Modified Lyth bound and implications of BICEP2 results

To reconcile the BICEP2 measurement on the tensor-to-scalar ratio r with Planck constraint, a large negative running of scalar spectral index n(s) is needed. So the inflationary observable such as ns should be expanded at least to the second-order slow-roll parameters for single-field inflationary models. The large value of r and the Lyth bound indicate that it is impossible to obtain the sub-Planckian excursion for the inflaton. However, we derive an absolutely minimal bound Delta phi/M-PI > root r/2 on the inflaton excursion for single-field inflationary models, which can be applied to non-slow-roll inflationary models as well. This bound excludes the possibility of the small-field inflation with Delta phi < 0.1M(P1) if the BICEP2 result on r stands, and it opens the window of sub-Planckian excursion with Delta phi < M-P1, even if r is as large as 0.1. To get the sub-Planckian excursion with Delta phi < 0.1M(P1), our modified bound requires r < 0.02. Using a fifth-order polynomial potential as an explicit example, we show that it not only agrees with the observational results, but also violates the Lyth bound.