Non-Minimal Coupling to Gravity

We then impose a field redefinition of φ, φ =¯ r 3

so that our kinetic terms are canonical to obtain the action,

S = 1

In this frame the additional degree of freedom in the action of Eqn. (7.1) due to the R² term is now manifest as a scalar field minimally coupled to gravity. Moreover the observational predictions from this model are (ns, r) = (1− 2/N , 12/N²) ( with (ns, r) = (0.967 , 0.003) for N = 60). Coincidentally this result also fits neatly in observational data [103].

Recently it has emerged that many, seemingly unrelated, inflationary models con-verge on this same result (up to some uncertainty on the effect of reheating). This has appeared in Higgs-type inflationary models with the non-minimal couplings to gravity [188] ξφ²R with ξ < 0, in chaotic inflation with non-minimal coupling to gravity [189, 190] with ξ > 0 and in the context of supersymmetric extensions in-volving k¨ahler potential [191–195] the study of which later evolved into the field of generalised α-attractors. A typical feature of α-attractor models is a pole that appears in the Laurent expansion of the kinetic term in the Einstein Frame. This is a similar property in a related family of models, the ξ-attractor models. It is this common pole which underlines the attractor properties of these models [194].

7.2. Non-Minimal Coupling to Gravity

In scalar-tensor theories the Jordan frame is referred to as the frame where the Lagrangian contains a coupling between the scalar field and the scalar curvature.

The frame where the expressions for observations take the usual form is the Einstein frame, where there is no coupling to gravity and where the weak energy condition is not violated [196]. The action in the Jordan frame takes the form,

SJ = 1 2

Z

d⁴x√

−g(M_pl² + f (φ^I))gµνR^µν− δIJ∂µφ^I∂^µφ^J− U(φ^I) , (7.5)

7.2: Non-Minimal Coupling to Gravity 107

where the U (φ) is the potential in the Jordan frame and the non-minimal coupling function, f (φ^I), will, in the following, take the form

f (φ^I) = X

with ξ_I⁽ⁿ⁾, the dimensionless non-minimal coupling parameters. Any physical answer should not depend on the frame you are working in (in the same way as coordinate invariance) and the ones we choose to calculate in is more a matter of convenience.

At the end of inflation (when scalar field stops evolving) the ζ in the two frames should be conserved and equivalent under a certain frame-covariant transforma-tion [197,198]. The transformation from the Jordan Frame to the Einstein frame is by means of a conformal transformation. A conformal transformation is a rescaling of the space-time metric gµν → ¯gµν, and as we have seen above this is often accom-panied by a rescaling of the scalar field φ→ ¯φ. Under the conformal transformation, gµν → Ω⁻¹(φ^I)gµν, Ω(φ^I) = 1 + f (φ^I) , (7.7) we obtain the action in the Einstein frame

SE = 1

In the case of more than one scalar field minimally attached to gravity it is not possible to canonicalize the model, so we are inevitably left with a field-space metric.

In the case of a single-field one could redefine the scalar field so that the kinetic terms become canonical and only the potential is modified. For multiple fields this is in general not possible however using the transport method and code we can these calculate directly with Eqn. (7.8).

If we take the single-field example, Eqn. (7.8) becomes,

SE = 1

In slow-roll, the inflationary dynamics are parameterized just by the potential, and by transforming to the Einstein frame from the Jordan frame we recover this pa-rameterization.

In addition, the larger the size of the coupling parameter ξ the flatter the potential in the Einstein frame becomes and the closer the potential become to resembling the potential in Eqn. (7.4). This is what is called attractor behaviour. Essentially what we believed to be a large collection of different models in the Jordan frame gets mapped into a smaller subset of models in the Einstein Frame with a narrow range of dynamics.

7.2: Non-Minimal Coupling to Gravity 108

7.2.1. The Palatini Formalism

Recently, however, in Ref. [84] it was shown that α-attractors are in fact not univer-sal but depend on the underlying theory of gravity in a subtle way. The non-minimal couplings of the type ξI(φ^I)ⁿg^µνRµν contain freedom to choose the space-time con-nection: one can either study the usual metric case where Rµν = Rµν(gµν), or choose an alternative approach, the so-called Palatini formulation of gravity, where the con-nection Γ and hence also Rµν = Rµν(Γ) are independent variables. In the metric formulation of gravity, the connection Γ is determined uniquely as a function of the metric tensor, i.e. it is ¯Γ = ¯Γ(gµν) with

Γ¯^λ_αβ = 1

2g^λρ(∂αgβρ+ ∂βgρα− ∂ρgαβ) , (7.10) the Levi-Civita connection. The application of the variational principle then gives rise to an extra equation for the connection, in addition to the one for the metric.

For the Einstein-Hilbert action, the extra equation forces the connection to have the usual Levi-Civita form, but in more general theories of gravity, such as f (R) theories, or in the presence of non-minimal couplings, this is no longer true in the Jordan frame. In the context of general theory of relativity, the metric formalism coincides with the one of Palatini, as minimizing the Einstein-Hilbert action with respect to the connection uniquely fixes it to be of the Levi-Civita form, Γ = Γ(gµν).

In more general models, however, especially in the ones involving matter fields that are non-minimally coupled to gravity, these two formalisms lead to two inherently different gravitational theories [10, 85, 199]. This means that inflationary models with non-minimal couplings to gravity cannot be characterized just by the inflaton field potential, but that the connection must also be specified. This was originally studied in [200–202], and has recently gained increasing attention, see [52, 84, 203–

209]. A non-minimally coupled scalar field model in the Jordan frame with Palatini gravity, when mapped to the Einstein frame, can be written as Einstein gravity where the fields are uncoupled from the Ricci scalar. This means instead of studying Einstein of Palatini with non-minimally coupled fields in the Jordan frame, we can study uncoupled fields in the Einstein frame. The field-space metric in the action in Eqn. (7.8) can be written for both theories as

GIJ = Ω⁻¹δIJ +3

2νM_pl²Ω⁻²dΩ dφ^I

dΩ

dφ^J , (7.11)

with ν = 1 in the metric case and ν = 0 in the Palatini case. With this conformal transformation, we have therefore transferred the dependence on the choice of grav-itational theory (choice of gravgrav-itational degrees of freedom) from the connection to the field-space metric.