Chaos in a three-body self-gravitating cosmological spacetime

We investigate the equal-mass three-body system in general relativistic lineal gravity in the presence of a cosmological constant $\ensuremath{\Lambda}$. The cosmological vacuum energy introduces features that do not have a nonrelativistic counterpart, inducing an expansion/contraction of space-time that competes with the gravitational self-attraction of the bodies. We derive a canonical expression for the Hamiltonian of the system and discuss the numerical solution of the resulting equations of motion. As for the system with $\ensuremath{\Lambda}=0$, we find that the structure of the phase space yields a rich variety of interesting dynamics that can be divided into three distinct regions: annulus, pretzel, and chaotic; the first two being regions of quasiperiodicity while the latter is a region of chaos. However, unlike the $\ensuremath{\Lambda}=0$ case, we find that a negative cosmological constant considerably diminishes the amount of chaos in the system, even beyond that of the $\ensuremath{\Lambda}=0$ nonrelativistic system. By contrast, a positive cosmological constant considerably enhances the amount of chaos, typically leading to KAM (Kolmogorov-Arnold-Moser) breakdown.