Open universe Grishchuk-Zel'dovich effect.

Abstract

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 a density parameter ${\mathrm{\ensuremath{\Omega}}}_{0}$1 with no cosmological constant, and concentrate on the recently discussed supercurvature modes. The effect is present in all of the low multipoles of the anisotropy, in contrast with the ${\mathrm{\ensuremath{\Omega}}}_{0}$=1 case where only the quadrupole receives a contribution. However, for no value of ${\mathrm{\ensuremath{\Omega}}}_{0}$ 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 wave number 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\ensuremath{\le}${\mathrm{\ensuremath{\Omega}}}_{0}$\ensuremath{\le}0.8, a curvature perturbation of order unity is permitted only for inverse wave numbers more than 1000 times the size of the observable universe. As ${\mathrm{\ensuremath{\Omega}}}_{0}$ tends to one, the constraint weakens to the flat space result that the inverse wave number be more than 100 times the size of the observable universe, whereas for ${\mathrm{\ensuremath{\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. \textcopyright{} 1995 The American Physical Society.