A general criterion to compare metric and Palatini formulations with arbitrary non-minimal couplings: the case of Natural Inflation

Abstract

We study Natural Inflation with several types of non-minimal couplings to gravity in metric and Palatini formalisms, focusing on how the choice of gravitational formulation affects inflationary observables. Starting from a Jordan-frame action with a general coupling function $γ^2(ϕ)$, we examine several representative cases: periodic couplings ($\sin$ and $\cos$) and power-law couplings ($ϕ^n$ with $n = 1, 2, -2$). To track differences between the two formalisms, we employ a diagnosis parameter $J$ defined as ${J}\equiv K_{\rm M}/K_{\rm P} -1$, where $K_{\rm M}$ and $K_{\rm P}$ are the kinetic functions in the Einstein frame in both formalisms. This parameter directly quantifies the relative enhancement of the kinetic sector in the metric formulation. For the aforementioned couplings, we compute predictions for the spectral index $n_s$ and tensor-to-scalar ratio $r$, compare them against Planck 2018 and ACT DR6 observational constraints in both formalisms. Extending the particular cases mentioned before, we derive a general classification of arbitrary coupling functions for small-field inflation depending on the behavior of $J$ at reference values during inflation: Case I with vanishing ${J}$, so, both formalisms become nearly degenerate; and Case II with non-vanishing ${J}$ for which residual differences persist when evaluated at field values consistent with inflationary expansion. We find that periodic couplings are disfavored by data, whereas power-law couplings can achieve agreement for specific parameter ranges. Notably, in all the cases we found that Natural Inflation consistent with slow-roll evolution has $J$ either vanishing or strongly suppressed; consequently, switching the gravitational formulation yields no improvement for its observational viability.