Stability of non-degenerate Ricci-type Palatini theories

We study the stability of theories where the gravitational action has arbitrary algebraic dependence on the three first traces of the Riemann tensor: the Ricci tensor, the co-Ricci tensor, and the homothetic curvature tensor. We collectively call them Ricci-type tensors. We allow arbitrary coupling to matter. We consider the case when the connection is unconstrained, and the cases when either torsion or non-metricity is assumed to vanish. We find which combinations of Ricci-type tensors lead to new degrees of freedom around Minkowski and FLRW space, and when there are ghosts. None of the theories with new degrees of freedom are healthy, except the previously known case when torsion is zero and the action depends only on the Ricci tensor. We find that projective invariance is not a sufficient condition for a theory to be ghost-free.


Introduction
Renormalisation in quantum field theory in curved spacetime requires higher order curvature terms, and they also appear in proposed ultraviolet completions of general relativity [1][2][3][4][5]. In the metric formulation of gravity, non-linear curvature terms generally lead to higher derivative equations of motion that have new degrees of freedom that suffer from the Ostrogradski instability [6][7][8][9][10][11]. (It is possible that this classical instability does not persist in quantum theory [12].) In 4 spacetime dimensions, non-linear curvature terms always (boundary terms aside) lead to higher derivative equations of motion and new degrees of freedom, but not all of them are unstable. In 4 dimensions, the most general healthy action with no matter and no dependence on derivatives of the Riemann tensor depends on the curvature only via Ricci scalar. With arbitrary algebraic dependence on the Ricci scalar, the action has one extra non-ghost scalar degree of freedom [13]. When matter is added, there are further stable cases, such as the Horndeski and beyond Horndeski theories that include scalar fields [14].
In the Palatini formulation, the metric and the connection are independent degrees of freedom [15][16][17][18]. The equations of motion are therefore second order for any action that is algebraic in the Riemann tensor and where the connection appears only via the Riemann tensor. 1 For this reason higher order curvature terms do not necessarily lead to new degrees of freedom, and new degrees of freedom do not involve the Ostrogradski instability. However, the theory can still be unstable because kinetic terms or gradient terms can have the wrong sign, or potentials can be tachyonic or unbounded from below [28][29][30][31][32]. (Fields with a kinetic term of the wrong sign may not be a problem in the quantum theory [33,34].) As in the metric case, requiring the theory to be stable greatly restricts the action. However, unlike in the metric case, the most general stable theory is not known even when there is no matter. In addition to the Riemann curvature tensor, the geometrical properties of the manifold are described by the non-metricity tensor and the torsion tensor. There are hence many more gravitational terms than in the metric case, and terms that individually correspond to an unstable theory can lead to a stable theory when combined [28,31,32].
Many Palatini theories have been considered in the literature. It is known that a theory with arbitrary algebraic dependence on the symmetric part of the Ricci tensor does not contain any new degrees of freedom [35][36][37][38]. If the torsion is taken to be zero, a theory with arbitrary algebraic dependence on the antisymmetric part of the Ricci tensor is equivalent to normal gravity with an additional healthy vector degree of freedom [29,39,40]. The stability properties of some parity-invariant theories quadratic in the curvature, non-metricity, and torsion and are also known [31,[41][42][43].
In the Palatini formulation, the Einstein-Hilbert action is invariant under the projective transformation of the connection [17]. It has been found that also in some extended actions requiring projective invariance guarantees stability [29,30,44]. Stable Palatini theories without projective symmetry are known, so it is not a necessary condition for stability [29,30,45]. Furthermore, it is expected that in general projective symmetry is not enough to guarantee stability, see e.g. ch. 8 of [46].
We extend previous work by considering the stability of theories where the connection enters only via the first contractions of the Riemann tensor. There are three independent first contractions: the Ricci tensor, the co-Ricci tensor, and the homothetic curvature tensor. We collectively call them Ricci-type curvature tensors, or just Ricci-type tensors.
Hamiltonian analysis is the definitive way to establish the dynamical content and stability of a theory [47]. However, with these complicated theories the analysis can be quite involved. We take a simpler route. We perform a Legendre transformation and introduce auxiliary fields to make the theory linear in the Riemann tensor. Expanding around Minkowski space or the Friedmann-Lemaître-Robertson-Walker (FLRW) universe to first order, we solve for the connection and insert the solution back into the action. Having reduced the theory to metric gravity with the Einstein-Hilbert action plus minimally coupled matter, we then look at the kinetic sector of the new matter fields. This method has its limits. We are restricted to non-degenerate theories, meaning theories for which the Legendre transformation is invertible. We also cannot find the dynamical content of the theory in general, only around specific backgrounds. If the theory is found to contain unstable degrees of freedom, this is sufficient to rule it out, but the reverse does not hold. It is possible for a theory to be well-behaved around one background but unstable around another. A theory may also contain new degrees of freedom around one background but not another. This can indicate a strong coupling problem [48][49][50][51]. We find examples of both situations. We also find a projectively symmetric theory where perturbations around Minkowski space are unstable, showing that projective symmetry is not a sufficient condition for stability.
In section 2 we first introduce the relevant geometrical quantities and the action. We then make the Legendre transformation and introduce the auxiliary fields. We look at the degrees of freedom and stability in three cases: the general case when the connection is unconstrained, the zero torsion case, and the zero non-metricity case. We consider perturbations around the Minkowski background, and in the general case also around the spatially flat FLRW background, and find when kinetic terms have the wrong sign. We summarise our findings and outline open questions in section 3.
2 Degrees of freedom 2.1 Geometrical quantities and the action

Torsion and non-metricity
We work in the Palatini formulation, where the metric and the connection Γ γ αβ are independent degrees of freedom. The connection is related to the covariant derivative as We can decompose the connection as where g Γ γ αβ is the Levi-Civita connection of the metric g αβ , and L γ αβ is the related distortion tensor. In this decomposition we can define the Levi-Civita connection with respect to any metric, it does not have to be the physical spacetime metric. Both the Levi-Civita connection and the distortion tensor depend on the choice of the metric, while their sum does not.
The non-metricity tensor and the two non-metricity vectors defined with the metric g αβ are where L γαβ ≡ g γµ L µ αβ . The torsion tensor and the torsion vector are defined independently of the metric as The distortion can be written in terms of non-metricity and torsion as where the disformation and the contortion are, respectively, β , with indices lowered with g αβ and raised with its inverse. The tensors L α βγ , J α βγ , and K α βγ all depend on the choice of the metric, but to avoid overburdening the notation we do not label them with g , as it is usually clear from the context which metric is used. Transformations of the metric shift pieces of the connection between g Γ γ αβ , J α βγ , and K α βγ .

The Ricci-type tensors
The Riemann tensor depends only on the connection, With the connection decomposition (2.1), we can write (2.6) in terms of Levi-Civita and distortion contributions as where g denotes a quantity defined with the Levi-Civita connection g Γ γ αβ . For a general connection the only symmetry of the Riemann tensor is antisymmetry in the last two indices. There are thus three independent first contractions of the Riemann tensor: the Ricci tensor, the co-Ricci tensor, and the homothetic curvature tensor, respectively defined as We collectively refer to these three quantities (and their linear combinations) as Ricci-type curvature tensors, or just Ricci-type tensors. The homothetic curvature tensorR α β depends on the metric, while the other two Ricci-type tensors depend only on the connection. The full contraction of the Riemann tensor is unique, since R ≡ g αβ R αβ = −R α α and g αβ ∼ R αβ = 0. Inserting the decomposition of the connection in terms of non-metricity and torsion (2.1)-(2.5) into the decomposition (2.7) of the Riemann tensor, we can write the co-Ricci tensor asR (2.9) Instead of the co-Ricci tensor, it will turn out to be more convenient to use the average of the co-Ricci tensor and the Ricci tensor, (2.10) Note that g αβ ∆ R αβ = 0. Similar decomposition for the homothetic curvature tensor gives So the homothetic curvature tensor is the exterior derivative of the non-metricity vector. This decomposition obscures the fact that ∼ R αβ is independent of the metric, as is transparent from the definition (2.8). We can choose to do the decomposition (2.11) with respect to any metric. So a sufficient (but not necessary) condition for ∼ R αβ to vanish is that the manifold is such that there exists any symmetric tensor q αβ that is non-degenerate (det q αβ ̸ = 0) and covariantly constant (∇ γ q αβ = 0).

Projective transformation
An important property of the Einstein-Hilbert action in the Palatini formulation is invariance under the projective transformation [17] where V α is an arbitrary vector. Under this transformation non-metricity and torsion change as Thus the non-metricity and torsion vectors change as (2.14) The traceless part of the non-metricity tensor and the part of the torsion tensor that does not contribute to T α are invariant. Therefore, if the theory is invariant under the projective transformation, we can completely exchange torsion for non-metricity if the torsion tensor can be written in terms of the torsion vector, Equivalently, if non-metricity is involved only via the non-metricity vectors g Q α and ĝ Q, and they are related by g Q α = 4 ĝ Q, it can be exchanged for torsion. 2 The Riemann tensor transforms as The symmetric parts of the Ricci-type tensors are invariant under the projective transformation. Also, the antisymmetric combination is projectively invariant; here α and β are any projectively invariant quantities.

Action and Legendre transformation
We consider the most general Palatini theory where the action depends algebraically on the Ricci-type tensors, and has no other dependence on the connection. We can take the action to depend on any set of independent combinations of the Ricci-type tensors. For convenience we choose R αβ , where g ≡ det g αβ , and Ψ and ∂Ψ collectively denote matter fields and their derivatives (which can be higher than first order), with arbitrary coupling to the Ricci-type tensors. In order to have all of the degrees of freedom of the Ricci-type tensors represented, we must have α ̸ = 0 in P R αβ defined in (2.17). Also, the terms in P R αβ proportional to β are already separately variables in the action. So without loss of generality we set α = 1, β = 0. We perform a Legendre transformation to make the action linear in the Ricci-type tensors [35,[53][54][55][56] The invertibility requirement imposes a constraint on the form of F . We need the same assumption also for the second partial derivatives of F with respect to ∆ αβ , Π αβ , and Θ αβ . We call theories where F satisfies these constraints non-degenerate. For such We now introduce the field redefinitions (we use units where the Planck mass is unity) where q ≡ 1/ det q αβ = det q αβ , with q αβ being the inverse of q αβ . (The inverse has to exist for the field redefinition (2.21) to be consistent.) In terms of these new fields, the action (2.19) reads where we have used the definitions (2.8) and (2.10) of the Ricci-type tensors to gather the factors multiplying the Riemann tensor into the term The field q αβ plays the role of the metric, and we have the symmetric field S αβ , and the antisymmetric fields B αβ , P αβ , and H αβ . The old fields g αβ , Σ αβ , ∆ αβ , Π αβ , and Θ αβ are solved algebraically in terms of the new fields q αβ , B αβ , S αβ , P αβ , H αβ , and the matter fields Ψ (and their derivatives) from the field redefinitions (2.21)-(2.23).
The action is linear in the Riemann tensor, which is the only place where the connection appears. So the action is quadratic in the connection. Variation with respect to the connection hence gives a linear equation of motion, which we can solve, and insert the solution back into the action. The kinetic terms for the remaining fields then indicate the number of degrees of freedom and whether they are ghosts or healthy. We look at the degrees of freedom in three cases: when the connection is unconstrained, when torsion is assumed to vanish, and when non-metricity is assumed to vanish.

Equation of motion
We start with the case when no constraints are imposed on the connection. Projective transformation of the new Legendre transformed action then plays an important role. Projective invariance of the new action (2.24) is distinct from possible projective invariance of the original action (2.18). Whether or not the original formulation of the theory is projectively invariant depends on which combinations of Ricci-type tensors the action contains. In the new action, part of the physics related to the connection in the original action has been transferred to the auxiliary fields, which are taken to be projectively invariant. Under a projective transformation, the Riemann tensor transforms as (2.15), so the new action (2.24) changes as where on the last line we have switched ∂ α V β to q ∇ α V β using the fact that the contribution of the Levi-Civita connection vanishes due to the antisymmetry f α βγδ = f α β[γδ] , done a partial integration, and dropped a boundary term. The Legendre transformed action is thus The field S αβ does not appear in the constraint, corresponding to the fact that only the antisymmetric parts of the Ricci-type tensors change under the projective transformation (2.16).
Variation of the new action (2.24) with respect to the connection gives where we have used the definition (2.6) of the Riemann tensor, applied the identity ∇ α √ −q = 1 2 √ −qQ α , and done a partial integration. Putting the variation to zero gives the connection equation of motion The δ α β trace of this equation gives the constraint (2.27). So the equation of motion imposes a constraint on the matter fields that makes the action projectively invariant. Correspondingly, the projective symmetry leaves undetermined one linear combination of the traces of the distortion L α βγ , corresponding to a linear combination of q Q α , q Q α , and T α . As equation (2.29) is linear, it is possible to solve L α βγ exactly. However, the general solution is rather messy. We consider perturbations around a background, and solve the equation of motion to first order.

Minkowski space
We expand the original metric as g αβ = η αβ + δg αβ , where η αβ is the Minkowski metric and δg αβ is a perturbation. The new metric is similarly expanded as q αβ = η αβ + δq αβ . The connection Γ γ αβ and the fields B αβ , S αβ , P αβ , and H αβ are first order small. To first order in perturbations, f α βγδ defined in (2.25) reads To first order, the connection equation of motion (2.29) simplifies to where indices are raised and lowered with the Minkowski metric. Solving (2.32) and choosing the torsion vector T α as the part of the connection left undetermined due to projective invariance, we get with the constraint Inserting the solution (2.33) into the action (2.24), expanding to second order, and using the constraint (2.34), we obtain the action Matter fields end up being minimally coupled because we use ∆ R αβ = 1 2 (R αβ + R αβ ) rather thanR αβ as a variable. Note that the undetermined vector T α does not appear in the action. Plugging in (2.31) and using the constraint (2.34) to eliminate ∂ α H αβ , we get The kinetic terms of B αβ (corresponding to R [αβ] ) give an unstable theory [57]. As there is no kinetic term for H αβ , which corresponds to ∼ R αβ , it makes no difference for this instability whether the action depends on R [αβ] or the projectively invariant combination 4R We see that the kinetic terms corresponding to different Ricci-type tensors can cancel each other, so it is not possible to conclude that one of them cannot be present in a stable theory without considering the others. As we do not get kinetic terms for S αβ , P αβ , and H αβ , we cannot draw conclusions about the stability of an action that contains the corresponding tensors ∆ R (αβ) , P R αβ , and ∼ R αβ . However, we next show that for ∆ R (αβ) this feature is peculiar to Minkowski space, as around a cosmological background it leads to news degrees of freedom, which may imply a strong coupling problem [48][49][50][51].

FLRW universe
We look at the kinetic terms of S αβ , P αβ , and H αβ (corresponding to ∆ R αβ , P R αβ , and ∼ R αβ , respectively) around a cosmological background. It is cumbersome to expand the action around a FLRW background in the case when we keep dependence on all Ricci-type tensors. We therefore consider a simplified case, where only g αβ R αβ , ∆ R (αβ) , P R αβ , and ∼ R αβ appear, With this simplification the metrics g αβ and q αβ are conformal to each other, which makes expansion around a FLRW background more tractable. Were we to allow F to depend on R (αβ) , the metrics would be related by a disformal transformation [35][36][37]. Performing the Legendre transformation and introducing field redefinitions as before, the auxiliary fields are defined in (2.22), (2.23), and where Σ is the auxiliary field corresponding to g αβ R αβ . Now the action becomes where on the second line we have taken into account that g αβ is conformal to q αβ . Because of this, the gravity sector of the action, where the kinetic terms can arise, does not explicitly depend on g αβ . We now expand the action (2.39) around a spatially flat FLRW background. The fields are split into background plus perturbations as q αβ =q αβ + δq αβ ,q αβ = a(η) 2 η αβ where overbar denotes a background quantity, and η is conformal time. We decomposeS αβ asS where u α = (a −1 , 0, 0, 0). As the fields P αβ and H αβ are antisymmetric, they are zero on the background, so P αβ = δP αβ and H αβ = δH αβ . When we substitute (2.42) into the action (2.39), the action turns out not to depend on S 1 . 3 We decompose the background of the distortion tensor as where ϵ αβγδ is the Levi-Civita tensor. We substitute this into the action (2.39), vary the action with respect to the functions L i , and solve the resulting equations to find the solution L γ αβ =L γ αβ (a, S 2 ). The details are given in appendix A. Inserting the background solution back into the action (2.39), the Riemann tensor term becomes where ′ ≡ d/dη, H ≡ a ′ /a 2 , andR(q) is the Levi-Civita Ricci scalar of the background metric q αβ . The kinetic term has the wrong sign when |S 2 | < 2, so the value S 2 = 0 is unstable, and the field runs to S 2 = ±2. This is transparent if we write the action in terms of a scalar field with a canonical kinetic term, χ = 3|S 2 2 − 4|/8. Let us now consider perturbations. We have to solve the connection to first order from (2.29). We sketch here the main points of the calculation and summarise the results, with details given in appendix A. The equation of motion for the connection to first order in perturbations has the form δL σ µν M σ µν α βγ (a, S 2 ) + ρ α βγ (a, S 2 , δq µν , δS µν , δP µν , δH µν ) = 0 , where M σ µν α βγ depends only on the background quantities, while the source term ρ α βγ also depends on the perturbations and their derivatives. The field δH αβ satisfies the constraint q ∇ α δH αβ = 0 due to projective invariance, where q ∇ is the background covariant derivative. Solving the connection is straightforward but tedious. Substituting the solution back into the action again gives the kinetic terms. The kinetic part of the action expanded to second order has the form depend on the background quantities. The fields P αβ and H αβ do not get kinetic terms.
To determine whether or not there are ghosts we have to diagonalise the kinetic terms. To facilitate this, we decompose the fields δq αβ and δS αβ into scalar, vector, and tensor modes. To second order, these decouple from each other, and we can look at each of them individually. If any are unstable, the theory is unhealthy. The tensor modes are the simplest: we have δq ij = h ij , with h i i = 0, ∂ i h ij = 0, and δS ij = s ij , with s i i = 0, ∂ i s ij = 0. Applying this decomposition in (2.46), we get the tensor sector where the spatial indices are lowered withq ij and raised with its inverse. Diagonalising, we obtain The factor in front of the kinetic term of s ij has the same form as the background solution, and has the wrong sign when |S 2 | < 2. In section 2.2.2 we found that S αβ does not have a kinetic term around Minkowski space. So whether S αβ leads to new degrees of freedom depends on the background, and a theory that contains it is unstable around some FLRW backgrounds. As S αβ corresponds to the projectively invariant tensor ∆ R (αβ) , we again see that projective invariance does not guarantee stability.
As in Minkowski space, we do not obtain kinetic terms for P αβ and H αβ . It thus remains an open question whether the corresponding tensors P R αβ and ∼ R αβ lead to new degrees of freedom, and if so, whether they are healthy.

Zero torsion
Let us now consider the case when the connection is taken to be symmetric a priori, i.e. torsion vanishes, as often assumed in the Palatini formulation. This condition does not depend on the choice of the metric. Any projective transformation generates torsion (as (2.14) shows), so this assumption breaks projective invariance. 4 With zero torsion, ∼ R αβ = 2R [αβ] , so we can drop the dependence on ∼ R αβ , setting H αβ = 0. When varying the action, we have to take into account that the connection is symmetric. From (2.28) we see that this amounts to symmetrising f α βγδ , and thus the equation of motion (2.29), with respect to β and δ. To first order in the perturbations around Minkowski space, the equation of motion is Substituting the solution (2.50) into the action gives Finally, substituting δf α βγδ from (2.31) with H αβ = 0 into the action (2.51), performing an integration by parts, and diagonalising the P αβ and B αβ kinetic terms with the field redefinitionB αβ = B αβ + 2P αβ , we get the action In contrast to the case when there was no a priori constraint on the connection, dependence on any of the Ricci-type tensors now leads to new degrees of freedom around Minkowski space. Individually (i.e. ignoring cross terms) the kinetic terms of B αβ (corresponding to R [αβ] ) are healthy, while the kinetic terms of P αβ and S αβ (corresponding to P R αβ and ∆ R (αβ) , respectively) lead to ghosts [57]. In the case P αβ = 0 ghost modes remain after diagonalising the B αβ and S αβ kinetic terms. In the case P αβ = S αβ = 0, it is possible to solve the connection exactly, and the extra degrees of freedom in B αβ correspond to one vector field with a positive kinetic term [29]. The theory can still be unstable if the field is tachyonic, which depends on the form of F .

Zero non-metricity
Let us now consider the case when non-metricity is taken to vanish a priori. This assumption is common in loop quantum gravity, and a theory that satisfies it is called Einstein-Cartan theory. Again, projective invariance is broken, and the constraint (2.27) does not apply. Unlike the zero torsion condition, the assumption that non-metricity is zero depends on which metric we use: the original metric g αβ or the new metric q αβ .
Let us first consider the case g Q γαβ = 0. This is the most natural option, as the theory is originally defined with the metric g αβ . The corresponding disformation is zero, J α βγ = 0, so L α β γ = K α β γ , where these quantities are defined with the decomposition (2.1) in terms of g αβ . Writing the homothetic curvature tensor (2.8) in terms of the Riemann tensor decomposition (2.7) and using the property K [α β γ] = 0 shows that ∼ R αβ = 0. We thus have H αβ = 0, just as in the zero torsion case. Furthermore, (2.10) shows that ∆ R αβ = 0, so S αβ = 0, and we can also set P αβ = 0, as P R αβ is not independent. So only B αβ remains. What if we instead take the non-metricity defined with the new metric to be zero, q Q γαβ = 0? We can now decompose the homothetic curvature tensor in terms of the new metric q αβ and again find that ∼ R αβ = 0, as discussed after (2.11). So we still have H αβ = 0.
In contrast, now ∆ R αβ does not vanish in general. However, the only difference to the previous case is that the traceR αβ of the Riemann tensor in (2.8) is defined with q αβ instead of g αβ . As we work to first order, only the background metric enters here, and if the background metrics are conformal to each other, we have to first order ∆ R αβ = 0. In particular, this is true for perturbations around Minkowski space. So we can again set S αβ = P αβ = 0, and only B αβ remains.
Varying the action with respect to the connection and taking into account that the symmetric part is determined by the antisymmetric part when non-metricity is zero gives, according to (2.28), Setting the variation to zero and expanding the equation of motion to first order in perturbations around Minkowski space, we get with the solution Inserting this back into the action and expanding to second order gives Substituting δf α βγδ from (2.31) with S αβ , P αβ , and H αβ set to zero, we get no kinetic terms for the new field B αβ . If the action is quadratic in R [αβ] , the full nonlinear theory has a ghost [28,[58][59][60]. There is also a possible strong coupling problem due to the difference in dynamical degrees of freedom between the linearised case and the full theory.

Conclusions
We have investigated the stability of non-degenerate actions that depend algebraically on Ricci-type tensors, i.e. the first traces (2.8) of the Riemann tensor. We performed a Legendre transformation and introduced auxiliary fields to make the action linear in the Riemann tensor. We then solved perturbatively for the connection and inserted it back into the action to shift the non-standard gravitational dynamics to the matter sector. We considered three cases: the general case when no constraints are imposed on the connection, the case with zero torsion, and the case with zero non-metricity. Our results are summarised in table 1.
In the general case, dependence on R [αβ] or the projectively invariant combination  [48][49][50][51]. When we impose constraints on the connection, the results are quite different.
In the case with zero torsion, ∼ R αβ is not an independent tensor. All other Ricci-type tensors apart from R (αβ) now lead to new degrees of freedom. Due to couplings between the fields, the kinetic sector is complicated. Taken in isolation, R [αβ] leads to a healthy vector field, as is well known, and true independent of the background [29,39,40]. In isolation, the other Ricci-type tensors P R αβ and ∆ R (αβ) lead to ghosts. If we drop P R αβ , the conclusions about R [αβ] and ∆ R (αβ) hold. We did not diagonalise the kinetic sector when P R αβ is also present. In the case with zero non-metricity we have ∼ R αβ = 0. If non-metricity is defined with respect to the original metric g αβ , we also have ∆ R αβ = 0. If it is defined with respect to the new metric q αβ , this holds to linear order around Minkowski space, but not in general. The only remaining Ricci-type tensor is R αβ . The symmetric part R (αβ) does not lead to new degrees of freedom [35][36][37][38], while R [αβ] does not give new degrees of freedom around Minkowski space, but is known to have a ghost vector in the full theory [28,[58][59][60].
The only case with stable new degrees of freedom is when torsion is zero and the action depends only on the Ricci tensor, which was known already. As our focus is on whether a theory is healthy, we did not enumerate the number and type of all degrees of freedom, only whether some of them are unstable. We have in all cases required that the action is non-degenerate, i.e. that the Legendre transformation is invertible. There can be degenerate actions that depend on some of the tensors excluded above but nevertheless give a stable theory.
The results demonstrate that in order to determine whether a theory can depend on some Ricci-type tensor, it is necessary to consider them in combination, as there can be cancellations. It remains open whether ∼ R αβ ever leads to new degrees of freedom, and if so, whether they are healthy. From the fact that we do not get kinetic terms for it around Minkowski space or FLRW space we cannot conclude that this never happens. However, if there are new degrees of freedom around a different background, this may point to a strong coupling problem. Wrong sign kinetic terms from Ricci-type tensors can be cancelled by other terms, such as those that depend explicitly on the torsion [28]. Our results show that projective invariance is not a sufficient condition for a theory to be ghost-free. The most general healthy Palatini theory remains to be determined. L 5 = 0, corresponding to the fact that there is no parity-violating source. From the rest we solve for L 1 , L 3 , and L 4 , obtaining .
Substituting these back into (2.43), we get the non-zero components of the background distortion:L Substituting this solution into the background action gives (2.44).