Consistency Relations for Large Field Inflation: Non-minimal Coupling

We derive the consistency relations for a chaotic inflation model with a non-minimal coupling to gravity. For a quadratic potential in the limit of a small non-minimal coupling parameter $\xi$ and for a quartic potential without assuming small $\xi$, we give the consistency relations among the spectral index $n_s$, the tensor-to-scalar ratio $r$ and the running of the spectral index $\alpha$. We find that unlike $r$, $\alpha$ is less sensitive to $\xi$. If $r<0.1$, then $\xi$ is constrained to $\xi<0$ and $\alpha$ is predicted to be $\alpha\simeq -8\times 10^{-4}$ for a quartic potential. For a general monomial potential, $\alpha$ is constrained in the range $-2.7\times 10^{-3}<\alpha<-6\times 10^{-4}$ as long as $|\xi|\leq 10^{-3}$ if $r<0.1$.


I. INTRODUCTION
In our previous paper, motivated by the possibility of large tensor-to-scalar ratio r [? ], we provided the consistency relations among spectral index n s , the tensor-to-scalar ratio r and the running of the spectral index α for several large field inflation models (chaotic with monomial potential, natural, symmetry breaking) [2]. Basic idea is to construct one relation out of two model parameters using three observables (n s , r and α).
In this paper, we investigate the stability of the consistency relation for chaotic inflation with a monomial potential which we have recently found. To do that, we consider a nonminimal coupling as a "perturber" of the model. Then, the number of model parameters becomes three and we need fourth observables (for example, the "running" of α), but it would introduce complication and the comparison with "unperturbed" relation would be difficult. So, in this paper we fix one of the model parameters and examine how introducing the non-minimal coupling affects the relation.

A. From Jordan to Einstein
We consider a single field inflation model with a non-minimal coupling to gravity. The action is given by where g µν is the Jordan frame metric and G * is the bare gravitational constant and we shall set 8πG * = 1 hence forth. As Ω(φ) and V (φ), we take where ξ is a non-minimal coupling parameter. In our convention, ξ = 1/6 corresponds to the conformal coupling.
As is well-known, by introducing the new metric called Einstein frame metric g µν = Ωg µν , the action can be rewritten as that of Einstein gravity with a scalar field [3]: where the hatted variables are defined by g µν and Ω ,φ = dΩ/dφ. Hence in terms of the canonically normalized scalar fieldφ defined by the system is reduced to the Einstein gravity plus a minimally coupled scalar field with the effective potential V defined by For Ω and V in Eq. (2), V with n = 4 becomes flat for large |ξφ 2 | with ξ < 0, which is the essence of the Higgs [4] (or Starobinsky [5]) inflation.
B. r, n s and α Hence, in order to compute the spectral index n s , the tensor-to-scalar ratio r and the running of the spectral index α, we only need to calculate slow-roll parameters in terms of φ and V : Then r, n s and α are given by In fact, for a single scalar field, the observables are independent of the conformal transformation [6,7].
For example, in the limit of small ξ, the slow-roll parameters become and r, n s and α are given by On the other hand, for n = 4 with large |ξ|φ 2 , we have and C. e-folding number Finally, we provide the relation for the e-folding number until the end of inflation N.
Since the scale factor and the proper time in the Jordan frame a and t are related to those in the Einstein frame a = Ω 1/2 a and dt = Ω 1/2 d t, the Hubble parameter in the Jordan frame H is related to that in the Einstein frame H by the relation [8,9] and the e-folding number N is given by Under the slow-roll approximation N becomes where f (φ) is defined by Eq. (4). The e-folding number is frame-invariant and can be calculated in either frame. For example, for |ξ| ≪ 1, N is given by N ≃ φ 2 /(2n), and for n = 4 and |ξ|φ 2 ≫ 1, N ≃ (1 − 6ξ)φ 2 /8.
(11)-(12), n and φ 2 are written in terms of r and n s and ξ Then from Eq. (13), α can be written as a function of r and n s which is too complicated to show here.

IV. SUMMARY
We have derived consistency relations for chaotic inflation with a nonminimal coupling ξ. For quadratic potential, we find that although the tensor-to-scalar ratio r is sensitive to ξ, the running of the spectral index α is rather insensitive to the change in ξ as long as ξ is small. For a quartic potential, we find that α is insensitive to ξ even for large |ξ|. We also find that the consistency relation for general monomial potential does not change so much by changing ξ as long as |ξ| ≤ 10 −3 .
Even for general monomial potential, r < 0.1 forces α in the range −2.7 × 10 −3 < α < −8 × 10 −4 for n s = 0.96 as long as |ξ| < 10 −3 . Since α is found to be insensitive ξ, this α may be regarded as the prediction for chaotic potential irrespective of the nonminimal coupling. The measurement of α with the precision of O(10 −3 ) by future observations of the 21 cm line [16] would be crucially important to pin down the inflation model.