Canonical single field slow-roll inflation with a non-monotonic tensor-to-scalar ratio

Abstract

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 ϵ(ϕ) and its derivatives ϵ′(ϕ) and ϵ′′(ϕ), thereby extracting general conditions on the tensor-to-scalar ratio r and the running nsk at ϕH where the perturbations are produced, some 50–60 e-folds before the end of inflation. We find quite generally that for models where ϵ(ϕ) 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 ξ2(ϕ). To accommodate a large tensor-to-scalar ratio that meets the limiting values allowed by the Planck data, we study a non-monotonic ϵ(ϕ) decreasing during most part of inflation. Since at ϕH the slow-roll parameter ϵ(ϕ) is increasing, we thus require that ϵ(ϕ) develops a maximum for ϕ > ϕH after which ϵ(ϕ) 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 Δϕe ≡ |ϕH−ϕe| is obtained with a sufficiently thin profile for ϵ(ϕ) which, however, should not conflict with the second slow-roll parameter η(ϕ). As a consequence of this analysis we find bounds for Δϕe, rH and for the scalar spectral index nsH. Finally we provide examples where these considerations are explicitly realised.