Bounds on slow roll and the de Sitter Swampland

The recently introduced swampland criterion for de Sitter [17] can be viewed as a (hierarchically large) bound on the smallness of the slow roll parameter 𝜖_V. This leads us to consider the other slow roll parameter η_V more closely, and we are lead to conjecture that the bound is not necessarily on 𝜖_V, but on slow roll itself. A natural refinement of the de Sitter swampland conjecture is therefore that slow roll is violated at O(1) in Planck units in any UV complete theory. A corollary is that 𝜖_V need not necesarily be O(1), if η_V≲ − O(1) holds. We consider various tachyonic tree level constructions of de Sitter in IIA/IIB string theory (as well as closely related models of inflation), which superficially violate [17], and show that they are consistent with this refined version of the bound. The phrasing in terms of slow roll makes it plausible why both versions of the conjecture run into trouble when the number of e-folds during inflation is high. We speculate that one way to evade the bound could be to have a large number of fields, like in N -flation.