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 $ε$ and its derivative $ε^{\prime }.$ To accommodate the recent (large) values of $r$ reported by the BICEP2 collaboration we advocate for a decreasing $ε$ during most part of inflation. However because at $ϕ_{\mathrm{H}}$, at which the perturbations are produced, some $50$ $-$ $60$ e-folds before the end of inflation, $ε$ is increasing we thus require that $ε$ develops a maximum for $ϕ> ϕ_{\mathrm{H}}$ 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 $Δϕ$ is obtained with a sufficiently thin $ε$ which, however, should not conflict with the curvature of the potential measured by the second slow-roll parameter $η$. The conclusion is that under these circumstances $Δϕ$ and the spectral index $n_{\mathrm{s}}$ are restricted to narrow windows of values.

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