Large-Field Inflation with Multiple Axions and the Weak Gravity Conjecture

TL;DR

The paper develops a general, geometry-based framework to study large-field inflation with many axions and evaluates how the effective axion decay constant $f_{ ext{eff}}$ can scale with the number of axions $N$ when $P$ instanton terms contribute to the potential. By deriving a recurrence relation that depends on the dihedral angles and facet distances of the $N$-polytope bounding the axion moduli space, it identifies regimes from no enhancement to exponential enhancement and connects these scaling behaviors to the number of dominant instantons and their alignment. It then translates these geometric insights into quantum-gravity constraints via the weak gravity conjecture, arguing that to prevent parametric enhancement at large $N$ either super-Planckian excursions must occur or the number of dominant instantons must grow faster than quadratically (potentially exponentially). The work discusses possible loopholes to the WGC, reviews implications for the swampland program, and calls for explicit string-theory realizations to test these model-independent conclusions. Overall, the results provide a rigorous, model-independent link between multi-axion inflation phenomenology and nonperturbative quantum gravity constraints, with significant implications for viability of large-field inflation in theories with many axions.

Abstract

In this note, we discuss the implications of the weak gravity conjecture (WGC) for general models of large-field inflation with a large number of axions $N$. We first show that, from the bottom-up perspective, such models admit a variety of different regimes for the enhancement of the effective axion decay constant, depending on the amount of alignment and the number of instanton terms that contribute to the scalar potential. This includes regimes of no enhancement, power-law enhancement and exponential enhancement with respect to $N$. As special cases, we recover the Pythagorean enhancement of $N$-flation, the $N$ and $N^{3/2}$ enhancements derived by Bachlechner, Long and McAllister and the exponential enhancement by Choi, Kim and Yun. We then analyze which top-down constraints are put on such models from the requirement of consistency with quantum gravity. In particular, the WGC appears to imply that the enhancement of the effective axion decay constant must not grow parametrically with $N$ for $N \gg 1$. On the other hand, recent works proposed that axions might be able to violate this bound under certain circumstances. Our general expression for the enhancement allows us to translate this possibility into a condition on the number of instantons that couple to the axions. We argue that, at large $N$, models consistent with quantum gravity must either allow super-Planckian field excursions or have an enormous, possibly even exponentially large, number of dominant instanton terms in the scalar potential.

Figures (13)

• Figure 1: The fundamental domain of the moduli space of two axions for $N$-flation, "naive" alignment and alignment including quantum gravity constraints. $N$-flation yields an effective axion decay constant $f_\text{eff}$ enhanced by a factor of $\sqrt{2}$, which is compatible with the WGC as long as $f_\text{eff}$ is not super-Planckian. From the bottom-up perspective, alignment can lead to a very large enhancement. However, unless axions can exploit a loophole, the WGC predicts that new terms then become relevant in the scalar potential and shorten the field range.
• Figure 2: Normal vectors and the angles between them in a simple model for $N=2$, $N=3$ and $N=4$.
• Figure 3: The enhancement of the effective axion decay constant for $N=10^2$ and $N=10^3$ and different choices for the dihedral angle $\alpha_2$. The enhancement is slow for angles $\alpha_2 < \frac{\pi}{2}$ and diverges for angles $\alpha_2 > \frac{\pi}{2}$, where the different regimes are separated by $\Delta \alpha_2 \sim \mathcal{O}(1/N)$. The red dots denote the $N$-flation case $\alpha_2=\frac{\pi}{2}$.
• Figure 4: The enhancement of the effective axion decay constant for fixed angle $\alpha_2= \frac{\pi}{2}+ \frac{1}{200}$ and different $N$. For small $N \ll 200$, we recover the Pythagorean $\sqrt{N}$ law (blue) since the deviation of $\alpha_2$ from $\frac{\pi}{2}$ is smaller than $1/N$. For larger $N$, the enhancement starts to grow linearly (green). Close to the limit $N = 200$ where $\alpha_2-\frac{\pi}{2} = 1/N$, the enhancement grows polynomially and later (super-)exponentially (violet). Note that there is nothing special to the value $N=200$: had we chosen a smaller (larger) angle, the divergence would have appeared at larger (smaller) $N$.
• Figure 5: The enhancement of the effective axion decay constant for fixed angle $\alpha_2= \frac{\pi}{2}- \frac{1}{20}$ and different $N$. For small $N \ll 20$, we recover the Pythagorean $\sqrt{N}$ law (blue), while for larger $N$ the enhancement grows more slowly and finally dies out completely (violet).