Annu. Rev. Astron. Astrophys. 1994. 32: 319-70 Copyright © 1994 by Annual Reviews. All rights reserved

7.4. Topology

The fluctuations at large angular scales can also be used to constrain the topology of the Universe, which in principle could be nontrivial (Zel'dovich 1973, Sokolov & Starobinskii 1976, Fang & Houjun 1987, Fang 1991). The Sachs-Wolfe spectrum of Cs is an integral over the power spectrum (Equation 14) in an ordinary simply-connected universe, but becomes a sum over modes that are harmonics of the box-size in a universe that has the topology of a three-torus (i.e. has periodic boundary conditions). This sum fails to accurately approximate the integral on scales approaching that of the box. Then the fact that COBE measures a roughly flat power spectrum on large scales means that the box must be roughly the horizon size or bigger, unless the initial power spectrum has a pathological increase for small multipoles. More detailed comparisons indicate that the scale of any such topology is 80% of the horizon size (Stevens et al 1993, Sokolov 1993, Starobinskii 1993).

For open universes, Gurzadyan and collaborators have argued that curvature effects may result in elongated shapes of anisotropies in the CMB (Gurzadyan & Kocharyan 1993, Gurzadyan & Torres 1993). The isotropy pattern of the CBR can also be used to place limits on the rotation of the Universe (Collins & Hawking 1973, Barrow et al 1985). Currently the limit on the dimensionless rotation is / H₀ 10^-6 (Smoot 1992), which is about 1" every 30 Gyr!