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. $r_{<em>,max}$, 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 $r_{</em>,max}\simeq 7\times 10<sup>{-4}$ for $n_s=0.9645$ without running and running of running, and $r_{*,max}$ is significantly relaxed to the order of ${\cal O}(10^{-2}\sim 10<sup>{-1})$ in the inflation model with $\alpha_s$ and/or $\beta_s\sim +{\cal O}(10<sup>{-2})$ which are marginally preferred by the Planck 2015 data.