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Density perturbations in generalized Einstein scenarios and constraints on nonminimal couplings from the Cosmic Microwave Background

We study cosmological perturbations in generalized Einstein scenarios and show the equivalence of inflationary observables both in the Jordan frame and the Einstein frame. In particular the consistency relation relating the tensor-to-scalar ratio with the spectral index of tensor perturbations coincides with the one in Einstein gravity, which leads to the same likelihood results in terms of inflationary observables. We apply this formalism to nonminimally coupled chaotic inflationary scenarios with potential $V=cϕ^p$ and place constraints on the strength of nonminimal couplings using a compilation of latest observational data. In the case of the quadratic potential ($p=2$), the nonminimal coupling is constrained to be $ξ>-7.0 \times 10^{-3}$ for negative $ξ$ from the $1σ$ observational contour bound. Although the quartic potential ($p=4$) is under a strong observational pressure for $ξ=0$, this property is relaxed by taking into account negative nonminimal couplings. We find that inflationary observables are within the $1σ$ contour bound as long as $ξ<-1.7 \times 10^{-3}$. We also show that the $p \ge 6$ cases are disfavoured even in the presence of nonminimal couplings.