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.
