Probability Distribution for $\Omega$ in Open-universe Inflation

Abstract

The problem of making predictions in eternally inflating universe that thermalizes by bubble nucleation is considered. A recently introduced regularization procedure is applied to find the probability distribution for the ensemble of thermalized bubbles. The resulting probabilities are shown to be independent of the choice of the time parametrization. This formalism is applied to models of open ``hybrid'' inflation with $\Omega <1$. Depending on the parameters of the model, the probability distribution for $\Omega $ is found to have a peak either very close to $\Omega =1$, or at an intermediate value of $\Omega $ in the range $0.03\lesssim \Omega <1$.