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 $\epsilon(\phi)$ and its derivatives $\epsilon^{\prime }(\phi)$ and $\epsilon^{\prime\prime }(\phi)$, thereby extracting general conditions on the tensor-to-scalar ratio $r$ and the running $n_{sk}$ at $\phi_{H}$ where the perturbations are produced, some $50$ $-$ $60$ $e$-folds before the end of inflation. We find quite generally that for models where $\epsilon(\phi)$ 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 $\xi_2(\phi)$. To accommodate a large tensor-to-scalar ratio that meets the limiting values allowed by the Planck data, we study a non-monotonic $\epsilon(\phi)$ decreasing during most part of inflation. Since at $\phi_{H}$ the slow-roll parameter $\epsilon(\phi)$ is increasing, we thus require that $\epsilon(\phi)$ develops a maximum for $\phi > \phi_{H}$ after which $\epsilon(\phi)$ 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 $\Delta\phi_e\equiv |\phi_H-\phi_e |$ is obtained with a sufficiently thin profile for $\epsilon(\phi)$ which, however, should not conflict with the second slow-roll parameter $\eta(\phi)$. As a consequence of this analysis we find bounds for $\Delta \phi_e$, $r_H$ and for the scalar spectral index $n_{sH}$. Finally we provide examples where these considerations are explicitly realised.