On the Lyth bound and single field slow-roll inflation

Understand

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.