Universal Constraints on Axions from Inflation

We consider the presence of an axion like particle, $\sigma$, with a generic $CP$ violating axial coupling of the form $ (\alpha \, \sigma/f) F \tilde{F}$, where $F_{\mu\nu}$ is the gauge field strength of a generic abelian $U(1)$ gauge group, not necessarily associated with the standard electromagnetism, and $f$ is the decay constant of the axion. It has previously been demonstrated that if the axion is identified with the inflaton, such an interaction can lead to measurable cosmological signatures (non-Gaussian modifications of the curvature perturbation spectrum) depending on the parameter $\xi= \alpha \dot{\sigma}/ (f H)$. In the present paper we will show that the generation of curvature perturbation at horizon crossing due to the axial coupling has a universal form and remains unmodified in terms of the $\xi$ parameter even if the axion, $\sigma$, is not identified with the inflaton. As a consequence, it does not appear to be possible to generate CMB tensor perturbations through this mechanism, larger than the vacuum ones, without violating the observational constraints unless we combine this mechanism with a curvaton or if the $\sigma$ field becomes heavy and decays during inflation. Even in this last case there are non-trivial constraints coming from the slow-roll evolution of the curvature perturbation on super horizon scales which should be taken into account. We also comment on implications for inflationary models where axions play an important role as, for example, models of natural inflation where more than one axion are included and models where the curvaton is an axion.