Density perturbations in generalized Einstein scenarios and constraints on nonminimal couplings from the Cosmic Microwave Background
Shinji Tsujikawa, Burin Gumjudpai
TL;DR
The paper develops a general perturbation theory for generalized gravity actions with a nonminimal coupling and proves the inflationary observables are invariant between the Jordan and Einstein frames, yielding the same spectra and consistency relations as in standard Einstein gravity. It then specializes to a nonminimally coupled inflaton with $F(\phi)=(1-ξκ^2\phi^2)/κ^2$ and potentials $V(φ)=cφ^p$, deriving the Einstein-frame potential, background evolution, and perturbation spectra as functions of $ξ$ and $ψ=ξκ^2φ^2$. For the canonical cases $p=2,4,6$, the authors provide analytic slow-roll limits and perform a likelihood analysis against current CMB and large-scale structure data, obtaining constraints such as $ξ> -7.0×10^{-3}$ (1σ) for $p=2$ and $ξ< -1.7×10^{-3}$ (1σ) for $p=4$, while $p≥6$ remains disfavoured. The results show that negative nonminimal couplings can improve the fit for $p=4$ (and the Fakir-Unruh scenario with large negative $|ξ|$ is observationally favored in that case), demonstrating how precise data constrain the strength of nonminimal couplings in early-universe models.
Abstract
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.