the potential walls break down, one can show that structure formation stops at the latest when the discreteness scale of quantum geometry is reached [80]. This can be seen as a consistency test of the theory, since structure below the discreteness could not be supported by quantum geometry.

We have thus a glimpse of the inhomogeneous situation with a complicated but consistent approach to a general classical singularity. The methods involved, however, are not very robust, since the BKL scenario, which despite some progress at analytical and numerical levels [249Jump To The Next Citation Point169] is even classically still at the level of a conjecture for the general case [39249], would need to be available as an approximation to quantum geometry. For more reliable results the methods need to be refined to take into account inhomogeneities properly.

Applications of the BKL picture are, however, in trouble, if bounces are a generic phenomenon of effective equations. The BKL scenario is an asymptotic statement about the behavior of solutions near a curvature singularity. When solutions bounce in the presence of quantum corrections, they may never reach a regime where asymptotic statements of this form apply, thus preventing the use of the BKL picture (although alternative pictures in the same spirit may become available). To be precise, some solutions may be tuned such that they do not bounce before an asymptotic regime is reached, where one could then use the BKL decoupling of different patches together with bounces in anisotropic models to conclude that even corresponding inhomogeneous solutions are non-singular. Since this requires special initial conditions to be able to enter an asymptotic regime before bounces occur, such a procedure could not provide general singularity removal. (See also [71Jump To The Next Citation Point] for a more detailed discussion of general singularities.)

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