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Naturalness in Higgs inflation in a frame independent formalism

We make use of the frame and gauge independent formalism for scalar and tensor cosmological perturbations developed in Ref. [1] to show that the physical cutoff for 2-to-2 tree level scatterings in Higgs inflation is above the Planck scale M_P throughout inflation. More precisely, we found that in the Jordan frame, the physical cutoff scale is $(Λ/a)_J \gtrsim \sqrt{M_P^2+ξϕ^2}$, while in the Einstein frame it is $(Λ/a)_J \gtrsim M_P$, where $ξ$ is the nonminimal coupling and $ϕ$ denotes the Higgs vev during inflation. The dimensionless ratio of the physical cutoff to the relevant Planck scale is equal to one in both frames, thus demonstrating the physical equivalence of the two frames. Our analysis implies that Higgs inflation is unitary up to the Planck scale, and hence there is no naturalness problem in Higgs inflation. In this paper we only consider the graviton and scalar interactions.