Scalar fields with derivative coupling to curvature in the Palatini and the metric formulation

We study models where a scalar field has derivative and non-derivative couplings to the Ricci tensor and the co-Ricci tensor with a view to inflation. We consider both the metric formulation and the Palatini formulation. In the Palatini case, the couplings to the Ricci tensor and the Ricci scalar give the same result regardless of whether the connection is unconstrained or the non-metricity or the torsion is assumed to vanish. When the co-Ricci tensor is included, the unconstrained case and the zero torsion case are physically different. We reduce all the actions to the Einstein frame with minimally coupled matter, and find the leading order differences between the metric case and the Palatini cases.

Generic actions with derivative couplings to the curvature, like generic actions with higher order curvature terms, lead to higher than second order equations of motion, which involve extra degrees of freedom that suffer from the Ostrogradsky instability [47].The most general scalar-tensor theories with second order equations of motion, called Horndeski theories, are explicitly known [48][49][50].They are, however, not the most general stable scalartensor theories, because it is possible that the theory is degenerate and some degrees of freedom are not physical.On the gravity side, the simplest example is f (R) theory [47].Degenerate higher order scalar-tensor theories (DHOST) have been explicitly catalogued up to terms cubic in the second derivatives of the field [49,50].The only such theories that are phenomenologically viable (with propagating gravitational waves and a Newtonian limit), at least at linear order in perturbation theory, are those that are related to Horndeski theories by an invertible disformal transformation [51] (see also [52,53]).Beyond DHOST are Udegenerate scalar-tensor theories, which are degenerate only in the unitary gauge, where the gradient of the scalar field has to be timelike [54][55][56][57][58][59].They have also been explicitly catalogued up to third order in second derivatives, and the procedure for determining whether a theory with arbitrary powers of second derivatives is DHOST or U-degenerate or neither is known.
These results are for the metric formulation of gravity.In other formulations that are equivalent for the Einstein-Hilbert action with minimally coupled matter but physically distinct for more complicated actions, the stability properties of non-minimally coupled scalar fields have not been completely categorised.(For Horndeski theories in teleparallel and symmetric teleparallel gravity, see [60,61].)One of the most common alternatives to the metric formulation is the Palatini formulation, where the connection is an independent variable [62,63]. 1 Higgs inflation, where the field couples directly to the Ricci scalar has been much studied in the Palatini formulation, and the predictions are different than in the metric case .Inflation in the case when derivatives of the field couple directly to the curvature has also been studied [91][92][93][94]; in [95], such a theory was used for quintessence (see also [96][97][98][99]).Unlike in the case when only the field couples directly to the curvature, in the derivative coupling case the results of the metric and the Palatini formulation are close to each other.We extend previous work by including the co-Ricci tensor in the cases when the connection is taken to be metric-compatible or torsion-free a priori.When parts of the connection are constrained in the action, the theory is in general different from the unconstrained case.For example, the theory with an Einstein-Hilbert term plus a term quadratic term in the antisymmetric part of the Ricci tensor is stable in the zero torsion case, but unstable in the unconstrained case [100,101].
In section 2 we give the geometrical background for the Palatini formulation and present the action.We shift to the Einstein frame with minimally coupled matter by making a disformal transformation followed by solving the remaining pieces of the connection from the equation of motion and inserting them back into the action.We calculate the leading order differences between the Palatini cases when the connection is unconstrained, when non-metricity or torsion is put to zero, and the metric case.In section 3 we summarise our findings and outline open questions.Some technical details are relegated to appendices A and B.
2 Non-minimal coupling to kinetic terms

Curvature, non-metricity, and torsion
In the Palatini formulation the metric g αβ and the connection Γ γ αβ are independent variables.The connection, defined with the covariant derivative as where Γγ αβ is the Levi-Civita connection of the metric g αβ .We denote quantities defined with the Levi-Civita connection by.In the second equality we have decomposed the distortion tensor L γ αβ into the disformation tensor J αβγ and the contortion tensor K αβγ , defined as where Q αβγ and T αβγ are the non-metricity and the torsion, respectively, defined as We have The Riemann tensor can be decomposed into the Levi-Civita and the distortion contributions as There are three independent first contractions of the Riemann tensor, called Ricci-type tensors, The first is the Ricci tensor, the second is the co-Ricci tensor, and the third is the homothetic curvature tensor.There is only one independent Ricci scalar, R = − R, R = 0. Instead of the co-Ricci tensor, it can be convenient to use the average of the co-Ricci tensor and the Ricci tensor.Using the definition (2.5) and the decompositions (2.1), (2.2), and (2.4), we see that the average vanishes when The Einstein tensor is where ϵ αβγδ is the Levi-Civita tensor.

The action
We consider a scalar field φ whose kinetic term X αβ ≡ ∂ α φ∂ β φ couples linearly to the first traces of the Riemann tensor, while φ can appear non-linearly.(General non-linear couplings have been studied in [101].)The homothetic curvature tensor does not appear because it is antisymmetric, so in the Palatini case, we have couplings to R αβ , Rαβ and R, and the action is where g = det g αβ , X ≡ g αβ X αβ , and L m (Ψ, φ, g αβ ) is a matter action2 , with Ψ denoting any matter degrees of freedom other than φ.
In the metric case Rαβ = −R αβ , so we can put α 3 = 0. Then when α 1 = − 1 2 α 2 , the action is of the Horndeski form, and there are no extra degrees of freedom, otherwise there is an extra ghost [48].If also α 2 > 0, the scalar degree of freedom corresponding to φ is healthy, otherwise it is a ghost [31].
In the Palatini case, the theory is different depending on which, if any, constraints are imposed on the connection.The case without constraints has been studied in [96,97].
Solving the connection equation obtained by varying (2.8) with respect to Γ γ αβ and inserting the solution into the action gives a metric theory with a modified scalar sector.For an action including (2.8) but more general, it was shown in [97] that the theory is at least U-degenerate (and can be DHOST or Horndeski).The reason is that it is symmetric under the projective transformation Γ γ αβ → Γ γ αβ + δ γ β V α , where V α is an arbitrary vector field.When the gradient of the scalar field is timelike, the ghost is subsumed in the unphysical projective mode. 3The results of [97] show that for the action (2.8), the theory is in the DHOST class.(For the case when X αβ couples only to the Ricci scalar and the Einstein tensor (2.7), i.e. α 3 = −α 2 , this was shown already in [96].) We will consider the case when either non-metricity or torsion is set to zero.

Disformal transformation
We could solve the connection separately in the cases with zero non-metricity or zero torsion and insert the solution back into the action.However, it is easier to first get rid of all non-minimal couplings except those to ∆ R αβ with a disformal transformation.This will also establish that the result is the same in the case when the connection is unconstrained and when non-metricity is put to zero, and that in the zero torsion case the difference arises only from ∆ R αβ .It has been shown that observables such as inflationary power spectra are invariant under disformal transformations at least for Horndeski theories [102][103][104][105][106] (see also [107]).
Applying the disformal transformation (2.9) to the action (2.8), using the above results, writing the co-Ricci tensor Rαβ in terms of ∆ R αβ defined in (2.6), and dropping the tildes on g αβ and X, we get The non-minimal couplings to R and R αβ are eliminated by choosing From (2.16) we can solve for γ 1 and γ 2 in closed form.The solutions are not very illuminating, so we do not write them down.For α 3 = 0 they simplify; the case α 1 = − 1 2 α 2 , α 3 = 0 is given in [93].
The disformal transformation is invertible and the original and transformed metric describe the same physics when These conditions set a limit on the values X can take.This is a limitation of the disformal transformation.Large spatial gradients such as may occur during preheating may mean that the coefficient of the Ricci tensor is not positive, so that even in the case α 3 = 0, the theory cannot be mapped to a minimally coupled Einstein frame with a disformal transformation.For study of slow-roll inflation in the super-Hubble regime, this is not a problem.
Inserting γ 1 and γ 2 back into the action (2.15), we get (dropping the matter Lagrangian) where we have defined (2.18) If α 3 = 0, then G 1 = 0.In this case the Ricci scalar is the only term that contains the connection, so the connection equation of motion gives the Levi-Civita connection.(In the case when there are no a priori constraints on the connection, it is determined only up to a projective transformation.)Inserting it back into the action we obtain a metric theory with a minimally coupled scalar field.The physics related to the distortion has been shifted to the modifications of the scalar field kinetic term and potential (and the matter Lagrangian).In the Einstein frame all matter couples to the scalar field and its kinetic term.We have not assumed anything about the connection, showing that if the co-Ricci tensor does not appear in the action, the physics is the same whether we keep the connection unconstrained or put non-metricity or torsion to zero.This is also the case in Palatini f (R) theory, which can be reduced to the Einstein-Hilbert plus minimally coupled matter form via field transformations [112][113][114][115].If the non-metricity is put to zero a priori, (2.6) shows that Rαβ = −R αβ , so the co-Ricci tensor is not independent, and there is no α 3 coupling.
The action (2.19) is manifestly in the Horndeski class.We noted in section 2.2 that based on the results of [97], the action is of the DHOST form.However, as written in the introduction, the only viable DHOST theories (at least to cubic order in second derivatives) seem to be those that are related to Horndeski theories by an invertible disformal transformation.There is no physical difference between Horndeski and DHOST theories as regards physical degrees of freedom and stability.

Zero torsion case with α 3 ̸ = 0
When α 3 ̸ = 0 and the non-metricity is non-zero, we have to solve the connection equation of motion and insert the solution back into the action.Let us consider the case with zero torsion.(The case with no constraints was considered in [97].)Varying the action (2.17) with respect to the distortion tensor (taking into account that it is symmetric in the last two indices) gives the equation of motion where X α ≡ ∂ α φ, Y αβ ≡ ∇α ∇β φ, and prime denotes partial derivative with respect to φ.
The general solution has the form Inserting (2.21) into (2.20),we solve for the coefficients l i (φ, X).The result is rather lengthy and is given in appendix A. Inserting l i into the action (2.17), we get the minimally coupled action where the coefficients B i (φ, X) and A i (φ, X) are again relegated to appendix A. If α 3 = 0, then B i = A i = 0, and (2.22) reduces to (2.19).The terms on the second line are non-Horndeski, but the functions A i satisfy the conditions for the theory to be DHOST [49,50].
In order to obtain a minimally coupled action, it was important to consider coupling to ∆ R αβ , which vanishes for the Levi-Civita connection, rather than Rαβ .
To second order in X αβ , the action (2.22) reads In [96,97] where the connection was unconstrained, the couplings to Rαβ were instead eliminated by writing them in terms of the commutator of the Levi-Civita covariant derivative, without transforming to the Einstein frame.This leads to a different form of the action; it is well known that a Horndeski or a DHOST theory can take quite different-looking forms.Transforming the action in [96,97] (keeping only the same original terms that we have) to the Einstein frame gives, to second order in X αβ , Comparing (2.23) and (2.24) shows that the theories agree for α 3 = 0, as then it makes no difference whether or not the torsion is constrained to be zero, as shown in section 2.3.For α 3 ̸ = 0 the theory with an unconstrained connection and the theory with zero torsion are physically inequivalent.
In [101] it was shown that an action that depends on Rαβ has a ghost around Minkowski space in the zero torsion case, and that in the unconstrained case there is a ghost around some FLRW backgrounds.This is not in contradiction with our result and the results of [96,97] that these cases are stable.In [101] it was assumed that the Legendre transformation to the Einstein frame is non-degenerate, which means that all degrees of freedom in Rαβ are included in the Einstein frame action.In our case with a scalar field, there are no vector or tensor modes.In [101] the FLRW ghost was in the tensor sector.

Metric case
Let us compare the Palatini case result (2.22) to the metric case.We are interested in the leading order differences in slow-roll inflation to see how the cases could be distinguished observationally.We again start with the action (2.8), now assuming that the connection is Levi-Civita.The action is of the Horndeski form when the kinetic term couples only to the Einstein tensor G αβ = R αβ − 1 2 g αβ R, not to the Ricci tensor and the Ricci scalar separately, otherwise it has a ghost.So we set α 1 = − 1 2 α 2 , α 3 = 0. We again shift to the Einstein frame with the disformal transformation (2.9).Now the calculation is more involved, because the Riemann tensor depends on the metric and its first and second derivative, unlike in the Palatini case.Hence, it is not invariant under the disformal transformation, which now introduces second order derivatives of φ.The transformation rules of the connection, the Riemann tensor, the Ricci tensor and the Ricci scalar are somewhat lengthy, and are given in appendix B. Inserting the result of the disformal transformation into the action (2.8), expanding to second order in X αβ and choosing the disformal functions γ 1 and γ 2 so that the non-minimal couplings vanish, we get the Einstein frame action To first order in X αβ and Y αβ , we recover the result of the original New Higgs Inflation paper [31], apart from the term involving F ′ .(The hybrid case with both F ′ ̸ = 0 and α 2 ̸ = 0 has been studied in [46].)Apart from the F ′ term, this leading order result agrees with the Palatini action (2.19) when α 1 = − 1 2 α 2 , α 3 = 0, as observed in [39].This is easy to understand: the distortion is sourced by F ′ and X αβ , and only appears in the action via the total derivative and quadratic terms in the Riemann tensor (2.4) and the coupling to the kinetic terms.So if F ′ = 0, the distortion only enters at second order in X αβ .The second order terms on the first line of (2.25) also agree with the Palatini result, which is less obvious.It is only the non-Horndeski terms that are different.
However, in the Palatini case we can obtain the same action to first (but not second) order in X αβ and Y αβ by coupling to just R, i.e. with α 2 = α 3 = 0.In the metric case such a coupling would lead to a ghost.Also, in the metric case the derivatives of F and α 2 enter, unlike in the Palatini case.The terms involving Y αβ are also different: in the Palatini case they appear only if α 3 ̸ = 0.If the dynamics are dominated by the derivative coupling, the differences are small in slow-roll, but if the non-derivative coupling is important, the theories can have quite different predictions, as comparison of [46] and [93] shows.

Conclusions
We have considered a theory where a scalar field kinetic term couples linearly to the Ricci tensor and the co-Ricci tensor, which appear linearly in the action, while the field itself can have non-linear non-minimal couplings.We look at both the Palatini formulation and the metric formulation.Extending previous Palatini work, we consider the case when either the non-metricity or the torsion is taken to vanish a priori.To establish the stability properties of the different cases and compare them side-by-side, we use a disformal transformation, followed by solving for the connection and inserting the solution back into the action.In this way we reduce the different cases to metric gravity with the Einstein-Hilbert action minimally coupled to matter.
We find that all the Palatini cases we consider are ghost-free: they are either in the Horndeski or DHOST class.If there is no coupling to the co-Ricci tensor, the Palatini result is independent of the assumptions about the connection.Otherwise, the case with unconstrained connection and the with zero torsion are physically different.(If nonmetricity is zero, the co-Ricci tensor vanishes.)We expand the actions up to second order in the scalar field kinetic term and compare the differences.
At leading order, the metric case and all the Palatini cases all agree with each other.However, in the Palatini case a much wider range of couplings is stable, for example it is possible to simply couple the Ricci scalar to the trace of the kinetic term, simplifying the model.At second order, the Horndeski terms agree in the Palatini and metric cases, but the beyond Horndeski terms are different.The detailed form of the terms beyond the leading order might appear contrived if written in the metric formulation to begin with, but in the original Palatini formulation they are simple.The Palatini formulation can be seen as a selection principle to determine which complicated derivative couplings should appear in a metric formulation action.
Higgs inflation driven by derivative couplings in the metric formulation does not have a unitarity problem, unlike the metric formulation of the original Higgs inflation scenario with a non-derivative coupling to the Ricci scalar alone [33,35,36,39].The theory is however sensitive to loop corrections [39].It would be interesting to see whether these features change in the derivative-driven or hybrid Palatini case.In the case with a non-derivative non-minimal coupling to the Einstein tensor alone, the unitarity problem is ameliorated in the Palatini formulation [69,79,80,[82][83][84][85][86].
It is an interesting question how to characterise the stability properties of theories in the Palatini formulation without reducing the theory a metric equivalent or calculating propagators.In [97] projective symmetry was used to show that a theory is U-degenerate, as the ghosts appear only in the unphysical projective mode.Projective invariance does not guarantee the absence of ghosts in general [101], only in particular cases [97,99,100].It would be interesting to understand better theories whose structure is tuned to the projective symmetry so that it makes them stable, and in particular whether projective symmetry (which has only a vectorial gauge mode) can prevent terms that would lead to tensor ghosts.

A Details of the solution for the connection in the zero torsion case
We give here details of the connection calculation in section 2.4 in the case when α 3 ̸ = 0 and the torsion is zero.The general solution of the connection equation of motion (2.20) in terms of the coefficients (2.21) is The coefficients of the final Einstein frame action (2.22) are

B Disformal transformation in the metric formulation
Under the disformal transformation (2.11), the Levi-Civita connection transforms as For the Riemann tensor we get