Non-minimal (self-)running inflation: metric vs. Palatini formulation

We consider a model of quartic inflation where the inflaton is coupled non-minimally to gravity and the self-induced radiative corrections to its effective potential are dominant. We perform a comparative analysis considering two different formulations of gravity, metric or Palatini, and two different choices for the renormalization scale, widely known as prescription I and II. Moreover we comment on the eventual compatibility of the results with the final data release of the Planck mission.


Introduction
According to the theory of cosmic inflation [1][2][3][4], our Universe underwent a period of exponential expansion during the initial moments of its life. Inflation has the merit of providing at the same time a solution to issues like the flatness and horizon problems of the Universe and a way to generate primordial inhomogeneities, whose power spectrum is currently being tested in several experiments [5][6][7][8][9]. In particular, the final data release of the Planck mission [9] casts strong constraints on the tensor-to-scalar ratio r, an observable related to the amplitude of primordial gravitational waves and to the scale of inflation. As a consequence, the predictions of the simple monomial inflation models are ruled out at 2σ level, leaving non-minimally coupled to gravity models as the most favorite ones. In this article we are going to study models of inflation with a non-minimal coupling to gravity of the type ξφ 2 R, where φ is the inflaton field, R the Ricci scalar and ξ a coupling constant. Similar models have been studied in a large number of works over the past decades (in e.g. ). These models are particular interesting, since non-minimal couplings should be interpreted as a generic ingredient of consistent model building, arising from quantum corrections in a curved space-time [38]. In particular, this is the case for the scenario where the Standard Model Higgs scalar is the inflaton field [15]. Comparisons of non-minimally coupled models of chaotic inflation were performed in e.g. [18][19][20][21][22][23]27]. In Refs. [19,22], it was shown that for large values of the non-minimal coupling, all models, independently of the original scalar potential, asymptote to a universal attractor: the Starobinsky model [1]. However, the presence of non-minimal couplings to gravity requires a discussion about the gravitational degrees of freedom. In the usual metric formulation of gravity the independent variables are the metric and its first derivatives, while in the Palatini formulation the independent variables are the metric and the connection. Using the Einstein-Hilbert Lagrangian, the two formalisms predict the same equations of motion and therefore describe equivalent physical theories. However, with non-minimal couplings between gravity and matter, such equivalence is lost and the two formulations describe different gravity theories [16] and lead to different phenomenological results, as recently investigated in e.g. . In particular, the attractor behaviour of the so-called ξ attractor models [22] is lost in the Palatini formulation [45] . It is important to remark that in [22,45] the role of quantum corrections is implicitily assumed to be subdominant. On the other side, it has been demonstrated that radiative corrections to inflationary potentials may play a relevant role [62][63][64][65], dynamically generating the Planck scale [66,67], predicting super-heavy dark matter [68,69] and leading to linear inflation predictions when a non-minimal coupling to gravity is added [43,46,67,[70][71][72]. However most of the previous studies are assuming that the leading contribution to radiative corrections is coming from some other additional particle rather than the inflaton itself. The aim of this work is to study instead non-minimal inflation when self-corrections are the dominant loop contribution and to present a comparative analysis of the possible gravity formulation (metric or Palatini).
The article is organized as follows. In section 2 we set the notation reintroducing the main concepts about running coupling constants and the effective potential. In section 3 we discuss the gravitational sector and the main differences between the metric and the Palatini formulation of a gravity theory. In section 4 we present the comparative study of the inflationary predictions. We conclude in section 5.

Model building and effective potential
Consider the following action for a scalar-tensor theory in the Jordan frame where M P is the reduced Planck mass, R is the Ricci scalar constructed from a connection Γ and V eff (φ) is the effective potential of the inflaton scalar. The tree-level inflaton potential is however our focus is on the 1-loop 1 improved effective inflaton potential. Assuming that only self-corrections are relevant during inflation, the improved potential is where the effective quartic coupling is The second part of eq. (2.4) is the contribution coming from the Coleman-Weinberg (CW) 1-loop correction [102] to the effective potential, while the first one comes from the renormalization group equation (RGE) [103,104] of the quartic coupling, whose solution is 1 While cosmological perturbations are invariant under frame transformations (see for instance [27,73]), the equivalence of the Einstein and Jordan frames at the quantum level is still to be established. In the present article we therefore apply the following strategy: first we compute the effective potential in the Jordan frame, eq. (2.3), and consequently we move to the Einstein frame for computing the slow-roll parameters. Given a scalar potential in the Jordan frame, the cosmological perturbations are then independent, in the slow-roll approximation, of the choice of the frame in which the inflationary observables are evaluated [27,73]. For further discussions on frames equivalence and/or loop corrections in scalar-tensor theories we refer the reader to Refs. [44,72,.
where λ 0 = λ(µ 0 ) is the boundary condition for the RGE. For convenience we choose µ 0 = M P and keep λ 0 as a free parameter. It is important to keep in mind that the solution (2.4) is correct at the order O(λ 2 (µ)), therefore any µ-dependence at higher order can be thrown away and the effective coupling can be safely truncated as The purpose of the RGE improved effective potential is to obtain a potential that, at a given perturbative order, is independent on the choice of µ (e.g. [105] and references therein). Therefore, in the regime of validity of the RGE, i.e. until λ(µ) is small enough, any choice of µ is equivalent and should not carry any physical meaning [105]. Any effect coming from the choice of µ should be due to the loss of validity of the 1-loop expansion in eq. (2.6) and the need for a result at least at 2-loops. However there are two choices which are quite popular and eventually convenient, which are [106][107][108][109] also known as prescription I and also known as prescription II. Prescription I [110] is the choice motivated by the scaleinvariant quantization in the Jordan frame, while prescription II [80][81][82] corresponds to the usual quantization in the Jordan frame and it is convenient because it cancels explicitly the CW part of (2.6), moving all the loop correction into the running of the quartic coupling.
For convenience later on we will use the following notation: It is also useful to notice that in case of very small λ 0 , the dependence on µ explicitly cancels away. Performing a Taylor expansion of eq. (2.6) till the 2nd order in λ 0 we get the following approximated effective coupling 9) and the dependence on µ is completely removed. We notice that such expression recalls the running quartic coupling used in [46] with δ = 9λ 0 8π 2 , therefore we expect that some of the results of [46] will be valid also here.
Because of perturbativity of the theory, the inflationary predictions of such a potential, in absence of a non-minimal coupling to gravity, are pretty similar to the ones of the treelevel quartic potential and already ruled out by data [9]. Such predictions are dramatically changed if a non-minimal coupling to gravity is added, as we do in Lagrangian (2.1). However, a modification of gravity calls for a discussion of what theory of gravity we are going to consider. This will be shortly discussed in the following section.

Non-minimal gravity
We discuss now the gravitational sector and its non-minimal coupling to the inflaton. In order to avoid repulsive gravity we assume f (φ) > 0. This feature is independent on the eventual gravity formulation (metric or Palatini). In the metric formulation the connection is determined uniquely in function of the metric tensor, i.e. it is the Levi-Civita connection Γ =Γ(g µν ) On the other hand, in the Palatini formalism both g µν and Γ are treated as independent variables, and the only constraint is that the connection is torsion-free, Γ λ αβ = Γ λ βα . By solving the equations of motion we obtain [16] where Since the connections (3.1) and (3.2) are different, the metric and Palatini formulations provide indeed two different theories of gravity. Alternatively we can understand the differences by studying the problem in the Einstein frame via the conformal transformation In the Einstein frame gravity looks the same in both the formulations (see also eq. (3.2)), however the matter sector (in our case φ) behaves differently. Performing the computations [16], the Einstein frame Lagrangian becomes where χ is canonically normalized scalar field in the Einstein frame, and its scalar potential is given by In the metric case, χ is derived by integrating the following equation where the first term comes from the transformation of the Jordan frame Ricci scalar and the second from the rescaling of the Jordan frame scalar field kinetic term. On the other hand, in the Palatini case, the field redefinition is induced only by the rescaling of the inflaton kinetic term i.e. ∂χ ∂φ where there is no contribution from the Jordan frame Ricci scalar. Therefore we can see that the difference between the two formalisms in the Einstein frame relies on the different definition of χ induced by the different non-minimal kinetic term involving φ.
In the following we will focus on one particular type of f function: which the usual Higgs-inflation [15,94] non-minimal coupling 2 where we relaxed the condition that the inflaton is the Higgs boson and allowed the possibility that inflation is driven by another scalar beyond the Standard Model particle content.

Inflationary results
In this section we investigate the phenomenological implications of the non-minimal coupling in eq. (3.9). Since a detailed discussion of reheating is beyond the purpose of the present article, we do not need to specify the exact shape of the potential around its minimum. It is sufficient to assume that during inflation the potential is well described by eqs. (2.3) and (2.6). The corresponding Einstein frame scalar potential is given by where λ(µ) is given in (2.5) and the difference between the metric and the Palatini formulations is given by the different solution of eqs. (3.7) and (3.8).
Assuming slow-roll, the inflationary dynamics is described by the usual slow-roll parameters and the total number of e-folds during inflation 3 . The slow-roll parameters are defined as and the number of e-folds as where the field value at the end of inflation, χ f , is defined via (χ f ) = 1. The field value χ i at the time a given scale left the horizon is given by the corresponding N e . To reproduce the correct amplitude for the curvature power spectrum, the potential has to satisfy [9] ln 10 10 A s = 3.044 ± 0.014 , where and the other two relevant observables, i.e. the spectral index and the tensor-to-scalar ratio are expressed in terms of the slow-roll parameters by n s 1 + 2η − 6 (4.6) respectively. Before performing a detailed numerical analysis, let us discuss the strong coupling limit, ξ → +∞. In this case the two formulations share a similar Einstein frame field redefinition where we set in a convenient way the value χ = 0 and q is either for metric gravity, or q = q P = ξ , (4.10) for Palatini gravity. In the strong coupling limit the Einstein frame potential behaves like a running cosmological constant If λ 0 is small (λ 0 1), we can replace λ eff with λ app and get We can see that in both gravity formulations the limit solution is linear inflation, with the only difference in the normalization factor q. As expected, this result is in agreement with [46] with δ = 9λ 0 8π 2 . For λ eff = λ I we have It is interesting to notice that eq. (4.13) is the same as λ app in eq. (2.9) with the replacements λ 0 →λ 0 and µ 0 →μ 0 . Therefore we expect that inflationary results for λ eff = λ app,I will be the same in the strong coupling limit, but for different values of λ 0 and ξ. And again this holds independently on the formulation of gravity. Finally, for λ eff = λ II we have and therefore where we used eq. (4.8) andχ In this case the potential does not resemble the behaviour of linear inflation, in contrast to what happens with λ eff = λ app,I . Moreover by using eq. (4.6) we get which means that n s 1 and the strong coupling limit of λ eff = λ II is ruled out. For completeness, we perform a full inflationary analysis considering also ξ values not in the strong coupling limit. We proceed in the following way. We first fix the gravity formulation (metric or Palatini) and then the effective coupling that we want to study (λ eff = λ app , λ I , λ II ). Assuming N e = 50, we remain with only two free parameters: λ 0 and ξ. We vary λ 0 between λ 0 ≈ 10 −13 (the usual value for quartic inflation) and λ 0 = 1 (naive upper limit set as a necessary, but not always sufficient, condition to ensure perturbativity of the theory during inflation). Therefore ξ is fixed in order to satisfy the constraint (4.4).
The corresponding results are given in Figs. 1, 2, 3 and 4. In Fig. 1 we are presenting the results for the metric formulation and plotting r vs. n s (a), r vs. ξ (b), ξ vs. n s (c) and λ 0 vs. ξ (d) for λ eff = λ app (cyan), λ eff = λ I (blue, dashed) and λ eff = λ II (light blue, dot-dashed) with N e = 50 e-folds. For reference we also plot predictions of quartic (brown), quadratic (black) and linear (yellow) inflation for N e ∈ [50,60]. The gray areas represent the 1,2σ allowed regions coming from Planck 2018 data [9]. In Fig.2 we show the same plots for r vs. n s (a) and λ 0 vs. ξ (b) as in Fig. 1, but respectively zoomed in the regions r ≤ 0.02 and λ 0 ≥ 0.05. Figs. 3 and 4 are the same as Figs. 1 and 2 but for the Palatini formulation of gravity and a zoom respectively for r ≤ 10 −3 and λ 0 ≤ 0.3.
The results of the two formulations share some similarities. First, for ξ 0, the predictions are compatible with the ones of standard quartic inflation. Then, by increasing ξ until ξ 10 3 , the predictions are aligned with the respective strong-coupling limits of the standard (without loop corrections) non-minimal inflation ( [22] for metric and [45] for Palatini). The attractor limit is well described in the vertical region around n s 0.96 in Figs. 1c and 3c. When λ 0 (or equivalently ξ) is small enough, the predictions for λ eff = λ app,I,II are overlapped. This happens for λ 0 0.1 (ξ 1.2 × 10 4 ) in metric gravity and λ 0 10 −3 (ξ 4 × 10 3 ) in Palatini gravity. Moreover, as anticipated before, it is impossible to discriminate between λ eff = λ app , λ I in the r vs. n s plots respectively in both gravity formulations for any values  of λ 0 and ξ. Differences are appreciable only in the actual values of those two parameters. This happens for λ 0 0.2 in metric gravity and λ 0 0.01 in Palatini gravity. On the other hand, there are several differences between the results of the two formulations. First of all, while in the Palatini formulation it is possible to reach the linear inflation limit [46] within perturbativity, this never happens in case of metric gravity. The same happens for the strong coupling limit of λ eff = λ II , which is only allowed in Palatini gravity. Moreover, the lower limit for r in the metric case coincides with the prediction of R 2 inflation, where quantum corrections are still sub-dominant, while for the Palatini case the lower limit is r 10 −7 , where the loop effects are relevant. Furthermore, only in the Palatini formulation for λ eff = λ II with λ 0 0.03 we encountered Landau poles in the inflationary region and therefore removed the corresponding points. In addition, comparing Figs. 1 and 3 we can see that while in the metric case ξ is monotonically increasing with λ 0 , in the Palatini one ξ increases until n s ∼ 1 is reached, then it decreases, and then it increases again. In the first and third region ξ behaves as expected, therefore let us focus on the region n s 1, where there results of λ 0 vs. ξ were unforeseen. As shown in Fig. 3d, in this region the lines of λ eff = λ app,I,II are still overlapped, therefore it is enough to study the λ eff = λ app case. In such a region we can still apply the strong coupling limit ξ → ∞ and approximate the Einstein frame potential again as a running cosmological constant (see eq. (4.11)). However now it is convenient to solve the field redefinition (3.8) as follows where we conveniently shifted the position of χ = 0. Therefore the Einstein frame potential becomes It can proven that in this case 9λ 0 √ ξχ 8π 2 M P 1 and therefore r 0 and n s 1. The constraint on the amplitude of the perturbation (4.4) implies (4.21) Now considering the small λ 0 limit, we get From this last equation we can see that λ 0 is inversely proportional to ξ 3 , in agreement with our numerical results for λ eff = λ app when n s 1 in the Palatini formulation. On the other hand it is also interesting to see separately the corresponding limit of the Einstein frame potential for λ eff = λ II . Such limit is already given in eq. (4.16) and the consequent constraint on the amplitude is which recovers the results of λ eff = λ app for small λ 0 and departs from them by increasing λ 0 and allowing the possibility of n s 1 (see eq. (4.18)). Therefore, in the Palatini formulation it is possible to discriminate between λ eff = λ I,II nearby n s ∼ 1 (λ 0 0.005), far away from the 2σ allowed region. On the other hand, in the metric case it is impossible to discriminate between λ eff = λ I,II within the 1σ region, but it is possible within the 2σ boundary from λ 0 0.15 (ξ 1.4 × 10 4 ) (see the zoomed plot in Fig. 2a). As we mentioned before, there should be no physical difference in λ eff = λ I,II , therefore this should be interpreted as a loss of accuracy in the expansion for the effective potential in eq. (2.3) and the need to consider higher order loop corrections. Finally we conclude remarking that, in agreement with the findings of [111], the impact of radiative corrections is stronger in the Palatini formulation rather than in the metric formulation, because the Jordan frame field excursion is larger in the Palatini formulation.

Conclusions
We studied a model of quartic inflation where the inflaton field φ is subject to relevant selfinduced radiative corrections and it is coupled non-minimally to gravity. We considered the Higgs-inflation-like non-minimal coupling. We studied the predictions of two different formulations of gravity, metric or Palatini, and the three possible versions of the effective quartic couplings λ eff (φ, µ) = λ app,I,II (φ): λ app is the case in which the tree-level quartic coupling λ 0 is very small, we can Taylor expand and explicitly remove from λ eff the dependence on the renormalization scale µ, while λ eff (φ, µ) = λ I,II (φ) corresponds to the prescription I,II choices given in eqs. (2.7) and (2.8). We showed that the formulations share several differences, as expected, but also some interesting similarities. We start with the last ones.
First of all, trivially, the predictions are compatible with the ones of standard quartic inflation for ξ 0. Then, by increasing ξ until ξ 10 3 , the predictions are substantially the same as the respective strong-coupling limits of the standard (tree-level) non-minimal inflation ( [22] for metric and [45] for Palatini). When λ 0 0.1 (ξ 1.2 × 10 4 ) in metric gravity, the predictions for λ eff = λ app,I,II are undistinguishable. The same holds for λ 0 10 −3 (ξ 4 × 10 3 ) in Palatini gravity. Moreover, we showed that for ξ 1, λ app and λ I can be mapped into each other just by varying λ 0 and ξ. Therefore it is impossible to distinguish between λ eff = λ app,I in the r vs. n s plots respectively in both gravity formulations. Eventual differences are appreciable only in the actual values of λ 0 and ξ.
On the other hand, the first difference that we notice is the possibility to reach within perturbativity the linear inflation limit [46] or the strong coupling limit (4.16) of λ II only in Palatini gravity. Moreover, the lower limit for r in the metric case coincides with the prediction of Starobinsky inflation, while for the Palatini case the lower limit is r 10 −7 . In the metric case ξ is everywhere monotonically increasing with λ 0 , while in the Palatini one ξ can also decrease around the value n s 1. Around such a value, still in Palatini gravity, it is also possible to discriminate between λ eff = λ I,II for λ 0 0.005. On the other hand, in the metric case it is impossible to discriminate between λ eff = λ I,II within the 1σ region, but it is possible within the 2σ boundary from λ 0 0.15 (ξ 1.4 × 10 4 ). As there should be no physical difference in λ eff = λ I,II , this should point out a loss of accuracy in the expansion for the effective potential in eq. (2.3) and the need to add higher order loop corrections. This will be considered in a separate work.