Canonical single field slow-roll inflation with a non-monotonic tensor-to-scalar ratio

Abstract

We take a pragmatic, model independent approach to single field slow-roll canonical inflation by imposing conditions, not on the potential, but on the slow-roll parameter $ϵ(\varphi )$ and its derivatives ${ϵ}^{\prime }(\varphi )$ and ${ϵ}^{\prime \prime }(\varphi )$, thereby extracting general conditions on the tensor-to-scalar ratio $r$ and the running $n_{sk}$ at ${\varphi }_H$ where the perturbations are produced, some $50$ $-$ $60$ $e$-folds before the end of inflation. We find quite generally that for models where $ϵ(\varphi )$ develops a maximum, a relatively large $r$ is most likely accompanied by a positive running while a negligible tensor-to-scalar ratio implies negative running. The definitive answer, however, is given in terms of the slow-roll parameter ${\xi }_2(\varphi )$. To accommodate a large tensor-to-scalar ratio that meets the limiting values allowed by the Planck data, we study a non-monotonic $ϵ(\varphi )$ decreasing during most part of inflation. Since at ${\varphi }_H$ the slow-roll parameter $ϵ(\varphi )$ is increasing, we thus require that $ϵ(\varphi )$ develops a maximum for $\varphi >{\varphi }_H$ after which $ϵ(\varphi )$ decrease to small values where most $e$-folds are produced. The end of inflation might occur trough a hybrid mechanism and a small field excursion $\Delta {\varphi }_e\equiv |{\varphi }_H-{\varphi }_e|$ is obtained with a sufficiently thin profile for $ϵ(\varphi )$ which, however, should not conflict with the second slow-roll parameter $\eta (\varphi )$. As a consequence of this analysis we find bounds for $\Delta {\varphi }_e$, $r_H$ and for the scalar spectral index $n_{sH}$. Finally we provide examples where these considerations are explicitly realised.

Comments: V2: 9 pages, 4 figures. Version accepted for publication in JCAP