Natural Inflation with non minimal coupling to gravity in $R^2$ gravity under the Palatini formalism

Natural Inflation with non-minimal coupling (NMC) to gravity, embodied by a Lagrangian term $\xi \phi^2 R $, is investigated in the context of an extended gravity of the form $R+ \alpha R^2$. The treatment is performed in the Palatini formalism. We discuss various limits of the model ``$\alpha \gg 1$'' and ``$\alpha \ll 1$'' in light of two scenarios of inflation: a ``Slow roll'' and a ``Constant roll'' scenario. By analyzing the observational consequences of the model, our results show a significant improvement regarding compatibility between the theoretical results of this model and the observational constraints from Planck 2018 and BICEP/Keck 2018, as exemplified by the tensor-to-scalar ratio and spectral index. Furthermore, a broader range for the parameter space of natural inflation is now compatible with the confidence contours of Planck \& BICEP/Keck results. The joint effects of the contributions of both the NMC to gravity and the $\alpha R^2$ make a significant improvement: $\alpha R^2$ gravity influences scalar-tensor ratio values, whereas NMC to gravity has a more significant impact on the spectral index values. Contributions from both terms allow more previously excluded intervals to be included being compatible now with observational data. These conclusions about the roles of NMC to gravity and, particularly, the extended gravity remain mainly valid with a periodic NMC similar in form to the natural inflation potential.