Prescription for probabilities in eternal inflation

Some of the parameters we call “constants of nature” may in fact be variables related to the local values of some dynamical fields. During inflation, these variables are randomized by quantum fluctuations. In cases when the variable in question (call it $\chi )$ takes values in a continuous range, all thermalized regions in the universe are statistically equivalent, and a gauge invariant procedure for calculating the probability distribution for $\chi$ is known. This is the so-called “spherical cutoff method.” In order to find the probability distribution for $\chi$ it suffices to consider a large spherical patch in a single thermalized region. Here, we generalize this method to the case when the range of $\chi$ is discontinuous and there are several different types of thermalized region. We first formulate a set of requirements that any such generalization should satisfy, and then introduce a prescription that meets all the requirements. We finally apply this prescription to calculate the relative probability for different bubble universes in the open inflation scenario.