Phenomenological consequences of the refined swampland conjecture

We discuss phenomenological consequences of the recently introduced refinements of the de Sitter swampland conjecture, which constrains the first and the second derivative of the scalar potential in terms of two $O(1)$ constants $c$ and $c^{\prime }$. Contrary to the original de Sitter swampland conjecture, the refinement has no constraints on spontaneous breaking scenarios, such as the Higgs, the chiral symmetry breaking, and the QCD axion. However, the refinement still strongly constrains inflation models. While we can achieve sufficient number of e-foldings, single-field inflationary models have trouble reproducing the observed valued of density perturbations, when $c$ and $c^{\prime }$ are $O(1)$. We point out that this constraint can be evaded for example by curvaton scenarios, which typically leads to detectable non-Gaussianities. Our work can also be regarded as bottom-up constraints on the values of $c$ and $c^{\prime }$.

Figure 1

For a plateau-type inflaton potential, we find that (1) near the bottom of the potential for reheating the condition [$\epsilon$] is satisfied (region colored red), and (2) the condition [$\eta$] is satisfied in the regions for accelerated expansion (region colored blue). Depending on the choice of the parameters there could be an intermediate region where neither condition is satisfied, as in the figure below.

Figure 2

The $n_s-r$ plane for the natural inflation (16), with e-folding between 50 and 60. The constraint (18) for $c=0.3$ excludes the region of the right side of the purple line, making the model inconsistent with current observational constraints [64] (blue regions, 68% and 95% CL). The black and solid (dotted) line shows the upper-bound for $r$ (12) with $c^{\prime }=0(0.01)$.

Figure 3

The constraint of $c$ and $c^{\prime }$ for the natural inflation model with e-folding 50. The regions below the lines are allowed.

Figure 4

Same as in Fig. 2 but for the $E$-model $\alpha$-attractor (21) with $n=1$, with e-folding between 50 and 60. This model contains the Starobinsky model (19) as a special case $\alpha =1$. The constraint (22) for $c=0.3$, 1, 3 excludes the region above the purple line.

Figure 5

Same as in Fig. 3 but for the $E$-model $\alpha$-attractor (21) with $n=1$.