Thermalized axion inflation: natural and monomial inflation with small $r$

A safe way to reheat the universe, in models of natural and quadratic inflation, is through shift symmetric couplings between the inflaton $\phi$ and the Standard Model (SM), since they do not generate loop corrections to the potential $V(\phi)$. We consider such a coupling to SM gauge fields, of the form $\phi F\tilde{F}/f$, with sub-Planckian $f$. In this case gauge fields can be exponentially produced already {\it during inflation} and thermalize via interactions with charged particles, as pointed out in previous work. This can lead to a plasma of temperature $T$ during inflation and the thermal masses $gT$ of the gauge bosons can equilibrate the system. In addition, inflaton perturbations $\delta \phi$ can also have a thermal spectrum if they have sufficiently large cross sections with the plasma. In this case inflationary predictions are strongly modified: (1) scalar perturbations are thermal, and so enhanced over the vacuum, leading to a generic way to {\it suppress} the tensor-to-scalar ratio $r$; (2) the spectral index is $n_s-1=\eta-4\epsilon$. After presenting the relevant conditions for thermalization, we show that thermalized natural and monomial models of inflation agree with present observations and have $r\approx 10^{-3} - 10^{-2}$, which is within reach of next generation CMB experiments.