Thermalized axion inflation

We analyze the dynamics of inflationary models with a coupling of the inflaton ϕ to gauge fields of the form ϕ 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 ξ= /(2 fH ), 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 ξ are alleviated. If the gauge fields have Standard Model interactions, which naturally provides reheating, they thermalize already at ξ≳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 ξ≳3.4; however, observations require ξ≳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 ≃ ξ H / where is the effective gauge coupling. Finally, we estimate the spectrum of perturbations if ϕ is thermal and find that the tensor to scalar ratio is suppressed by H /(2 T ), if tensors do not thermalize.