Billiard representation for multidimensional cosmology with intersecting p-branes near the singularity

Multidimensional model describing the cosmological evolution of n Einstein spaces in the theory with l scalar fields and forms is considered. When electromagnetic composite p-brane ansatz is adopted, and certain restrictions on the parameters of the model are imposed, the dynamics of the model near the singularity is reduced to a billiard on the (N−1)-dimensional Lobachevsky space HN−1, N=n+l. The geometrical criterion for the finiteness of the billiard volume and its compactness is used. This criterion reduces the problem to the problem of illumination of (N−2)-dimensional sphere SN−2 by pointlike sources. Some examples with billiards of finite volume and hence oscillating behavior near the singularity are considered. Among them examples with square and triangle two-dimensional billiards (e.g., that of the Bianchi-IX model) and a four-dimensional billiard in “truncated” D=11 supergravity model (without the Chern–Simons term) are considered. It is shown that the inclusion of the Chern–Simons term destroys the confining of a billiard.