Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order

We present all scalar-tensor Lagrangians that are cubic in second derivatives of a scalar field, and that are degenerate, hence avoiding Ostrogradsky instabilities. Thanks to the existence of constraints, they propagate no more than three degrees of freedom, despite having higher order equations of motion. We also determine the viable combinations of previously identified quadratic degenerate Lagrangians and the newly established cubic ones. Finally, we study whether the new theories are connected to known scalar-tensor theories such as Horndeski and beyond Horndeski, through conformal or disformal transformations.


Introduction
General Relativity (GR) is the unique consistent classical theory for a massless, selfinteracting spin two field in four dimensional spacetime [1]. It describes accurately gravitational phenomena spanning a large interval of scales, from short distances probed by table top experiments, to large distances probed by astronomy and astrophysics [2]. By including JHEP12(2016)100 a positive cosmological constant term to the Einstein-Hilbert action, GR can also describe the current acceleration of the universe, but only if one is willing to accept the enormous fine tuning that observations require on the value of the cosmological constant [3]. Attempts to avoid such fine tuning motivate the study of gravitational theories more general than GR, the simplest option being scalar-tensor theories of gravity (see e.g. [4] for a review). Theories that involve derivative scalar interactions, in the family of Galileons [5], are characterised by interesting screening effects, as for example the Vainsthein mechanism [6], which are able to reduce the strength of the scalar fifth force to a value compatible with present constraints on deviations from GR. Intriguingly, although the subject has been studied for many decades by now, we still do not know the structure of the most general consistent scalar-tensor theory, i.e. a theory describing a scalar interacting with a spin-2 tensor field in four dimensions. Horndenski [7] analysed the most general actions for scalar-tensor theories which lead to second order equations of motion (EOMs), and avoid Ostrogradsky instabilities [8,9]. In four dimensional spacetime, this condition allows one to consider actions which contain at most three powers of second derivatives of the scalar field. However, as realised only recently, there also exist viable theories "beyond Horndeski" [10][11][12], which do not suffer from the Ostrogradsky instability even though the corresponding Euler-Lagrange equations are higher order. Such theories have interesting consequences for cosmology and astrophysics. In particular, they lead to a breaking of the Vainshtein mechanism inside matter, which can modify the structure of nonrelativistic stars [13][14][15][16][17][18], as well as that of relativistic ones [19].
The aim of the present paper is to determine the maximal generalization of Horndenski theories in four dimensions, by which we mean all scalar-tensor theories that contain at most three powers of second derivatives of the scalar field, and that propagate at most three degrees of freedom.
As demonstrated in [20], a systematic way to identify scalar-tensor theories that contain at most three degrees of freedom, i.e. without Ostrogradsky ghost, is to consider Lagrangians that are degenerate, i.e. whose Hessian matrix -obtained by taking the second derivatives of the Lagrangian with respect to velocities -is degenerate. For scalar-tensor theories, such a degeneracy can depend on the specific coupling between the metric and the scalar field. From the Hamiltonian point of view, the degeneracy of the Lagrangian translates into the existence of constraints on phase space, in addition to the usual Hamiltonian and momentum constraints due to diffeomorphism invariance, and explains why one degree of freedom is eliminated, even if the equations of motion are higher order. A detailed Hamiltonian analysis confirms the direct link between this degeneracy and the elimination of the Ostrogradsky ghost [21]. For Lagrangians depending on the accelerations of several variables, the degeneracy of the Lagrangian is not sufficient to eliminate the multiple Ostrogradsky ghosts and extra conditions must be imposed, as shown in [22] for classical mechanics systems (see also [23] for a slightly different approach, reaching the same conclusion). The singularity of the Hessian matrix (this time obtained by taking the second derivatives of the Lagrangian with respect to the lapse and shift) finds application also in other contexts like massive gravity: indeed, it is this condition that provides the tertiary constraint necessary to remove the Boulware-Deser ghost mode [24].

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The degeneracy criterium, which provides a powerful and simple method to identify viable theories, was used in [20] to identify all scalar tensor theories whose Lagrangian depends quadratically on second order derivatives of a scalar field. Degenerate higher derivative Lagrangians, later dubbed EST (Extended Scalar Tensor) in [26], or DHOST (Degenerate Higher Order Scalar Tensor) in [27], include Horndeski theories as well as their extensions "beyond Horndeski". As stressed in [20] and [25], only specific combinations of Horndeski theories and of their extensions beyond Horndeski are (Ostrogradsky) ghostfree. Quadratic degenerate theories are further studied in [26][27][28], in particular how they change under disformal transformations of the metric.
In the present work, we extend the systematic classification of degenerate theories to include Lagrangians that possess a cubic dependence on second order derivatives, so to find the most general extension of Horndenski scalar-tensor theory of gravity. We also allow for non-minimal couplings with gravity and show that the only viable Lagrangian, among all possible ones involving the Riemann tensor contracted with the second derivative of the scalar field, is of the form G µν ∇ µ ∇ ν φ. The class of theories we consider thus encompasses Horndeski Lagrangians and our analysis confirms that all Horndeski theories are degenerate, as expected. We also find new classes of cubic Lagrangians that are degenerate. In total, we identify seven classes of minimally coupled cubic theories, and two classes of nonminimally coupled cubic theories. We study in which cases it is possible to combine any of these cubic theories with the previously identified quadratic theories to obtain more general Lagrangians. We investigate which cubic theories admit a well-defined Minkowski limit, i.e. when the metric is frozen to its Minkowski value. We also study whether the new cubic theories are related to known Lagrangians through conformal or disformal transformations. Technical appendixes contain details of the calculations leading to the results we present in the main text.

Degenerate Scalar-Tensor theories
Scalar-Tensor theories involving second order derivatives of the scalar field in the action are generally plagued by an Ostrogradsky instability, unless the Lagrangian is degenerate, i.e. there is a primary constraint that leads to the removal of the additional undesired mode. 1 In order to study these theories, it is useful to recast the action into ordinary first order form via the introduction of a suitable auxiliary variable. This can be done by replacing all first order derivatives ∇ µ φ by the components of a vector field A µ , as first explained in [20], and by imposing the relation In this paper we do not perform a full Hamiltonian analysis (see refs. [21] and [29] for Hamiltonian formulations of beyond Horndeski theories); however, we expect that, in general, the primary constraint is second-class and leads to a secondary constraint that is also second-class, so that both constraints remove one degree of freedom, as shown explicitly in [21] for the quadratic case. Note that the primary constraint can also be first-class in some very particular cases.

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using a Lagrangian multiplier. Therefore, after introducing the general action we investigate, we will focus on its kinetic structure by identifying the time derivatives of the fields contained in ∇ µ A ν .

Action
In this paper we consider the most general action involving quadratic and cubic powers of the second derivative of the scalar field: where the functions f 2 and f 3 depend only on φ and X ≡ ∇ µ φ∇ µ φ (we use a mostly plus convention for the spacetime metric). The tensors C (2) and C (3) are the most general tensors constructed with the metric g µν and the first derivative of the scalar field φ µ ≡ ∇ µ φ.
As we will see in detail in the next subsection, when written in terms of the auxiliary variable A µ , each second derivative of φ yields terms linear in velocities. By contrast, the curvature depends quadratically on the velocities of the metric and one can introduce terms non-minimally coupled to gravity, such as f 2 R and f 3 G µν φ µν , leading to second or third powers in velocities respectively. A priori, one could also envisage many more terms of this kind involving the Riemann tensor contracted in various ways. However, as shown in appendix A, the only viable Lagrangians among all the possible ones with appropriate powers in velocities, turn out to be these two (up to integrations by parts).
Note that one could also include in our general action (2.2) terms of the form P (X, φ) or terms depending linearly on φ µν . We have not included such terms explicitly because they do not modify the degeneracy conditions, but one should keep in mind that they can always be added to the Lagrangians that will be identified in our analysis.
Due to the way the tensors C (2) and C (3) are contracted in the action, one can always impose, without loss of generality, the symmetry relations: (2) and C µνρσαβ . (2.3) As a consequence, they can be expressed as C µνρσ where the functions a's and b's depend only on φ and X. The notation . . . means that these expressions are symmetrised so to satisfy eq (2.3). Explicitly, we have and L Introducing the auxiliary variable A µ as in (2.1), the general action (2.2) becomes where the tensors C µνρσ (2) and C µνρσαβ (3) are now expressed in terms of A µ and φ. Clearly, the two Lagrangians (2.2) and (2.9) are equivalent.
Although we do not perform explicitly a Hamiltonian analysis here, 2 let us briefly comment about the role of the Lagrangian multipliers λ µ and the relations they enforce. Since the action (2.9) does not involve the velocities of λ µ , the corresponding conjugate momenta p µ appear in the total Hamiltonian H T as primary constraints that weakly vanish. The evolution of p i gives the secondary constraints φ i −A i ≈ 0. By contrast, the evolution of p 0 allows one to solve for the multiplier used in H T to impose the other primary constraint π − λ 0 ≈ 0, where π is the momentum of φ. The evolution of π − λ 0 , on the other hand, fixes the multiplier associated with p 0 . All these constraints are second class and therefore can be consistently imposed in the Hamiltonian analysis; in particular the constraints φ i − A i ≈ 0 enables us to eliminate the velocity of A i in favour of the spatial derivative of A 0 , as explained in detail in the next section. It is thus clear that the constraints that follow from the λ µ in (2.9) do not get mixed up with the (potential) extra primary constraint necessary to eliminate the Ostrogradsky mode, which characterises degenerate theories.

Covariant ADM decomposition
In order to study the kinetic structure of the action (2.9), we must perform a 3 + 1 decomposition of its building blocks. We now assume the existence of an arbitrary slicing of spacetime with 3-dimensional spacelike hypersurfaces. We introduce the unit vector n µ normal to the spacelike hypersurfaces, which is time-like and satisfies the normalization condition n µ n µ = −1. This induces a three-dimensional metric, corresponding to the projection tensor on the spatial hypersurfaces, defined by h µν ≡ g µν + n µ n ν . (2.10)

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Following the construction of [20], we define the spatial and normal projection of A µ , respectivelyÂ Let us now introduce the time direction vector t µ = ∂/∂t associated with a time coordinate t that labels the slicing of spacelike hypersurfaces. One can always decompose t µ as thus defining the lapse function N and the shift vector N µ orthogonal to n µ . We also define the "time derivative" of any spatial tensor as the spatial projection of its Lie derivative with respect to t µ . In particular, we havė where V * ≡ n µ ∇ µ A * , K µν is the extrinsic curvature tensor and D µ denotes the 3-dimensional covariant derivative associated with the spatial metric h µν . Instead of using the velocitiesḣ µν andȦ * , it is convenient to work with the covariant objects K µν and V * and interpret them as "covariant velocities" associated with the fields h µν and A * . Working with these covariant quantities allows us to avoid dealing with the lapse and the shift vector.
Using these definitions, as well as the property ∇ µ A ν = ∇ ν A µ , the 3+1 covariant decomposition of ∇ µ A ν is given by where a µ ≡ n ν ∇ ν n µ is the acceleration vector. One can rewrite (2.15) as These two tensors fully characterise the velocity structure of the building block ∇ µ A ν that appears in the action (2.9) and will play an essential role in deriving the degeneracy conditions.

Horndeski Lagrangians and kinetic structure of the action
As an example of theories of the type (2.2), and as a useful step for the general case, let us first consider the particular case of the so-called quartic and quintic Horndeski Lagrangians 3 which correspond, respectively, to a quadratic and a cubic Lagrangian in our terminology. Indeed, they are of the form (2.2), with and It is instructive to extract the kinetic part of these two Lagrangians as the result will be useful for the general case. The kinetic structure of the original Lagrangians (2.18) and (2.19) is the same as the following ones where A 2 ≡ A µ A µ ,Â 2 ≡Â µÂ µ and we have introduced the projection tensor (orthogonal to the directions n µ andÂ µ )P Notice that the tensors (2.23) and (2.24) are orthogonal to the vector n µ , therefore the kinetic terms do not contain the velocity V * . This is the peculiarity of Horndeski Lagrangians which reflects in second order equations of motions.
Let us now turn to the general action (2.2). In order to extract its kinetic part, it is convenient to re-express the curvature terms in the action as Horndeski Lagrangians so that one can use the results above. The action (2.2) is thus rewritten as

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where the tensorsC µνρσ (2) andC µνρσαβ (3) are of the form (2.4)-(2.5) with the new functions while all the other functions remain unchanged. Replacing the Lagrangians L H 4 and L H 5 in (2.26) with the kinetically equivalent ones (2.22), one finds that the kinetic structure of the total action is described by the tensors C µνρσ and C µνρσαβ . (2.29) Only these tensors are relevant for the degeneracy conditions, which we derive below.

Degeneracy conditions and primary constraints
We now introduce the Hessian matrix of the Lagrangian with respect to the velocities V * and K ij . This matrix can be written in the form (introducing a factor 1/2 for convenience) The degeneracy of the theory is associated with the degeneracy of its Hessian matrix, i.e. detH = 0. Equivalently, one can find a non trivial null eigenvector These conditions translate into the existence of a primary constraint, which takes the form v 0 π * + V ij π ij + · · · ≈ 0 , (2.33) where we have introduced the "covariant momenta" conjugated respectively to A * and h µν , and the dots indicate momentum-independent terms, involving only the fields and their spatial derivatives. Note that we will always assume v 0 = 0 since we are interested in removing the Ostrogradsky mode: therefore in the following we will fix v 0 = 1 without loss of generality.
It is important to keep in mind that the primary constraint (2.33) is a scalar constraint involving only the scalar components of π ij , i.e. V ij π ij . It is indeed responsible for removing the scalar Ostrogradsky mode. However, there could still be extra primary constraints in the vector sector of π ij , which can further reduce the number of degrees of freedom (dof) (as JHEP12 (2016)100 pointed out in [27] and further stressed in [28]). Indeed, as we will show in what follows, some classes of theories that possess the constraint (2.33), also enjoy the two following primary constraints:Â iPjk π ij + · · · ≈ 0 , (2.35) where we have used the projector (2.25). These constraints remove the two helicity-2 dof present in the metric sector, leaving the theory with only one dof. In order to compute the Hessian matrix of (2.26), one needs to keep all terms quadratic and cubic in the velocities. The Hessian matrix decomposes into its quadratic and cubic contributions denoted , , Introducing the tensor On the other hand, the cubic Hessian matrix contains velocities. Therefore, the degeneracy conditions must be satisfied for arbitrary values of φ αβ . This implies that φ αβ can be "factorised" and the conditions (2.32) in the cubic case are analogous to the quadratic ones, namely The above equations mean that, in order to get a degenerate Lagrangian, the projections of the tensors C µνρσ (2) L ρσ or C µνρσαβ L ρσ , respectively via λ µν and Λ ij µν , must vanish. As shown in appendix B, this implies that these tensors are necessarily of the form where σ, σ 1 and σ 2 are arbitrary scalar quantities. By solving the conditions (2.42), one recovers the quadratic theories identified in [20]; they are summarised in appendix C. Conditions (2.43) are solved in detail in appendix D and in the next section we report the various classes of purely cubic theories. Then, we will consider the possibility to merge quadratic and cubic theories. In this case the additional condition to impose is that L µν is the same in (2.42) and (2.43), i.e. we have to use the same V ij .

JHEP12(2016)100 3 Classification of cubic theories
The degeneracy conditions for quadratic theories (i.e. with f 3 = b i = 0) have already been solved and the corresponding theories identified in [20]. These quadratic theories were then examined in more details in [26][27][28]. In this section we thus focus our attention on the purely cubic theories, i.e. characterized by f 2 = a i = 0. Solving the degeneracy conditions here is much more involved than in the quadratic case and rewriting them in the tensorial form (2.41) is instrumental to obtain the full classification. Below, we simply present the full classification, indicating for each class the free functions among the b i and the constraints satisfied by the other functions. All the cubic theories we identify are summarised at the end of the section in table 1. The details of how we have identified these classes are given in appendix D, where the reader can also find the explicit expression of the null eigenvectors associated with the degeneracy. The latter are indispensable to identify the healthy combinations of quadratic and cubic Lagrangians, which will be given in the next section.

Minimally coupled theories
We start with the minimally coupled case, corresponding to f 3 = 0. There are seven different classes of theories.
This class includes the pure quintic beyond Horndeski Lagrangian: which corresponds to the choice of functions

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The above combination is special as it leaves the Lagrangian linear in V * , therefore in (2.38) A 3 = 0.
All the other functions are determined in terms of b 1 as follows: ◮ 3 M-V. Two free functions, b 1 and b 4 , while the other functions are given by There is only one (scalar) dof that propagates due to the primary constraints (2.35), and their associated secondary constraints.
All the other functions vanish: Again, there is only one (scalar) dof that propagates. (3.6)

Non-minimally coupled theories
We now consider the purely cubic Lagrangians with f 3 = 0. There are two classes of theories.
In addition to f 3 , the functions b 1 and b 4 are free (with the only restriction b 1 = 0). The other functions are determined as follows: Quintic Horndeski (2.19), as well as the combination of quintic Horndeski plus quintic beyond Horndeski (3.2), is included in this class of models.

Minkowski limit
Here we discuss which ones among the classes of theories described above admit a healthy Minkowski limit, i.e. the limit where the metric is given by g µν = η µν and the metric fluctuations are ignored. In this limit, only the scalar sector is dynamical and the Hessian matrix reduces to its purely scalar component, i.e. A. For cubic theories, the degeneracy is thus expressed by the condition A (3) = 0, which imposes the relations The only classes that satisfy these conditions are This shows that there is a new theory, 3 N-II, which propagates three degrees of freedom on curved spacetime and has a healthy Minkowski limit. On the other hand, theories that do not satisfy (3.8) could still have a healthy decoupling limit around a non-trivial background.

Merging quadratic with cubic theories
In this section we wish to determine all the theories of the form (2.2), i.e. quadratic plus cubic Lagrangians, that are degenerate. Adding two degenerate Lagrangians does not always yield a degenerate one. This is the case only if the null eigenvectors associated with the two Lagrangians coincide. Therefore, in order to see whether the combination of two Lagrangians is viable, one needs to compare their eigenvectors, which are all listed in appendix C for quadratic theories and in appendix D for cubic ones, and check when they are equal. We present four tables describing all the different possibilities for merging quadratic and cubic theories. We indicate with theories that can be freely combined, with X theories that cannot be combined, and with (n) theories that can be combined imposing the additional condition(s) (n) listed below each table.
Minimally coupled quadratic plus minimally coupled cubic theories.
The quartic beyond Horndeski theory L bH 4 is included in 2 M-I, while the quintic beyond Horndeski theory L bH 5 (3.2) is included 3 M-I. They satisfy the condition (1) thus the combination L bH 4 + L bH 5 is still viable [20,25].
Non-minimally coupled quadratic plus minimally coupled cubic theories.
The quartic Horndeski theory L H 4 (2.18) is included in 2 N-I. The combination L H 4 +L bH 5 does not satisfy the conditions (1) and (3), thus this combination is not degenerate [20,25].
The quintic Horndeski theory L H 5 (2.19) is included in 3 N-I. As can be seen from the table, it is not possible to combine 3 N-I and 2 M-I thus the combination L H 5 + L bH 4 is not viable [25].
Non-minimally coupled quadratic plus non-minimally coupled cubic theories.
The classes 2 N-I and 3 N-I contain three free functions each, thus the combination 2 N-I + 3 N-I contains four free functions due to the conditions (1). In the next section, we show that this theory can be obtained by the generalised conformal and disformal transformation from L H 4 + L H 5 .

Conformal and disformal transformation
We now investigate which ones among the cubic theories can be obtained from known Lagrangians through conformal and disformal transformations. The same analysis for quadratic theories can be found in [26,27]. First we identify the class of theories minimally coupled with gravity (i.e. f 3 = 0) that can be obtained from beyond Horndeski (3.2) by a conformal transformation. Then, we study the class of theories that can be obtained from Horndeski theory (2.19) by a conformal together with a disformal transformation.

Conformal transformation on Beyond Horndeski
It was shown in [25] that under the generalised disformal transformation  5) and the otherb i vanish. In terms of the total b i , this gives These b's satisfy conditions (3.1), thus this theory is included in class 3 M-I.

Conformal and disformal transformation on Horndeski
The generalised conformal and disformal transformation transforms the Horndenski action as  16) .  [26,27], we can show that this theory corresponds to the combination of 2 N-I and 3 N-I and satisfies the condition (1). This theory has four free functions, which correspond tof 2 ,f 3 , Ω and Γ. Thus this theory can be regarded as the "Jordan frame" version of the Horndenski theory where the gravitational part of the Lagrangian is described by Hordenski with the metricḡ µν , , while the matter is non-minimally coupled through g µν . By performing the generalised conformal and disformal transformation, the gravitational action is described by the combination of 2 N-I and 3 N-I and the metric is minimally coupled to matter.

Conclusions
In this paper, we presented all Ostrogradsky ghost-free theories that are at most cubic in the second derivative of the scalar field, and that propagate at most three degrees of freedom. Extending Horndeski's results, we have found new Lagrangians, which lead to higher order equations of motion but avoid Ostrogradsky instabilities by means of constraints that prevent the propagation of dangerous extra degrees of freedom.
In order to achieve our results, we used the degeneracy criterium introduced in [20], and classified the Lagrangians that are degenerate, i.e. whose Hessian matrix, obtained by taking the second derivatives of the Lagrangian with respect to velocities, is degenerate. In total, we identified seven classes of minimally coupled cubic theories and two classes of non-minimally coupled cubic theories, which contain as subclasses all known scalar-tensor theories which are cubic in second derivatives of the scalar field. We also investigated which cubic theories admit a well-defined Minkowski limit, i.e. when the metric is frozen to its Minkowski value. Our results are summarised in the table 1.
We then studied in which cases it is possible to combine any of these cubic theories with the previously identified quadratic ones. Note that one can also add arbitrary terms of the form P (X, φ) and Q(X, φ) φ without changing the degeneracy of the total Lagrangian. We confirmed the previous finding that the combination of quartic or quintic JHEP12(2016)100 beyond Horndeski with a different Horndeski is not viable. Finally, we studied whether our cubic theories are related to known Lagrangians through generalised conformal or disformal transformations. We identified the theory, with four free functions, that is obtained by the generalised conformal and disformal transformation from the combination of quartic and quintic Horndeski Lagrangians.
Various interesting developments are left for the future. First, phenomenological aspects of these new theories should be investigated, in particular studying the existence of stable cosmological FLRW solutions -possibly self-accelerating -and their properties, by using for instance the effective description of dark energy (see e.g. [30] for a review and [31] for a recent generalization that includes non-minimal couplings to matter). It would also be worth analysing possible distinctive features of screening mechanisms in these set-ups. Secondly, on the theory side, it would be interesting to analyse further generalizations of scalar-tensor theories containing higher powers of second derivatives of the scalar field. Such theories do not admit a well-defined Minkowski limit, and some explicit examples have been discussed in [26] and in [32]. A more complete classification using the techniques we presented should be feasible, and left for future investigations.

A Curvature dependent Lagrangians
Curvature tensors depend quadratically on the extrinsic curvature so, according to the kinetic structure presented in section 2.2, their combination with the second derivative of the scalar field yields cubic powers in velocities. All the possible quadratic and cubic terms in velocities involving the curvature are where and that we studied in the main text.
In this appendix we concentrate separately on the four remaining cubic non-minimally coupled Lagrangians (A.4)-(A.5). What characterises these Lagrangians in comparison with (A.7) is that they all feature time (and space) derivatives of the extrinsic curvature. This indicates the possible presence of additional Ostrogradsky modes, this time coming from the metric sector of the theory, unless there are suited extra primary constraints that remove them.
The covariant 3+1 decomposition of (A.4)-(A.5) shows that the only components of the extrinsic curvature that acquire time derivatives are the scalar ones: Their covariant velocities appear in the form in analogy to what we encountered in section 2.2. Therefore, applying the same kind of field redefinition used for the scalar field (2.1), Lagrangians (A.4)-(A.5) generally propagate two more Ostrogradsky modes, E and F , in addition to A * . To avoid their propagation, we need two more primary constraints. Defining the conjugate momenta associated to the new fields for the set of Lagrangians (A.4)-(A.5) we obtain .12) and the dots in π * represent non relevant terms. From the form of the momenta (A.11), it is clear that a total of three primary constraints can only be obtained in the trivial way, i.e.
Hence α = β = 0 and, due to the Lorentz invariance of f i , relations (A. 12) give Indeed, decomposing (B.1) in the directions n α n β , h i α h j β and n α h i β leads to the equations (2.40). As a consequence, C ρσ (2) (L) is necessarily in the kernel of M viewed as an operator acting on symmetric 4 dimensional matrices. A matrix V µν is in the kernel of M when Furthermore, the only available vector V µ in the theory which is orthogonal to n µ is in the directionÂ µ . Hence, there exists a scalar σ such that which is the relation (2.42). The generalization to cubic theories is rather immediate. Let us show that (2.43) and (2.41) are equivalent. Following the same strategy as previously, we first show that (2.41) is equivalent to Hence, due to symmetries, C (3) (L) can we written as where m γδ is a symmetric matrix. Due to the covariance, the symmetric matrix m is necessarily of the form m γδ = 4σ 1 g γδ + 4σ 2 A γ A δ where σ 1 and σ 2 are scalars. Notice that there is no components of the form A (γ n δ) nor of the form n γ n δ in m γδ because V µν γ n γ = 0. As a conclusion, (B.6) is true if and only if there exist scalars σ 1 and σ 2 such that:
C.1 Minimally coupled theories ◮ 2 M-I. Three free functions a 1 , a 2 , and a 3 , together with We assume a 2 = −a 1 /3. This case includes beyond Horndeski theory. The corresponding null eigenvector is given by .
◮ 2 M-II. Three free functions a 1 , a 4 , a 5 and The corresponding null eigenvector is given by .
◮ 2 M-III. Four free functions a 2 , a 3 , a 4 , a 5 and the unique condition The eigenvector is given by This class was called M-III in [26] and IIIc in [27].
For minimally coupled quadratic theories, the vector components of π ij (i.e.Â iPjk π ij ) are proportional to a 1 , therefore, as noticed in [28], this class propagates only one scalar dof.

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C.2 Non-minimally coupled theories ◮ 2 N-I. Three free functions f 2 , a 1 and a 3 . The conditions are The combination of Horndeski and beyond Horndeski theories is included in this class. The corresponding null eigenvector is given by This class was called N-I in [26] and Ia in [27].
Three free functions f 2 , a 4 , a 5 and The corresponding null eigenvector is given by (C.14) This class was called N-II in [26] and Ib in [27].
For non-minimally coupled quadratic theories, the vector components of π ij are instead proportional to f 2 − Xa 1 , hence also here there are not tensor modes [28].
Three free functions f 2 , a 1 and a 2 . The conditions are a 1 + a 2 = 0 , and
Three free functions f 2 , a 2 and a 3 . The conditions are a 1 + a 2 = 0 (C.21) The corresponding null eigenvector is given by This class was called N-III (ii) in [26] and IIb in [27].
Due to the condition f 2 − Xa 1 = 0, only one scalar dof is present in this class [28].

D Identifying cubic theories
In this appendix we solve in details the conditions (2.43) for purely cubic theories. Let us first note that, since V ij lies in the hyper-surface orthogonal to n µ , it can be decomposed as follows where v 1 and v 2 are scalar quantities. The tensor L µν , introduced in (2.39), can thus be written as
◮ ℓ 1 = 0 and T = 0 In that case v 2 is related to v 1 by .

(D.18)
The matrix A is highly degenerate with rank=3 and Ab + = 0 produces 3 conditions. Two of them give 19) and the third one is (D.32) The components b 5 , b 7 , b 8 and b 10 are free.

D.2 Non-minimally coupled theories
The resolution follows the same strategy as in the minimally coupled case. First, we write the generalised conditions in a form analogous to (D.14) where Σ is a matrix given by (D.34) Hence, the solution forb = (b + ,b − ) is the sum of the general solution of the homogeneous equation (with f 3 = 0) and a particular solution. Again, we solve them according to whether ℓ 1 and T vanish or not. When ℓ 1 = 0 necessarily v 1 = v 2 = 0, which would imply in turn that ℓ 1 = 1. This is an inconsistency, hence there is no solution when ℓ 1 = 0.