Chaotic Inflation and Reheating in Generalized Scalar-Tensor Gravity

In the present work, we study slow-roll inflation in scalar-tensor gravity theories in the presence of both the non-minimal coupling between the scalar field and curvature, and the Galileon self-interaction of the scalar field. Furthermore, we give predictions for the duration of reheating as well as for the reheating temperature after inflation. After working out the expressions for the power spectra of scalar and tensor perturbations in the case of a general non-minimal coupling function that depends solely on the scalar field and a general scalar potential, we focus on the special cases of the power-law coupling function and chaotic quadratic inflation. Thus, under the slow-roll approximation we confront the predictions of the model with the current PLANCK constraints on the spectral index $n_s$ and the tensor-to-scalar ratio $r$ using the $n_{s}-r$ plane. We found that the combination of the non-minimal coupling and Galileon self-interaction effects allows us to obtain better results for $r$ than in the case in which each effect is considered separately. Particularly, we obtained that the predictions of the model are in agreement with the current observational bounds on $n_{s}$ and $r$ within the $95 \%$ C.L region and also slightly inside the $68 \%$ C.L region. Also, we investigate the oscillatory regime after the end of inflation by solving the full background equations, and then we determine the upper bound for the Galileon and non-minimal coupling parameters under the condition that the scalar field oscillates coherently during reheating. Finally, after approximating reheating by a constant equation of state, we derive the relations between the reheating duration, the temperature at the end of reheating, its equation of state, and the number of $e$-folds of inflation and then we relate them all to the inflationary observables.