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 ${\cal O} (1)$ constants $c$ and $c'$. 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 $c \sim {\cal O} (10^{-1})$ and $c' \sim {\cal O} (10^{-2})$ 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 ${\cal O} (1)$ when measured as a distance in the target space geometry, we further restrict $c \sim {\cal O} (10^{-2})$, while the constraint on $c'$ remains unchanged. We comment on how to accommodate the required small values of $c$ and $c'$.
Highlights
Inflation is the leading paradigm for explaining the behavior of the quasi–de Sitter expansion in the very early Universe
It has been noted that, while the distance conjecture by itself does not pose a significant challenge for single-field inflationary models, the de Sitter conjecture is in direct tension with slow-roll inflationary potentials favored by cosmic microwave background (CMB) data [46,47,48,49,50,51,52,53,54]
To be able to match such a large value of δ, we must explore the subtleties of the distance conjecture, which asserts that, for any infinite field distance limit, an infinite tower of states becomes exponentially light, and, effective field theories (EFTs) are valid only for finite scalar field variations [21,22,23,24,25]
Summary
CMB data favor standard slow-roll single-field inflationary models with plateaulike potentials V, for which Vφφ < 0, over power-law potentials; here, φ is the dilaton or inflaton and Vφ ≡ dV=dφ [13]. It has been noted that, while the distance conjecture by itself does not pose a significant challenge for single-field inflationary models (and corresponds observationally to a suppressed r through the well-known Lyth bound [45]), the de Sitter conjecture is in direct tension with slow-roll inflationary potentials favored by CMB data [46,47,48,49,50,51,52,53,54].
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