Modulated Natural Inflation

TL;DR

Natural (axionic) inflation faces challenges from requiring flat potentials and potential conflicts with the mild weak gravity conjecture when embedding in string theory. The authors develop a two-axion alignment framework with modular corrections from the Dedekind eta-function, yielding an effective, modulated potential $V=\Lambda^4\left(1-\cos\left[\frac{\varphi}{f}\right]\right)\left(1-\delta\cos\left[\frac{\varphi}{f_{\text{mod}}}\right]\right)$ that can realize trans-Planckian $f$ while preserving slow-roll. The modulations shift the scalar spectral index $n_s$ and induce running, producing spectra compatible with Planck data via full Mukhanov-Sasaki numerics; subleading instantons ensure the mild Weak Gravity Conjecture is satisfied, a feature tied to the string-theory origin of the potential. This work provides a string-consistent route to large-field inflation that remains viable under current CMB constraints and motivates explicit compactification realizations.

Abstract

We discuss some model-independent implications of embedding (aligned) axionic inflation in string theory. As a consequence of string theoretic duality symmetries the pure cosine potentials of natural inflation are replaced by modular functions. This leads to "wiggles" in the inflationary potential that modify the predictions with respect to CMB-observations. In particular, the scalar power spectrum deviates from the standard power law form. As a by-product one can show that trans-Planckian excursions of the aligned effective axion are compatible with the weak gravity conjecture.

Figures (6)

• Figure 1: Convex hull for the model \ref{['eq:2fields']} including the leading and all instantons. To avoid clutter, subleading instantons are only shown up to order 10 in the expansion of the $\eta$-function. The upper panel refers to a case of decay constant misalignment, the lower panel to alignment. In the alignment case, the convex hull condition is clearly violated if one considers only the leading instantons. However, by taking into account all subleading instantons, the convex hull dramatically increases. The weak gravity conjecture can thus be satisfied even in the alignment case. For illustrative purposes we have included a representative 'unit ball'. See Brown:2015iha for discussion of the size of the 'unit ball'.
• Figure 2: Slope of the axion potential for three exemplary parameter choices. The alignment increases from top to bottom. For too strong alignment the potential (and its derivatives) is very "wiggly" and unsuitable for inflation.
• Figure 3: Trajectories of the axions and saxions during inflation.
• Figure 4: Slow roll parameters $\epsilon$ (blue) and $\eta$ (orange) as a function of e-folds. For the dotted lines we neglected subleading instantons (which cause the wiggles on the potential).
• Figure 5: Upper panel: spectral index and tensor-to-scalar ratio for the benchmark scenario assuming $N_*=50-60$ (solid orange line). The dotted orange line is obtained by neglecting the wiggles on the potential. Also shown is the natural inflation band and the most recent Planck constraints. We depict the Planck constraints in the presence of a running spectral index which are weaker than the constraints without running. Lower panel: spectral index and running of the spectral index for the benchmark scenario (orange line) together with the Planck constraints. In pure natural inflation the running is negligible (green line).