Statistical properties of cosmological billiards

Belinski, Khalatnikov, and Lifshitz pioneered the study of the statistical properties of the never-ending oscillatory behavior (among successive Kasner epochs) of the geometry near a spacelike singularity. We show how the use of a “cosmological billiard” description allows one to refine and deepen the understanding of these statistical properties. Contrary to previous treatments, we do not quotient the dynamics by its discrete symmetry group (of order 6), thereby uncovering new phenomena, such as correlations between the successive billiard corners in which the oscillations take place. Starting from the general integral invariants of Hamiltonian systems, we show how to construct invariant measures for various projections of the cosmological-billiard dynamics. In particular, we exhibit, for the first time, a (non-normalizable) invariant measure on the “Kasner circle” which parametrizes the exponents of successive Kasner epochs. Finally, we discuss the relation between: (i) the unquotiented dynamics of the Bianchi-IX ($a$, $b$, $c$ or mixmaster) model; (ii) its quotienting by the group of permutations of ($a$, $b$, $c$); and (iii) the billiard dynamics that arose in recent studies suggesting the hidden presence of Kac-Moody symmetries in cosmological billiards.

Figure 1

The hyperbolic billiard on the unit hyperboloid ${\mathcal{H}}_2$. The big-billiard chamber defined by the three gravitational walls is sketched.

Figure 2

The disk model ${\mathcal{B}}_2$ of the hyperbolic billiard. Both the big billiard (with walls $a$, $b$, $c$) and the small billiard (with walls $G$, $B$, $R$) are sketched. The six fundamental Kasner intervals are indicated on the boundary of the disk, which is identified with the Kasner circle.

Figure 3

The Poincaré model ${\mathcal{P}}_2$ of the hyperbolic billiard. Both the big billiard and the small billiards are sketched.

Figure 4

Kasner epochs of the big billiard in the disk model ${\mathcal{B}}_2$ of the hyperbolic billiard.

Figure 5

Kasner epochs of the big billiard in the Poincaré model ${\mathcal{P}}_2$ of the hyperbolic billiard.

Figure 6

Billiard phase space in the $u^{-}{u}^{+}$ parametrization: the epoch hopscotch court. The $B_{xy}$ regions are sketched, and filled with different colors (shades of gray).

Figure 7

The era hopscotch court in the $(u^{-},u^{+})$ parametrization. The six starting boxes $F_{xy}$ are sketched, and filled with different colors (shades of gray), according to Fig. 6, as explained in Table . For each starting box $F_{xy}$, the starting subboxes corresponding to $k$-epoch eras (as described in Table ) are indicated for $k=1,2,3$; namely $F_{xy}^1$, $F_{xy}^2$, and $F_{xy}^3$. The three $(u^{-},u^{+})$ points of the simplest periodic orbit are denoted as asterisks. Please note that, for typographical reasons, the $F_{xy}^k$’s are in this figure indicated as $Fkxy$.

Figure 8

The phase space of the small billiard in the $u^{-}$, $u^{+}$ parametrization. The regions $RG$ and $RB$ are delimited by solid thick lines; the regions $BR$ and $GR$ are delimited by dashed thick lines. The Kasner intervals are indicated on both axes by thin gray lines. The subdomains corresponding to the first few eras of $RG$ and $RB$ are also sketched.