Lyth bound revisited

Abstract

Imposing that the excursion distance of inflaton in field space during inflation be less than the Planck scale, we derive an upper bound on the tensor-to-scalar ratio at the CMB scales, i.e. $r_{\ast ,max}$, in the general canonical single-field slow-roll inflation model, in particular the model with non-negligible running of the spectral index ${\alpha }_s$ and/or the running of running ${\beta }_s$. We find that $r_{\ast ,max}\simeq 7\times 10^{-4}$ for $n_s=0.9645$ without running and running of running, and $r_{\ast ,max}$ is significantly relaxed to the order of $\mathcal{O}(10^{-2}\sim 10^{-1})$ in the inflation model with ${\alpha }_s$ and/or ${\beta }_s\sim +\mathcal{O}(10^{-2})$ which are marginally preferred by the Planck 2015 data.

Comments: 5 pages, 2 figures; minor correction, clarifications and refs added