Inflation models selected by the swampland distance conjecture with the Lyth bound

TL;DR

The paper investigates whether the Swampland Distance Conjecture (SDC) can theoretically constrain large-field inflation by bounding the tensor-to-scalar ratio $r$ as a function of the inflaton excursion $\Delta\phi$, complementing the Lyth bound. It derives the SDC bound $H \le m_{\rm pl} e^{-{\lambda_{\mathrm{dc}} \Delta\phi / m_{\rm pl}}}$ and translates it into a $\Delta\phi$–$r$ relation, exploring three representative towers with $\lambda_{\mathrm{dc}}=1$, $\sqrt{3/2}$ (KK tower), and $1/\sqrt{2}$ (string tower). The authors analyze four large-field models—chaotic, natural, hilltop, and $\alpha$-attractors—to obtain model-specific $r(\Delta\phi)$ predictions and compare them to both the SDC and Lyth bounds as well as current CMB constraints. They find that, in certain parameter regimes (notably for $\lambda_{\mathrm{dc}}=\sqrt{3/2}$), the SDC can impose stricter upper limits on $r$ than present observations and can even exclude some models, illustrating the swampland criteria as a valuable theoretical discriminator beyond empirical data.

Abstract

We investigate the extent to which the Swampland Conjecture can be employed to constrain large-field inflationary models from the perspective of quantum gravity consistency. In particular, we focus on the Swampland Distance Conjecture, which imposes an upper bound on the amplitude of primordial gravitational waves predicted by large-field inflation scenarios. This provides a striking contrast with the well-known Lyth bound, which yields a lower bound on the tensor-to-scalar ratio in such models. The two bounds thus play complementary roles in assessing the viability of inflationary scenarios. We demonstrate that, for certain representative large-field inflation models, the Swampland Distance Conjecture alone can impose more stringent upper limits on the tensor-toscalar ratio than current observational constraints from the cosmic microwave background. These findings highlight the utility of Swampland criteria as a theoretical discriminator among competing inflationary models, independent of empirical data.

Figures (6)

• Figure 1: Upper bounds on the tensor-to-scalar ratio $r$ as a function of the excursion distance of the inflaton $\Delta\phi$. Regarding the reference values for $\lambda_{\mathrm{dc}}$, we take $1$, $\sqrt{\frac{3}{2}}$, and the minimum $\frac{1}{\sqrt{2}}$ in the d=4 dimensions. The vertical green dashed line is the observational upper bound on the tensor-to-scalar ratio, $r < 0.036$ (95$\%$ CL) by the observational data of the cosmic microwave background (CMB) BICEP:2021xfzTristram:2021tvh.
• Figure 2: Lower bound on $\Delta \phi$ as a function of $r$ from the Lyth bound. Here we put $\Delta \phi = 60$ for a representative value in large field models.
• Figure 3: Allowed regions for four inflation models enclosed by the swampland distance conjecture (blue) and the Lyth bound (brown). We adopted the parameter $\lambda_{\mathrm{dc}} = 1$ for the swampland Distance Conjecture. For the theoretical calculations, we plotted the cases for chaotic inflation with the red lines for $p=1, 4/3, 2, 3$, natural inflation (black), hilltop inflation (green) and $\alpha$ attractors (cyan). Two lines mean $N$=47 (lower one) and $N$=62 (upper one), respectively. The vertical green dashed line is the observational upper bound on $r$ by the data of the cosmic microwave background (CMB) BICEP:2021xfzTristram:2021tvh.
• Figure 4: The same as that of Fig. \ref{['fig:sdc_lb_e1']}, but for $\lambda_{\mathrm{dc}} =\sqrt{\frac{1}{2}}$.
• Figure 5: The same as that of Fig. \ref{['fig:sdc_lb_e1']}, but for $\lambda_{\mathrm{dc}} =\sqrt{3/2}$.