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

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 $ε(ϕ)$ and its derivatives $ε^{\prime }(ϕ)$ and $ε^{\prime\prime }(ϕ)$, thereby extracting general conditions on the tensor-to-scalar ratio $r$ and the running $n_{sk}$ at $ϕ_{H}$ where the perturbations are produced, some $50$ $-$ $60$ $e$-folds before the end of inflation. We find quite generally that for models where $ε(ϕ)$ 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 $ξ_2(ϕ)$. To accommodate a large tensor-to-scalar ratio that meets the limiting values allowed by the Planck data, we study a non-monotonic $ε(ϕ)$ decreasing during most part of inflation. Since at $ϕ_{H}$ the slow-roll parameter $ε(ϕ)$ is increasing, we thus require that $ε(ϕ)$ develops a maximum for $ϕ> ϕ_{H}$ after which $ε(ϕ)$ 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 $Δϕ_e\equiv |ϕ_H-ϕ_e |$ is obtained with a sufficiently thin profile for $ε(ϕ)$ which, however, should not conflict with the second slow-roll parameter $η(ϕ)$. As a consequence of this analysis we find bounds for $Δϕ_e$, $r_H$ and for the scalar spectral index $n_{sH}$. Finally we provide examples where these considerations are explicitly realised.