Large scale perturbations in the open universe

When considering perturbations in an open (${\mathrm{\Omega }}_0$<1) 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 supercurvature 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 subcurvature 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 ponted out that if the perturbations originate as a vacuum fluctuation of a scalar field there will be no supercurvature modes in nature. Finally the effect on the CMB of any supercurvature contribution is considered, which generalizes to ${\mathrm{\Omega }}_0$<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 ${\mathrm{\Omega }}_0$=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.