Chaos, Symmetry and String Cosmology

We review the finding that the general solution of many physically relevant (bosonic) Einstein-matter systems in the vicinity of a cosmological, i.e. space-like, singularity exhibit a remarkable mixture of chaos and symmetry. On the one hand, the asymptotic behavior, as one approaches the singularity, of the general solution is found to be describable, at each (generic) spatial point, as a billiard motion in an auxiliary Lorentzian space. After a suitable “radial” projection, this evolution is described by a billiard motion on hyperbolic space (which is “chaotic” in many physically interesting cases). On the other hand, for certain Einstein-matter systems, notably for pure Einstein background (Ricci tensor = 0) in any space-time dimension D and for particular Einstein-matter systems arising in string theory, the billiard tables describing asymptotic cosmological behavior are found to be identical to the Weyl chambers of some Lorentzian Kač–Moody algebras. In the case of the bosonic sector of supergravity in 11-dimensional space-time the underlying Lorentzian algebra is that of the hyperbolic Kač–Moody group E10, and there exists some evidence of a correspondence between the general solution of the Einstein-threeform system and a null geodesic in the infinite-dimensional coset space E10/K(E10), where K(E10) is the maximal compact subgroup of E10.