Lyth bound revisited

TL;DR

This paper revisits the Lyth bound in single-field slow-roll inflation, deriving an upper limit on the tensor-to-scalar ratio considering field excursion constraints and spectral index running, with implications for Planck data.

Contribution

It provides a revised upper bound on the tensor-to-scalar ratio accounting for spectral index running and field excursion limits in inflation models.

Findings

Upper bound on r_* is about 7×10^{-4} without running.

Allowing spectral index running relaxes the bound to 10^{-2}–10^{-1}.

Results are consistent with Planck 2015 data preferences.

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