Eternal Inflation, Bubble Collisions, and the Disintegration of the
  Persistence of Memory
  We compute the probability distribution for bubble collisions in an inflating
false vacuum which decays by bubble nucleation. Our analysis generalizes
previous work of Guth, Garriga, and Vilenkin to the case of general
cosmological evolution inside the bubble, and takes into account the dynamics
of the domain walls that form between the colliding bubbles. We find that
incorporating these effects changes the results dramatically: the total
expected number of bubble collisions in the past lightcone of a typical
observer is N ~ \gamma V_f / V_i, where \gamma is the fastest decay rate of the
false vacuum, V_f is its vacuum energy, and V_i is the vacuum energy during
inflation inside the bubble. This number can be large in realistic models
without tuning. In addition, we calculate the angular position and size
distribution of the collisions on the cosmic microwave background sky, and
demonstrate that the number of bubbles of observable angular size is N_{LS}
\sim \sqrt{\Omega_k} N, where \Omega_k is the curvature contribution to the
total density at the time of observation. The distribution is almost exactly
isotropic.
