THE GRISHCHUK-ZELDOVICH EFFECT IN THE OPEN UNIVERSE

Abstract

When considering perturbations in an open 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 {\em homogeneous Gaussian random field}, despite the fact that any square integrable {\em 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 $Ω_0<1$ the analysis given by Grishchuk and Zeldovich in 1978.