Open Universes from Finite Radius Bubbles

The interior of a vacuum bubble in de Sitter space may give an open universe with sufficient homogeneity to agree with observations. Here, previous work by Bucher, Goldhaber and Turok is extended to describe a thin bubble wall with nonzero radius and energy difference across the wall. The vacuum modes present before formation of the bubble propagate into the interior of open universe and the power spectrum of the resulting gauge invariant gravitational potential is calculated. It is found to become scale invariant on small scales, with onset at about the same scale as that found in the zero radius case. There is sensitivity to the radius and energy difference at large scales, but it is expected that they cannot be strongly constrained because of cosmic variance. As the prediction of a scale invariant spectrum is robust with respect to variation of these parameters at small scales, it is a generic feature of the contribution of these modes for these thin wall models.