  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.
