Bounds on Slow Roll and the de Sitter Swampland

The recently introduced swampland criterion for de Sitter (arXiv:1806.08362) can be viewed as a (hierarchically large) bound on the smallness of the slow roll parameter $epsilon_V$. This leads us to consider the other slow roll parameter $eta_V$ more closely, and we are lead to conjecture that the bound is not necessarily on $epsilon_V$, but on slow roll itself. A natural refinement of the de Sitter swampland conjecture is therefore that slow roll is violated at ${cal O}(1)$ in Planck units in any UV complete theory. A corollary is that $epsilon_V$ need not necessarily be ${cal O}(1)$, if $eta_V lesssim -{cal O}(1)$ 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 arXiv:1806.08362, and show that they are consistent with this refined version of the bound. The phrasing in terms of slow roll makes it plausible why bo