Hamilton–Jacobi method in non-minimal coupling inflation: metric vs. Palatini

Abstract The Hamilton–Jacobi approach offers a natural framework for analyzing inflationary dynamics, relying on the specified Hubble parameter rather than the potential, particularly in extended gravity theories. In this study, we apply this method to investigate inflation with non-minimal coupling, comparing the metric and Palatini formulations. Using a power-law Hubble parameter in the Jordan frame, we find that both formalisms satisfy the attractor condition, with a slight suppression in the Palatini case. Our results align closely with the latest observational data, demonstrating the Palatini formalism’s heightened sensitivity to coupling constants and model parameters. Furthermore, we show that within the model’s viable parameter space, the reheating process can achieve sufficiently high temperatures to support successful leptogenesis.