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International audience; We show that solitonic cosmological gravitational waves propagated through the Friedmann universe and generated by the inhomogeneities of the gravitational field near the Big Bang can be responsible for increase of cosmological distances.

In this second part of the book, we develop the billiard description of the BKL behavior. We show how, in the BKL limit, the equations of motion of systems containing gravity can be recast at each spatial point as equations of motion for a billiard ball moving in a region of hyperbolic space. The emerging billiards that capture the motion near a spacelike singularity are called “cosmological billiards.” We start with pure gravity in four space-time dimensions. The inclusion of matter fields, and the extension to higher dimensions, are considered in Chapter 6. In most papers devoted to cosmological billiards, index conventions different from those used so far are adopted. These different conventions are that Greek indices are space-time indices while Latin indices are spatial indices. Thus, in four space-time dimensions, α, β, · · · = 0, 1, 2, 3, and a, b, · · · = 1, 2, 3. Furthermore, indices in non-coordinate frames are not distinguished with a bar from coordinate indices. In order to ease the reading of the cosmological billiard literature, we shall switch to these different conventions. No confusion should arise as the context is always clear. The derivation of the billiard description of the BKL limit given in this second part of the book was originally worked out in a collaboration of the second author with Thibault Damour and Hermann Nicolai [45, 46, 47, 49, 51]. One of its main ingredients is the use of the Iwasawa decomposition of the metric for handling the influence of the off-diagonal metric components on the dynamics of the scale factors. Chapters 5 and 6 are largely based on the ideas developed in those works, and in particular draw very much from the review [51], from which we adopt in particular the notations. Hamiltonian Form of the Action The cosmological billiard picture is based on the Hamiltonian formalism, which we first briefly review. For more information, we refer the reader to the original papers [65, 5] as well as to the book [133], Chapter 21.

Stochasticity of the Oscillatory Regime In the asymptotic vicinity of the singularity, the oscillatory regime described above acquires a stochastic (chaotic) character. Of course, any system of differential equations, no matter how complicated it is, has one and only one definite solution for each set of initial data. From this point of view we have no place for stochasticity. When we are speaking about stochastic behavior of the solution of differential equations we mean a quite different situation, namely the case when the initial data are not known exactly but are distributed in accordance to some statistical law. Together with the initial data, all characteristics of the solution at any subsequent time will also be distributed in some way. In the framework of such a statistical approach there are two essentially different types of behavior. The first corresponds to the non-chaotic systems for which the initial distribution being picked around some points in the phase space generates an evolution which remains picked around the corresponding trajectories (i.e., usual classical solutions) starting from these points. The systems of the second type are chaotic in the sense that, for them, any initial distribution is going to spread out over the whole phase space in the course of the evolution, independently of how sharply localized the distribution was at the initial instant (some authors say that such systems have a source of stochasticity). Moreover, in the chaotic case, the final asymptotic regime which arises at a late time has a universal character completely independent of the form of the initial distribution. We stress that it is this last property (known as mixing ) that represents the basic feature of chaos. For complicated nonlinear systems of differential equations, the chaotic behavior (if it is present) should be considered as a rather useful property because it gives the qualitative character of the general solution. Indeed for such a system only approximate forms of the general solution can be obtained and all parameters in such forms have no exact sense; they are unavoidably uncertain and can be interpreted as statistically distributed.

Introduction In the preceding sections we have established the existence of a general cosmological solution of the gravitational equations with a singularity with respect to time, and have studied the asymptotic properties of this solution near the singularity. We saw that in this limit the solution may be described by an infinite alternation of Kasner epochs. The notion of a Kasner epoch and the transition between two of these are the key elements in the dynamics of the oscillatory regime. We have derived and studied these properties in the example of empty space in four space-time dimensions. In this chapter, we investigate how the analysis can accommodate matter sources, as well as a change in the space-time dimension. We will successively consider the following systems. First, we consider perfect fluids in four space-time dimensions. Next, we consider gauge fields of the Yang-Mills and electromagnetic types and scalar fields, also in four space-time dimensions. Then, we discuss pure gravity in higher dimensions. Finally, we outline some results on the character of the cosmological singularity in presence of a viscous matter source. In Appendix C, we solve the case of a commuting spinor field. This case is mathematically interesting but its physical significance is not clear (because the spinors are treated as classical c -numbers). We shall see that in certain important cases the influence of matter upon the solution in the vicinity of the singularity appears to be qualitatively negligible. This is, for instance, true for the case of standard perfect fluid. For other (less simple) types of the energy–momentum tensor, the influence of matter proves to be more important, but it can still be analyzed analytically with sufficient clarity and along the same general lines. We will also find the surprising result that in space-time dimensions equal to or higher than 11, D ≥ 11, gravity near a cosmological singularity ceases to be chaotic [63, 62]. We follow in this chapter the original BKL approach. The same analysis can be performed using the billiard description. It is given in Chapter 6; see in particular Section 6.6.

In the present letter we indicate an extension of the pure gravity inverse scattering integration technique to the case when fermions (introduced on the base of supersymmetry) are present. In this way the integrability technique for simple ($N=1$) supergravity in two space-time dimensions coupled to the matter fields taking values in the Lie algebra of $E_{8\left( +8\right) }$ group is developed. This theory contains matter living only in one Weyl representation of $SO\left( 16\right) $ and represents the reduction to two dimensions of the three-dimensional simple supergravity constructed in [1]. Our spectral linear problem use superspace and covers the complete set of principal bosonic and fermionic equations of motion. This linear system, as in pure gravity, contains only the first order poles with respect to the spectral parameter. The procedure of constructing the exact super-solitonic solutions is outlined., Version 1611.02924v1 was withdrawn due to groundless assumption on the existence of the relation (43). In footnote 7 it was stressed that its proof remains to be seen. Analysis showed that such relation indeed is doubtful. Then the extension of the pure gravity inverse scattering method to superspace surely can cover only the simple integrable supergravity with E8(8) symmetry

The long story of the oscillatory approach to the initial cosmological singularity and its more recent incarnation in multidimensional universe models is told., Comment: The invited paper for Proceedings of the XIII Marcel Grossmann Meeting (Stockholm, 2012) by reason of the Marcel Grossmann Award to V.A. Belinski and I.M. Khalatnikov