Microwave background anisotropy in a toroidal universe

Large-scale cosmic microwave background temperature fluctuations are calculated for a universe with the topology of a 3-torus. In such a universe only perturbations which are harmonics of the fundamental mode are permitted. By comparison with data from the Cosmic Background Explorer satellite, we find that the minimum (comoving) scale of a cubic toroidal universe is 2400${\mathit{h}}^{\mathrm{-}1}$ Mpc for an n=1 inflationary model. This is approximately an order of magnitude greater than previous limits and 80% of the horizon scale, implying that a topologically ‘‘small’’ universe is no longer an interesting cosmological model.