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 during inflation. 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 4, 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 natural inflation with N-k greater than or similar to 60 e-folds, longer than usually assumed, is preferred. Since the duration of inflation, for any specific inflaton potential, is linked to reheating parameters, including the duration N-re, temperature T-re, and equation-of-state cu re parameter during reheating, we imposed the effects of reheating to the inflationary predictions to put further constraints. The results show that reheating consideration impacts the duration of inflation N-k. If reheating occurs instantaneously for which N-re = 0 and omega(re) = 1/3, the duration of natural inflation is about N-k similar or equal to 57 e-folds, where the exact value is less sensitive to the model parameter Delta compatible with the CMB data. The duration of natural inflation is longer (or shorter) than N-k similar or equal to 57 e-folds for the equation of state larger (or smaller) than 1/3 hence N-re not equal 0. The maximum tem- perature at the end of reheating is T-re(max) similar or equal to 3 x 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.