On the Lyth bound and single field slow-roll inflation

Abstract

We take a pragmatic, model independent approach to single field slow-roll inflation by imposing conditions to the slow-roll parameter $ϵ$ and its derivative ${ϵ}^{\prime }.$ To accommodate the recent (large) values of $r$ reported by the BICEP2 collaboration we advocate for a decreasing $ϵ$ during most part of inflation. However because at ${\varphi }_{\mathrm{H}}$, at which the perturbations are produced, some $50$ $-$ $60$ e-folds before the end of inflation, $ϵ$ is increasing we thus require that $ϵ$ develops a maximum for $\varphi >{\varphi }_{\mathrm{H}}$ and then decrease to small values where most e-folds are produced. The end of inflation might occur trough a hybrid field and a small $\Delta \varphi$ is obtained with a sufficiently thin $ϵ$ which, however, should not conflict with the curvature of the potential measured by the second slow-roll parameter $\eta$. The conclusion is that under these circumstances $\Delta \varphi$ and the spectral index $n_{\mathrm{s}}$ are restricted to narrow windows of values.

Comments: 5 pages, 2 figures; V2: added discussion on power spectrum constraints, 3 new references