For completeness, we provide in this section a short guide to the work that has been accumulated since the late 1960’s to consolidate the BKL phenomenon.

As we have indicated, there are two aspects to the BKL conjecture:

- The first part of the conjecture states that spatial points decouple as one goes to a spacelike singularity in the sense that the evolution can be described by a collection of systems of ordinary differential equations with respect to time, one such system at each spatial point. (“A spacelike singularity is local.”)
- The second part of the conjecture states that the system of ordinary differential equations with respect to time describing the asymptotic dynamics at any given spatial point can be asymptotically replaced by the billiard equations. If the matter content is such that the billiard table has infinite volume, the asymptotic behavior at each point is given by a (generalized) Kasner solution (“Kasner-like spacelike singularities”). If, on the other hand, the matter content is such that the billiard table has finite volume, the asymptotic behavior at each point is a chaotic, infinite, oscillatory succession of Kasner epochs. (“Oscillatory, or mixmaster, spacelike singularities.”)

A third element of the original conjecture was that the matter could be neglected asymptotically. While generically true in four spacetime dimensions (the exception being a massless scalar field, equivalent to a fluid with the stiff equation of state ), this aspect of the conjecture does not remain valid in higher dimensions where the -form fields might add relevant walls that could change the qualitative asymptotic behavior. We shall thus focus here only on Aspects 1 and 2.

- In the Kasner-like case, the mathematical situation is easier to handle since the conjectured asymptotic behavior of the fields is then monotone and known in closed form. There exist theorems validating (generically) this conjectured asymptotic behavior, starting from the pioneering work of [3] (where the singularities with this behavior are called “quiescent”), which was extended later in [49] to cover more general matter contents. See also [18, 108] for related work.
- The situation is much more complicated in the oscillatory case, where only partial results exist.
However, even though as yet incomplete, the mathematical and numerical studies of the BKL
analysis has provided overwhelming support for its validity. Most work has been done in four
dimensions.
The first attempts to demonstrate that spacelike singularities are local were done in the simpler context of solutions with isometries. It is only recently that general solutions without symmetries have been treated, but this has been found to be possible only numerically so far [87]. The literature on this subject is vast and we refer to [2, 87, 147] for points of entry into it. Let us note that an important element in the analysis has been a more precise reformulation of what is meant by “local”. This has been achieved in [163], where a precise definition involving a judicious choice of scale invariant variables has been proposed and given the illustrative name of “asymptotic silence” – the singularities being called “silent singularities” since propagation of information is asymptotically eliminated.

If one accepts that generic spacelike singularities are silent, one can investigate the system of ordinary differential equations that arise in the local limit. In four dimensions, this system is the same as the system of ordinary differential equations describing the dynamics of spatially homogeneous cosmologies of Bianchi type IX. It has been effectively shown analytically in [151] that the Bianchi IX evolution equations can indeed be replaced, in the generic case, by the billiard equations (with only the dominant, sharp walls) that produce the mixmaster behavior. This validates the second element in the BKL conjecture in four dimensions.

The connection between the billiard variables and the scale invariant variables has been investigated recently in the interesting works [92, 162].

Finally, taking for granted the BKL conjecture, one might analyze the chaotic properties of the billiard map (when the volume is finite). Papers exploring this issue are [30, 32, 121, 132] (four dimensions) and [68] (five dimensions).

Let us finally mention the interesting recent paper [40], in which a more precise formulation of the BKL conjecture, aimed towards the chaotic case, is presented. In particular, the main result of this work is an extension of the Fuchsian techniques, employed, e.g., in [49], which are applicable also for systems exhibiting chaotic dynamics. Furthermore, [40] examines the geometric structure which is preserved close to the singularity, and it is shown that this structure has a mathematical description in terms of a so called “partially framed flag”.

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