The Open Universe Grishchuk-Zel'dovich Effect

The Grishchuk--Zel'dovich effect is the contribution to the microwave background anisotropy from an extremely large scale adiabatic density perturbation, on the standard hypothesis that this perturbation is a typical realization of a homogeneous Gaussian random field. We analyze this effect in open universes, corresponding to density parameter $\Omega_0<1$ with no cosmological constant, and concentrate on the recently discussed super-curvature modes. The effect is present in all of the low multipoles of the anisotropy, in contrast with the $\Omega_0=1$ case where only the quadrupole receives a contribution. However, for no value of $\Omega_0$ can a very large scale perturbation generate a spectrum capable of matching observations across a wide range of multipoles. We evaluate the magnitude of the effect coming from a given wavenumber as a function of the magnitude of the density perturbation, conveniently specified by the mean-square curvature perturbation. From the absence of the effect at the observed level, we find that for $0.25\leq\Omega_0\leq 0.8$, a curvature perturbation of order unity is permitted only for inverse wavenumbers more than one thousand times the size of the observable universe. As $\Omega_0$ tends to one, the constraint weakens to the flat space result that the inverse wavenumber be more than a hundred times the size of the observable universe, whereas for $\Omega_0 < 0.25$ it becomes stronger. We explain the physical meaning of these results, by relating them to the correlation length of the perturbation. Finally, in an Appendix we consider the dipole anisotropy and show that it always leads to weaker constraints.