Thermalized Axion Inflation

We analyze the dynamics of inflationary models with a coupling of the inflaton $\phi$ to gauge fields of the form $\phi F \tilde{F}/f$, as in the case of axions. It is known that this leads to an instability, with exponential amplification of gauge fields, controlled by the parameter $\xi= \dot{\phi}/(2fH)$, which can strongly affect the generation of cosmological perturbations and even the background. We show that scattering rates involving gauge fields can become larger than the expansion rate $H$, due to the very large occupation numbers, and create a thermal bath of particles of temperature $T$ during inflation. In the thermal regime, energy is transferred to smaller scales, radically modifying the predictions of this scenario. We thus argue that previous constraints on $\xi$ are alleviated. If the gauge fields have Standard Model interactions, which naturally provides reheating, they thermalize already at $\xi\gtrsim2.9$, before perturbativity constraints and also before backreaction takes place. In absence of SM interactions (i.e. for a dark photon), we find that gauge fields and inflaton perturbations thermalize if $\xi\gtrsim3.4$; however, observations require $\xi\gtrsim6$, which is above the perturbativity and backreaction bounds and so a dedicated study is required. After thermalization, though, the system should evolve non-trivially due to the competition between the instability and the gauge field thermal mass. If the thermal mass and the instabilities equilibrate, we expect an equilibrium temperature of $T_{eq} \simeq \xi H/\bar{g}$ where $\bar{g}$ is the effective gauge coupling. Finally, we estimate the spectrum of perturbations if $\phi$ is thermal and find that the tensor to scalar ratio is suppressed by $H/(2T)$, if tensors do not thermalize.