Density Perturbations in Chain Inflation
  We consider the model of ``Chain Inflation,'' in which the period of inflation in our universe took the form of a long sequence of quantum tunneling events. We find that in the simplest such scenario, in which the tunneling processes are uniform, approximately 10^4 vacua per e-folding of inflation are required in order that the density perturbations produced are of an acceptable size. We arrive at this conclusion through a combination of analytic and numerical techniques, which could also serve as starting points for calculations with more general sets of assumptions. 
Brian Feldstein, Brock Tweedie