Dispersion, topological scattering, and self-interference in multiply connected Robertson-Walker cosmologies

Abstract

We investigate scattering effects in open Robertson-Walker cosmologies whose spacelike slices are multiply connected hyperbolic manifolds. We work out an example in which the 3-space is infinite and has the topology of a solid torus. The world-lines in these cosmologies are unstable, and classical probability densities evolving under the horospherical geodesic flow show dispersion, as do the densities of scalar wave packets. The rate of dispersion depends crucially on the expansion factor, and we calculate the time evolution of their widths. We find that the cosmic expansion can confine dispersion: The diameter of the domain of chaoticity in the 3-manifold provides the natural, time-dependent length unit in an infinite, multiply connected universe. In a toroidal 3-space manifold this diameter is just the length of the limit cycle. On this scale we find that the densities take a finite limit width in the late stage of the expansion. In the early stage classical densities and conformally coupled fields approach likewise a finite width; nonconformally coupled fields disperse. Self-interference occurs if the dispersion on the above scale is sufficiently large, so that the wave packet can overlap with itself. Signals can be backscattered through the topology of 3-space, and we calculate their recurrence times.