The Overshoot Problem in Inflation after Tunneling

TL;DR

This paper demonstrates that the typical overshoot problem in small-field inflation models is absent when inflation is initiated by a CDL tunneling event, especially at low inflationary scales, and provides bounds on overshoot based on potential shape.

Contribution

It establishes conditions under which the overshoot problem is eliminated in small-field inflation following CDL tunneling, extending results to general potentials and initial velocities.

Findings

Overshoot is absent for $n extgreater=4$ in monomial potentials.

Overshoot is bounded by the barrier width after tunneling.

Initial conditions for small- and large-field inflation are unified in the string landscape.

Abstract

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.