Large scale perturbations in the open universe.

When considering perturbations in an open (Ω0 < 1) universe, cosmologists retain only sub-curvature modes (defined as eigenfunctions of the Laplacian whose eigenvalue is less than −1 in units of the curvature scale, in contrast with the super-curvature modes whose eigenvalue is between −1 and 0). Mathematicians have known for almost half a century that all modes must be included to generate the most general homogeneous Gaussian random field, despite the fact that any square integrable function can be generated using only the sub-curvature modes. The former mathematical object, not the latter, is the relevant one for physical applications. The mathematics is here explained in a language accessible to physicists. Then it is pointed out that if the perturbations originate as a vacuum fluctuation of a scalar field there will be no super-curvature modes in nature. Finally the effect on the cmb of any super-curvature contribution is considered, which generalizes to Ω0 < 1 the analysis given by Grishchuk and Zeldovich in 1978. A formula is given, which is used to estimate the effect. In contrast with the case Ω0 = 1, the effect contributes to all multipoles, not just to the quadrupole. It is important to find out whether it has the same l dependence as the data, by evaluating the formula numerically.