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.
