Spinodal Instabilities and Super-Planckian Excursions in Natural Inflation

Abstract

Models such as Natural Inflation that use Pseudo-Nambu-Goldstone bosons (PNGB's) as the inflaton are attractive for many reasons. However, they typically require trans-Planckian field excursions $\Delta \Phi >M_{\mathrm{Pl}}$, due to the need for an axion decay constant $f>M_{\mathrm{Pl}}$ to have both a sufficient number of e-folds {\em and} values of $n_s,r$ consistent with data. Such excursions would in general require the addition of all other higher dimension operators consistent with symmetries, thus disrupting the required flatness of the potential and rendering the theory non-predictive. We show that in the case of Natural Inflation, the existence of spinodal instabilities (modes with tachyonic masses) can modify the inflaton equations of motion to the point that versions of the model with $f<M_{\mathrm{Pl}}$ can still inflate for the required number of e-folds. The instabilities naturally give rise to two separate phases of inflation with different values of the Hubble parameter $H$ one driven by the zero mode, the other by the unstable fluctuation modes. The values of $n_s$ and $r$ typically depend on the initial conditions for the zero mode, and, at least for those examined here, the values of $r$ tend to be unobservably small.

4 pages, 2 figures. V2: Final version accepted for publication in PRL. Changes: discussion clarified including regarding the scope of our result and the Hartree approximation, no changes to technical results or conclusions

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