Natural inflation with multiple sub-Planckian axions

TL;DR

The paper addresses the challenge of achieving a super-Planckian effective axion decay constant for natural inflation by extending the Kim-Nilles-Peloso (KNP) alignment mechanism to $N$ sub-Planckian axions. By analyzing a general $N$-axion potential with an anomaly-matrix structure, it shows that $f_{ m eff} \gg f_i$ can arise with ${\cal O}(1)$ anomaly coefficients when $N$ is large enough such that $N \ln N \gtrsim 2\ln(f_{\rm eff}/f_i)$, and that $f_{\rm eff}/f_i$ scales as $\sqrt{N!}\,n^{N-1}$ for typical coefficient size $n$. The work also provides two explicit constructions—one yielding a multiple-axion monodromy with $f_{\rm eff}/f_i \sim \prod_{j=2}^N n_j$ and another giving $f_{\rm eff}/f_i \sim 2^{N-1}$ with ${|n_{ij}|\le1$}—demonstrating that exponential enhancements are achievable with a modest number of axions. Compared to the N-flation approach, this framework achieves large $f_{\rm eff}$ without requiring an excessively large number of fields. The results expand the model-building toolkit for natural inflation within a UV-complete setting and offer concrete pathways to realize measurable tensor modes without invoking super-Planckian fundamental decay constants.

Abstract

We extend the Kim-Nilles-Peloso (KNP) alignment mechanism for natural inflation to models with $N>2$ axions, which obtains a super-Planckian effective axion decay constant $f_{\textrm{eff}}\gg M_{Pl}$ through an alignment of the anomaly coefficients of multiple axions having sub-Planckian fundamental decay constants $f_0\ll M_{Pl}$. The original version of the KNP mechanism realized with two axions requires that some of the anomaly coefficients should be of the order of $f_{\textrm{eff}}/f_0$, which would be uncomfortably large if $f_{\rm eff}/f_0 \gtrsim {\cal O}(100)$ as suggested by the recent BICEP2 results. We note that the KNP mechanism can be realized with the anomaly coefficients of $\mathcal{O}(1)$ if the number of axions $N$ is large as $N\ln N\gtrsim 2\ln (f_{\textrm{eff}}/f_0)$, in which case the effective decay constant can be enhanced as $f_{\rm eff}/f_0 \sim \sqrt{N !}\,n^{N-1}$ for $n$ denoting the typical size of the integer-valued anomaly coefficients. Comparing to the other multiple axion scenario, the N-flation scenario which requires $N \sim f_{\textrm{eff}}^2/f_0^2$, the KNP mechanism has a virtue of not invoking to a too large number of axions, although it requires a specific alignment of the anomaly coefficients, which can be achieved with a probability of ${\cal O}(f_0/f_{\rm eff})$ under a random choice of the anomaly coefficients. We also present a simple model realizing a multiple axion monodromy along the inflaton direction.

Figures (2)

• Figure 1: Flat direction in the fundamental domain of axion fields in the limit $\Lambda_2=0$. Even though the fundamental domain is sub-Planckian with $f_i\ll M_{Pl}$, the flat direction can have a super-Planckian length if one (or both) of $n_i/{\rm gcd}\, (n_1,n_2)$ is large enough. The right panel depicts the flat direction in the fundamental domain for which the axion periodicity is manifest.
• Figure 2: Multiple monodromy structure for the three-axion model with $n_2=n_3=2$. The solid red line represents the inflaton direction in the fundamental domain of three axions. Note that $\Delta \phi_2=2\pi f_2$ along the inflaton direction requires $\Delta \phi_1=2\pi n_2f_1$, and $\Delta \phi_3=2\pi f_3$ requires $\Delta \phi_2=2\pi n_3 f_2$. As a result, $\Delta \phi_3=2\pi f_3$ along the inflaton direction yields $\Delta \phi_1=2\pi n_2n_3 f_1$.