The reheating constraints to natural inflation in Horndeski gravity
  For the subclass of Horndeski theory of gravity, we investigate the effects
of reheating on the predictions of natural inflation. In the presence of
derivative self-interaction of a scalar field and its kinetic coupling to the
Einstein tensor, the gravitational friction to inflaton dynamics is enhanced.
As a result, the tensor-to-scalar ratio $r$ is suppressed. We place the
observational constraints on a natural inflation model and show that the model
is now consistent with the observational data for some plausible range of the
model parameter $\Delta$, mainly due to the suppressed tensor-to-scalar ratio.
To be consistent with the data at the $1\sigma$ ($68\%$ confidence) level, a
slightly longer $N_k\gtrsim60$ duration of inflation than usually assumed is
preferred. Since the duration of inflation, for any specific inflaton
potential, is related to reheating parameters, including the duration $N_{re}$,
temperature $T_{re}$, and equation-of-state $\omega_{re}$ parameter during
reheating, we imposed the effects of reheating to the inflationary predictions
to put further constraints. The results show that the duration of inflation
$N_k$ is affected by considerations of reheating, mainly by the $\omega_{re}$
and $T_{re}$ parameters. If reheating occurs instantaneously for which
$N_{re}=0$ and $\omega_{re}=1/3$, the duration of inflation is estimated to be
$N_k\simeq57$, where the exact value is less sensitive to the model parameter
$\Delta$ compatible with the CMB data. The duration of inflation is longer (or
shorter) than $N_k\simeq57$ for the equation of state larger (or smaller) than
1/3 hence $N_{re}\neq0$. The maximum temperature at the end of reheating is
$T_{re}^\text{max}\simeq3\times 10^{15}$ GeV, which corresponds to the
instantaneous reheating. The low reheating temperature, as low as a few MeV, is
also possible when $\omega_{re}$ is closer to $1/3$.
