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 (F) over tilde /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 = (phi)over dot/(2f H), 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 greater than or similar to 2.9, 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 greater than or similar to 3.4; however, observations require xi greater than or similar to 6, 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 similar or equal to xi H/(g) over bar where (g) over bar 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.