Metric perturbations from quantum tunneling in open inflation

We study the effect that quantum fluctuations produced during the nucleation of a single-bubble open inflationary universe have on the amplitude of temperature anisotropies in the microwave background. We compute the instanton action for the quantum tunneling between the false and true vacua in open inflation models and show that the amplitude of quantum fluctuations of the bubble wall is very sensitive to the gravitational effects of the true vacuum. We study the spectrum of quantum fluctuations of the bubble wall and confirm that there is only an inhomogeneous (k(2)= -3) discrete mode associated with transverse traceless fluctuations of the bubble wall. This supercurvature mode could, in principle. distort the anisotropy of the microwave background. We compute the amplitude of the gauge-invariant metric perturbations induced by the bubble wall fluctuations on a comoving hypersurface, and calculate the induced amplitude of temperature fluctuations in the microwave background, for arbitrary values of st,. we find that in the limit Ohm(0) similar or equal to 1, the quadrupole dominates the angular power spectrum, such as in the usual Grishchuk-Zel'dovich effect. The resulting bounds on the amplitude of quantum fluctuations of the bubble wall from the absence of such an effect in the observed microwave background anisotropies are quite strong. We also study the contribution from a discrete long wavelength supercurvature mode (k(2) similar or equal to 2m(2)/3H(2)) that appears in the spectrum of open de Sitter vacuum fluctuations. We constrain the parameters of the models of open inflation so that these modes do not distort the observed temperature anisotropy.