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.
