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

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