Non-linear curvature perturbation in multi-field inflation models with non-minimal coupling

Abstract

Abstract. Using the δN formalism we consider the non-linear curvature perturbation in multi-field models of inflation with non-minimal coupling. In particular, we focus on the relation between the δN formalism as applied in the conformally related Jordan and Einstein frames. Exploiting results already known in the Einstein frame, we give expressions for the power spectrum, spectral tilt and non-gaussianity associated with the Jordan frame curvature perturbation. In the case that an adiabatic limit has not been reached, we find that in general these quantities differ from those associated with the Einstein frame curvature perturbation, and also confirm their equivalence in the absence of isocurvature modes. We then proceed to consider two analytically soluble examples, the first involving a non-minimally coupled 'spectator' field and the second being a non-minimally coupled extension of the multi-brid inflation model. In the first model we find that predictions can easily be brought into agreement with the recent Planck results, as the tensor-to-scalar ratio is generally small, the spectral tilt tuneable and the non-gaussianity suppressed. In the second model we find that predictions for all three parameters can differ substantially from those predicted in the minimally coupled case, and that the recent Planck results for the spectral tilt can be used to constrain the non-minimal coupling parameters.

Quantum corrections in Higgs inflation: the real scalar case

“…At the classical level it can be shown explicitly that the two frames are related by a 1-to-1 mapping of the fields, and thus lead to the same physics. At the quantum level they give the same results for the 2-and 3-point function [17][18][19][20]. 6 5 The Higgs (and GB, see section 5) contributions to the mid-field regime are negligible in comparison to additional gauge fields and fermions.…”

“…The two-and three-point functions of the perturbations during inflation are the same in both frames [15][16][17][18][19][20]. We shall demonstrate that this equivalence also holds for the 1-loop effective potential, and, derived from that, for the beta functions.…”

“…Then, for the large field, βλ = β λ /ξ 2 and we can give the beta function over the whole range: 20) for the small, mid-, and large field regimes, taking the conservative results for the mid-field. For completeness, we also state the corresponding full beta function for ξ:…”

Equivalence of the Einstein and Jordan frames

“…It was shown that the curvature perturbation is frameindependent [11,12,13,14,15] in the absence of curvature perturbations. Hoewever, framedependent definitions were introduced in the presence of isocurvature perturbations obscuring the equivalence of the two frames [22,23]. Applying the results of the previous section, we can express the perturbations spectrum in terms of the barred fields, which manifestly shows their frame-independence.…”

Derivative-dependent metric transformation and physical degrees of freedom

“…It is thus rather important to understand how the description of a theory changes from one frame to the other, especially from the Einstein frame to the matter frame, and vice versa. Along this direction, the invariance of physical observables under conformal and disformal transformations is intensively studied in the literature [2,3,[6][7][8][9][10][11][12][13][14][15][16][17].…”