The Grishchuk-zeldovich Effect in the Open Universe

When considering perturbations in an open universe, cosmologists retain only subcurvature 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. This article summarizes recent work with A. Woszczyna. The mathematics is briefly explained in a language accessible to physicists. Then 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.