S-dual Inflation and the String Swampland

Abstract

The Swampland de Sitter conjecture in combination with upper limits on the tensor-to-scalar ratio $r$ derived from observations of the cosmic microwave background endangers the paradigm of slow-roll single field inflation. This conjecture constrains the first and the second derivatives of the inflationary potential in terms of two $\mathcal{O}(1)$ constants $c$ and $c^{\prime }$. In view of these restrictions we reexamine single-field inflationary potentials with $S$-duality symmetry, which ameliorate the unlikeliness problem of the initial condition. We compute $r$ at next-to-leading order in slow-roll parameters for the most general form of $S$-dual potentials and confront model predictions to constraints imposed by the de Sitter conjecture. We find that $c\sim \mathcal{O}(10^{-1})$ and $c^{\prime }\sim \mathcal{O}(10^{-2})$ can accommodate the 95\% CL upper limit on $r$. By imposing at least 50 $e$-folds of inflation with the effective field theory description only valid over a field displacement $\mathcal{O}(1)$ when measured as a distance in the target space geometry, we further restrict $c\sim \mathcal{O}(10^{-2})$, while the constraint on $c^{\prime }$ remains unchanged. We comment on how to accommodate the required small values of $c$ and $c^{\prime }$.

Comments: Extensive discussion on the ambiguity of the definition of slow-roll-parameters in relation to how this uncertainty may help ameliorate the tension between single-field inflationary models and the de Sitter Swampland conjecture. Accepted for publication in Phys. Rev. D