The reheating constraints to natural inflation in Horndeski gravity

Chen-Hsu Chien , Seoktae Koh, and Gansukh Tumurtushaa 2022 Eur. Phys. J. C 268

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 $Δ"> Δ$ , mainly due to the suppressed tensor-to-scalar ratio. To be consistent with the data at the $1σ"> 1 σ$ ( $68%"> 68 %$ confidence) level, a slightly longer natural inflation with $Nk≳60"> N k ≳ 60$ 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 $Nre"> N r e$ , temperature $Tre"> T r e$ , and equation-of-state $ωre"> ω r e$ 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 $Nk"> N k$ . If reheating occurs instantaneously for which $Nre=0"> N r e = 0$ and $ωre=1/3"> ω r e = 1 / 3$ , the duration of natural inflation is about $Nk≃57"> N k ≃ 57$ e-folds, where the exact value is less sensitive to the model parameter $Δ"> Δ$ compatible with the CMB data. The duration of natural inflation is longer (or shorter) than $Nk≃57"> N k ≃ 57$ e-folds for the equation of state larger (or smaller) than 1/3 hence $Nre\neq0"> N r e \neq 0$ . The maximum temperature at the end of reheating is $Tremax≃3\times1015"> T max r e ≃ 3 \times 10^15$ GeV, which corresponds to the instantaneous reheating. The low reheating temperature, as low as a few MeV, is also possible when $ωre"> ω r e$ is closer to 1/3.