Eur. Phys. J. C (2022) 82: 268
https://doi.org/10.1140/epjc/s10052-022-10236-w

Regular Article - Theoretical Physics

The reheating constraints to natural inflation in Horndeski gravity

¹ Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, 10617, Taipei, Taiwan, ROC
² Department of Science Education, Jeju National University, 63243, Jeju, Korea

^c gansuh.mgl@gmail.com

Received: 23 December 2021
Accepted: 20 March 2022
Published online: 28 March 2022

Abstract

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 $\Delta$, 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 natural inflation with $N_k\gtrsim 60$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 $N_{\mathit{re}}$, temperature $T_{\mathit{re}}$, and equation-of-state ${\omega }_{\mathit{re}}$ 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 $N_{\mathit{re}}=0$ and ${\omega }_{\mathit{re}}=1/3$, the duration of natural inflation is about $N_k\simeq 57$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 $N_k\simeq 57$e-folds for the equation of state larger (or smaller) than 1/3 hence $N_{\mathit{re}}\ne 0$. The maximum temperature at the end of reheating is $T_{\mathit{re}}^{\text{max}}\simeq 3\times {10}^{15}$ GeV, which corresponds to the instantaneous reheating. The low reheating temperature, as low as a few MeV, is also possible when ${\omega }_{\mathit{re}}$ is closer to 1/3.