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

Figure 1

The behavior of the potential $V(\varphi )$ and the corresponding slow-roll parameters $\epsilon (\varphi )$, $\eta (\varphi )$, ${\xi }^2(\varphi )$, ${\sigma }^3(\varphi )$, and ${\delta }^4(\varphi )=M_{\text{Pl}}^8(V_{\varphi }{)}^3{V}_{\varphi \varphi \varphi \varphi \varphi }/V^4(\varphi )$ for the polynomial potential with the coefficients ${\lambda }_1=-0.162$, ${\lambda }_2=0.00161$, ${\lambda }_3=-0.0132$, ${\lambda }_4=0.01$, and ${\lambda }_5=-0.00146$.

Figure 2

The “$+$” stands for the results on $n_s$, $r$, and $n_s^{\prime }$ from the polynomial potential with ${\lambda }_1=-0.162$, ${\lambda }_2=0.00161$, ${\lambda }_3=-0.0132$, ${\lambda }_4=0.01$, and ${\lambda }_5=-0.00146$. We also show the 68% and 95% contours constrained from the combination of $\mathrm{Planck}+\mathrm{WP}+\mathrm{highL}+\mathrm{BICEP}2$ data.