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 $n_s$ 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 $\Delta\phi/M_{\rm Pl}>\sqrt{r/2}$ 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 $\Delta\phi<0.1 M_{\rm Pl}$ if the BICEP2 result on $r$ stands and it opens the window of sub-Planckian excursion with $\Delta\phi<M_{\rm Pl}$ even if $r$ is as large as $0.1$. To get the sub-Planckian excursion with $\Delta\phi<0.1 M_{\rm Pl}$, 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. 
Qing Gao, Yungui Gong and Tianjun Li
gr-qc astro-ph.CO hep-th