On the Lyth bound and single field slow-roll inflation
  We take a pragmatic, model independent approach to single field slow-roll
inflation by imposing conditions to the slow-roll parameter $\epsilon$ and its
derivative $\epsilon^{\prime }.$ To accommodate the recent (large) values of
$r$ reported by the BICEP2 collaboration we advocate for a decreasing
$\epsilon$ during most part of inflation. However because at
$\phi_{\mathrm{H}}$, at which the perturbations are produced, some $50$ $-$
$60$ e-folds before the end of inflation, $\epsilon$ is increasing we thus
require that $\epsilon$ develops a maximum for $\phi > \phi_{\mathrm{H}}$ 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 $\Delta\phi$ is
obtained with a sufficiently thin $\epsilon$ 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 $\Delta\phi$
and the spectral index $n_{\mathrm{s}}$ are restricted to narrow windows of
values.
