Modified Lyth bound and implications of BICEP2 results

Abstract

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 $\Delta \varphi /M_{\mathrm{Pl}}>\sqrt{r/2}$ 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 $\Delta \varphi <0.1M_{\mathrm{Pl}}$ if the BICEP2 result on $r$ stands and it opens the window of sub-Planckian excursion with $\Delta \varphi <M_{\mathrm{Pl}}$ even if $r$ is as large as $0.1$. To get the sub-Planckian excursion with $\Delta \varphi <0.1M_{\mathrm{Pl}}$, 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.

v2: major revision, more discussion on the derivation of the modified Lyth bound is added,main conclusions remain unchanged. PRD in press

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