The Overshoot Problem in Inflation after Tunneling
  We show the absence of the usual parametrically large overshoot problem of
small-field inflation if initiated by a Coleman-De Luccia (CDL) tunneling
transition from an earlier vacuum in the limit of small inflationary scale
compared to the tunneling scale. For low-power monomial exit potentials
$V(\phi)\sim\phi^n, n<4$, we derive an expression for the amount of overshoot.
This is bounded from above by the width of the steep barrier traversed after
emerging from tunneling and before reaching a slow-roll region of the
potential. For $n\geq 4$ we show that overshooting is entirely absent. We
extend this result through binomials to a general potential written as a series
expansion, and to the case of arbitrary finite initial speed of the inflaton.
This places the phase space of initial conditions for small-field and
large-field inflation on the same footing in a landscape of string theory vacua
populated via CDL tunneling.
