Ostrogradsky mode in scalar-tensor theories with higher-order derivative couplings to matter

A metric transformation is a tool to find a new theory of gravity beyond general relativity. The gravity action is guaranteed to be free from a dangerous Ostrogradsky mode as long as the metric transformation is regular and invertible. Various degenerate higher-order scalar-tensor theories (DHOST) without extra degrees of freedom have been found through the metric transformation with a scalar field and its derivatives. In this work, we examine how a matter coupling changes the degeneracy for a theory generated from the Horndeski theory through the metric transformation with the second derivative of a scalar field, taking a minimally-coupled free scalar field as the matter field. When the transformation is invertible, this theory is equivalent to the Horndeski theory with a higher-order derivative coupling to the matter scalar field. Working in this Horndeski frame and the unitary gauge, we find that the degeneracy conditions are solvable and the matter metric must have a certain structure to remove the Ostrogradsky mode.


I. INTRODUCTION
Ostrogradsky has shown that Hamiltonian is bounded neither below nor above if the equation of motion of a system without the degeneracy is higher than second order [1,2]. This result implies that there would be instability in the system. A scalar-tensor theory is a natural extension of the theory of gravity, general relativity, where a scalar degree of freedom is present in addition to the degree of freedom of gravitational waves. The Ostrogradsky mode appears in this type of theory when the Lagrangian includes quadratic or even more terms of the second-order derivative of the scalar field unless some mechanisms to suppress the Ostrogradsky mode are introduced. Recent studies revealed the existence of scalar-tensor theories, dubbed Degenerate Higher-order Scalar-Tensor (DHOST) theories, which are free from the Ostrogradsky mode thanks to the degeneracy while their Lagrangian include quadratic or even more terms of the second-order derivative of the scalar field [3][4][5][6][7][8][9]. Equations of motion of these theories are of higher than second order in general, and then these theories are clearly beyond the Horndeski theory [10][11][12], whose equations of motion are up to second order. For a review of these recent developments of scalar-tensor theories, see Refs. [13,14]. More recently, scalar-tensor theories with more than second-order derivatives in Lagrangian are discussed, focusing on metric transformations with higher derivatives of the scalar field [15][16][17][18][19] (see also Refs. [20,21]).
A metric transformation is a tool to find a new healthy theory of gravity beyond general relativity free from the Ostrogradsky mode. The first example of theories beyond Horndeski was found in Ref. [3] via a metric transformation of the Einstein-Hilbert action. It was studied in Refs. [4,5,[7][8][9]22] how different DHOST theories are related via the conformal and disformal transformations [23]. For example, the so-called DHOST class I theory is shown to be equivalent to the Horndeski theory with a metric transformation. Further extension of the DHOST theories along this direction was made in Refs. [15][16][17][18][19] with a higher-derivative generalization of the disformal transformation. As long as the metric transformation is regular and invertible, the metric transformation is nothing but a rewriting of physical metric in a different manner and hence the physics is essentially the same [24][25][26][27][28] (see Ref. [29] for a recent discussion on exceptional cases). Therefore, the new theory is guaranteed to be healthy if the original theory is healthy. However, in the presence of matter, the situation completely changes. When the new metric is assumed to be minimally coupled to matter fields, the new theory is no longer equivalent to the original theory with minimally-coupled matter fields. Then, one may wonder whether the tamed Ostrogradsky mode due to the degeneracy may revive once the matter coupling is taken into account [30].
Provided the weak equivalence principle holds, there should be a metric universally coupled to matter fields. Then the natural question would be "to which metric the matter fields are minimally coupled?" One can consider the gravity theory described by a metric g µν and matter fields σ a minimally coupled to this metric, which we dub G(minimal): Another gravity theory L G [g µν ] can be generated through a metric transformation T : g µν →ḡ µν as (2) Let us first assume that the matter fields are coupled to the metricḡ µν . As a theory for the metric g µν , this new (modified) gravity theory has a non-minimal coupling to matter. We call this theory G(non-minimal): However, when the metricḡ µν is considered to be an independent variable, the G(non-minimal) theory simply reduces to G(minimal): One can also consider a theory where the gravity theory is given by L G [g µν ] while the matter fields are coupled to the metric g µν , which we call G(minimal): As a theory for the metric g µν , the new gravity theory has a minimal coupling to matter. When the metricḡ µν is considered to be an independent variable, this theory reduces to the original theory but the matter is now nonminimally coupled: In the last expression, g µν in the middle equation is considered as a function ofḡ µν through the inverse transformation: The G(minimal) theory is different from G(minimal) as well as G(non-minimal). The matter metric g µν contains higher derivatives when we consider the metric transformation with higher derivatives. In this case, the last expression in Eq. (6) implies that, in the presence of matter, the theory is non-degenerate in general and then so is the equivalent theory (5), while the gravity sector L G [g µν ] satisfies the degeneracy condition and the matter sector does not contain higher derivatives. Even if the gravity sector itself is well-behaved, once we take into account a matter sector, a careful analysis is needed since the Ostrogradsky mode (higher-derivative mode) can revive through the matter coupling [30].
In this paper, we investigate how the matter coupling changes the degeneracy of a theory generated from the Horndeski theory through the metric transformation T : g µν →ḡ µν with which contains the second derivatives of a scalar field φ. Here, we defined and φ µ ≡ ∇ µ φ and X µ ≡ ∇ µ X. This is the metric transformation studied in Ref. [19] where the invertibility conditions of the transformation are derived for the functions F a (a = 0, 1, 2, 3). For F 0 = F 0 (φ, X), F 1 = F 1 (φ, X), and F 2 = F 3 = 0, the transformation is known as a disformal transformation [23]. Taking a free scalar field as the matter sector, we will show that the degeneracy conditions are solvable in the unitary gauge φ = t and restrict the form of the transformation (7) as follows: where and The functions F U a (φ, X, W ) (a = φ, , ⊥) and V U (φ, X, Y, W ) can be chosen independently. They are related to the functions F a (φ, X, Y, Z) in Eq. (7) as The dependence on the higher-derivative terms Y and Z in the functions F a (a = 0, 1, 2, 3) is tightly restricted as the above relations to achieve the cancellation between the higher-derivative terms in the metric (9). Therefore, the matter coupling changes the degeneracy of the theory. The paper is organized as follows: In Section II, we define the models analyzed in this paper and their expression in the unitary gauge. In Section III, we show how the degeneracy conditions restrict the form of the metric (7). We then derive the restrictions when the invertibility conditions are further imposed in Section IV and give the covariant form of the resultant restricted metric in Section V. We finally summarize the results in Section VI. Some useful expressions for the computation are presented in two Appendices A and B. In Appendix C, we discuss how the structure of degeneracy changes in different frames for a simple example of a mechanical system.

A. Action
We consider the following action for the Horndeski theory with a non-minimal coupling to a matter scalar field σ, where the matter metricḡ µν is assumed to be a function of g µν and φ in the form (7), Here, L H and L m represent the Lagrangian densities of the Horndeski theory and matter, respectively. When the transformation T : g µν →ḡ µν is invertible, this theory is equivalent to any new theory of gravity generated from the Horndeski action through the metric transformation in the form (13) with a minimal coupling to the matter scalar field σ: provided the inverse transformation T −1 : g µν →ĝ µν , we find In the last expression,ĝ µν in the middle equation is considered as a function of g µν :ĝ µν = T −1 [g µν ]. Thus, the action (12) incorporates the DHOST class I theories and their generalizations proposed in Ref. [19]. We investigate whether there exists a matter metricḡ µν for which the new theory of gravity (12) is healthy, other than the trivial caseḡ µν = g µν . For simplicity, we assume that the matter field is given by a free scalar field Introducing the matter sector depends on the metricḡ µν only through g µν : B. The matter metric in the unitary gauge To analyze the degeneracy conditions of the theory (12), we work in the so-called unitary gauge, where the clock is synchronized with the scalar field φ, and the ADM decomposition of the metric g µν , This means that we will consider the U-degenerate conditions in Ref. [31]. The covariant degeneracy conditions are not necessarily to be imposed to eliminate the Ostrogradsky mode. As we shall see in the following, the degeneracy conditions in the unitary gauge are much simpler than the covariant one in the theory (12).
In the unitary gauge, φ µ and X are given by where n µ is the unit normal of a time-constant hypersurface. Then, we find that X µ , Y , and Z are given by Using these relations, g µν in Eq. (16) can be expressed as the form where U a (a = 0, 1, 2, 3) are functions of the following spatial scalars: These three scalar quantities are related to X, Y, Z as The explicit relations between the functions U a and F a are shown in Appendix A. For later convenience, we also give explicit forms of the metric components: where we introduced new functions and new variables D i N ≡ γ ij ∇ j N . In Eqs. (29) and (30), we used the fact that ∇ i N is decomposed as As we will see, it is convenient to introduce the vector which satisfies n µ N µ = 0 and N µ N µ = X s . Then, the expression (25) for g µν can be written as in terms of the functions V 0 and V 1 defined above as well as U 0 and U 3 .

III. DEGENERACY CONDITIONS
A. Kinetic structure of the action In the Lagrangian density (12), the Horndeski term √ −gL Horndeski does not containṄ norσ [13,14] and the matter term √ −ḡL matter contains onlyṄ andσ as the time-derivative terms. Hence, the kinetic matrix of the theory (12) is given by a block diagonal matrix composed of (i) the kinetic matrix of the Horndeski term and (ii) the kinetic matrix forṄ andσ obtained from the matter term:Ṅσ * where for the matter Lagrangian (15). This simple form of the kinetic matrix is an advantage of working in the unitary gauge (18). In the covariant gauge, both the Horndeski term and the matter term depend onφ and thus the kinetic matrix has a more complicated structure. The structure of the kinetic matrix changes due to the matter term aṡ This shows that the primary constraint π N ≡ ∂( √ −gL)/∂Ṅ = 0 in the Horndeski theory is lost due to the matter term. To keep the number of primary constraints, the reduced kinetic matrix (36) should be degenerate: when there is a primary constraint between the conjugate momenta of N and σ, for arbitrary δq b . Hence, must be satisfied. Therefore, the matrix (36) has a zero eigenmode. Moreover, we require ∂Φ/∂π N = 0 because the constraint function Φ must depend on π N to remove the would-be Ostrogradsky mode. We also require that the matter scalar field σ is dynamical and thus rank(K) = 1. Using the fact that g µν does not depend onσ, Eq. (36) is evaluated as 1 Its determinant is given by and the degeneracy condition is The conditions ∂Φ/∂π N = 0 and rank(K) = 1 require that ∂Φ/∂π a ∝ δ aσ should not be the zero eigenmode in Eq. (40). From Eq. (41), this requirement is satisfied if and only if It is noteworthy that the analysis would become complicated if we did not use the Horndeski frame (12) but the new gravity frame (14) (i.e., the minimal coupling frame for the invertible case). In the Horndeski frame (12), it is apparent that the primary constraint π N ≡ ∂( √ −gL)/∂Ṅ = 0 for the Horndeski theory is lost due to the matter coupling. Therefore, the Ostrogradsky mode appears unless the functions F a are appropriately chosen. On the other hand, in the new gravity frame (14), the number of the primary constraints is unchanged because the matter term does not contain the derivative of the gravity fields. These facts suggest that the matter term changes the higher-order (secondary, tertiary,...) constraints in the new gravity frame. To find the same restrictions, we would need to perform the full constraint analysis not only the kinetic matrix (see Appendix C).

B. Solutions of the degeneracy conditions
As usual, we assume that the degeneracy condition (43) is satisfied for all configurations. This is equivalent to requiring that the unwanted Ostrogradsky mode should not appear for any configuration in the theory. 2 Under this assumption, Eq. (43) is satisfied for arbitrary ∇ µ σ and thus must be identically satisfied. Because this equation is satisfied for any value ofṄ , we can consider it as a differential equation forṄ . As shown in Eqs. (28)-(30), the components of g µν depend onṄ only through ρ defined in Eq. (26). Therefore, Eq. (45) is equivalent to whose independent variables are ρ as well as the scalars N and X s in Eq. (26). In the following, we will see how this equation restricts the forms of the functions U a (a = 0, 1, 2, 3) in Eq. (25).

Restrictions on g µν
We solve each component of Eq. (46) as follows: The 00 component of Eq. (46) is given by and it can be solved for g 00 as a differential equation for ρ. Recasting Eq. (47) as and we can easily integrate it as where Γ , A (t) are integration constants depending on N and X s . All the integration constants below are also functions of N and X s unless explicitly stated. A special solution 1/ρ is expressed as the limit |A (t) | → ∞ with Γ/A (t) → const. A (t) = 0 follows fromĝ 00 = 0.
• (µ, ν) = (0, i): The 0i component of Eq. (46) is given by By substituting Eq. (49), this equation can be integrated once as where A i is an integration constant. Further integration gives where C i is an integration constant. From the spatial covariance, the integration constants A i , C i should be decomposed as Here, the coefficients A (s) , A (l) , C (s) , C (l) are the functions of N and X s . Their superscripts indicate that the coefficients with (s) and (l) correspond to N i (shift) and D i N (lapse), respectively.
The ij component of Eq. (46) is given by For any value of Γ, this equation becomes and its solution is given by Here, A ij , C ij are integration constants and can be decomposed as Here, the coefficients are functions of N and X s . The superscripts of the coefficients are assigned in a similar manner to Eq. (53).
2. Restrictions on the functions U0, U3, V0, and V1 The restrictions on g µν above can be translated to those on the functions U a in Section II B. First, from Eq. (28), the 00 component fixes V 0 in Eq. (31) as Substituting this result to Eqs. (29), the solution of g 0i [Eq. (52)] implies Decomposing A i and C i as Eq. (53), we find by comparing the coefficients. Since these relations are identities for ρ, the functions A (s) , C (s) are fixed as from the first equation. The function V 1 is fixed as Substituting the above results (58), (61), and (62) to Eqs. (30) and (56), we find for the Γ = 0 case and for the Γ = 0 case. Decomposing A ij and C ij as Eq. (57) and comparing the coefficients, we find and the functions U 0 , U 3 are fixed as Gathering all the results in this section, we find the following restrictions on the functions U 0 , U 3 , V 0 , V 1 . The corresponding restriction on the matter metric (13) can be read from the relations in Appendix A. We do not give its explicit form because it is cumbersome and not insightful. In section V, we will give a simpler form after imposing the invertibility conditions.

IV. INVERTIBILITY CONDITIONS
In this section, we examine the conditions to make the transformation g µν →ḡ µν invertible for the metric (13). As explained in Section II A, this corresponds to assuming the existence of the minimal coupling frame. The invertibility conditions have been derived in Ref. [19] (Eq. (29) therein) as where X, Y , Z are defined for the new metricḡ µν as Since the degeneracy conditions in the previous section only restrict the ρ-dependence, we first consider the invertibility conditions related to ρ. In the unitary gauge,X is simply given byX =ḡ 00 . Switching the independent variables from X, Y, Z to N, ρ, X s we thus find thatḡ 00 does not depend on ρ from the forth condition in Eq. (75): Recalling that g µν ≡ √ −ḡḡ µν , this condition is expressed as a condition on g µν : With g 00 = −V 0 = 0 (Eqs. (28) and (44)), this equation can be integrated as where C g is a function of N and X s . Using the expression of det(g µν ) in Appendix B, we obtain the following condition for the invertibility with g ≡ det(g µν ).
1. Restrictions on the functions U0, U3, V0, and V1 We examine how the condition (80) restricts the functions U 0 , U 3 , V 0 , and V 1 separately for Γ = 0 and Γ = 0. The remaining conditions in Eq. (75) will be examined in the next section.
• Γ = 0 case: In this case, V 0 does not depend on ρ (see Eq. (69)). Therefore, the invertibility condition (80) implies that is independent of ρ. Taking into account that U 0 , U 3 , V 1 are polynomials of ρ (see Eqs. (67)-(70)), we also find that should be independent of ρ. Thus, Applying these restrictions, U 0 , V 0 , V 1 are given by and U 3 is determined by V 1 as where we have introducedC (ll) ≡ C (ll) + [C (l) ] 2 /A (t) , which is independent of ρ. Then, C g becomes independent of ρ, as required.
• Γ = 0 case: In this case, V 0 ∝ (1 + Γρ) −1 (see Eq. (73)). Therefore, the invertibility condition (80) implies that From Eqs. (71)-(74), the quantities U 0 V 0 and V 2 1 + V 0 U 3 in the bracket are expressed as Notably, there is no term proportional to (1 + Γρ) −2 due to a cancellation. Since U 0 is the linear function of ρ, we find that the quantity on the left-hand side should identically vanish to satisfy the condition (88). This leads to However, this result violates the invertivility condition det(ḡ µν ) = 0 in Eq. (75). Therefore, there is no solution of the degeneracy and invertivility conditions in the Γ = 0 case.
While g µν depends on ρ through V 1 , its determinant is independent of ρ.

Restriction on the matter metricḡµν
From the result (92) and g µν = √ −ḡḡ µν (det(g µν ) =ḡ), we can find the form of the matter metricḡ µν with the degenerate and invertibility conditions in the unitary gauge. First, the inverse matter metricḡ µν is written as by using Eqs. (92) and (93). Then, the matter metricḡ µν is computed as where the metric γ (⊥) µν represents the components of g µν orthogonal to n µ and N µ : It might be insightful to write down the line element for the matter metric (95): where dx ≡ N µ dx µ / √ X s . Here, we have defined We see that spacetime is divided into two parts in the line element: the linear span of {n µ , N µ } and its orthogonal complement. This reflects the fact that the disformal part (the last three terms) in the transformation (13) is spanned by two vectors φ µ and X µ (see Eq. (101)). Then, the line element of the orthogonal complement (the last term) is only conformally transformed in Eq. (97). On the other hand, all components are independently transformed for Span{n µ , N µ }. A distinctive feature is that the ρ dependence only appears in the shift vector because the coefficients U, C (tr) , A (l) , and C (l) are functions of N and X s . Therefore, it is easy to see that the invertible condition (77) is satisfied.

V. COVARIANT FORM OF THE MATTER METRIC WITH THE DEGENERATE AND INVERTIBILITY CONDITIONS
In this section, we write down the covariant form of the matter metricḡ µν , i.e., the functions F a (a = 0, 1, 2, 3) in Eq. (13), that satisfies both the degeneracy conditions and the invertible conditions.
The covariant form is obtained by replacing the variables in the unitary gauge as and Since the coefficients in Eq. (98) are independent of ρ, all the covariantized coefficients should be functions of only φ, X, and the specific combination W ≡ Z − Y 2 /X (see Eq. (100)). Therefore, the covariant form of the matter metric g µν can be expressed as 3 where Here, the metric γ (⊥) µν denotes the components orthogonal to φ µ and X µ . We can explicitly write it as in terms of φ µ and which corresponds to the components of X µ orthogonal to φ µ . The functions F U a (φ, X, W ) (a = φ, , ⊥) and V U (φ, X, Y, W ) can be chosen independently. Comparing the metric (102) with Eq. (13), we find that the functions F a (φ, X, Y, Z) are given by The dependence on the higher-derivative terms Y and Z in the functions F a (a = 0, 1, 2, 3) is restricted as the relations above: the functions F a cannot be freely chosen. For example, F 1 = 0 (F 2 = 0) leads to F 2 = F 3 = 0 (F 3 = 0) because the relations above require in this case. We examine the remaining invertibility conditions in Eq. (75). The conditions X Y = X Z = 0 imply that X = g µν φ µ φ ν is independent of W as well as Y . Substituting the restricted form of the matter metric (102), we find X = [F U φ ] −1 . Thus, the invertibility conditions X Y = X Z = 0 are satisfied when The other conditions state that some quantities do not vanish. Therefore, they are satisfied except for specific forms of the functions F U φ , F U , F U ⊥ , and V U . Since the explicit forms are cumbersome, we do not show them for a general case. We will closely examine them only for a special case below.
Finally, we would like to remark that our result contains that in Ref. [32], 4 which is obtained by requiring that the matter metricḡ µν is independent ofṄ in the unitary gauge: when the function V U is chosen to satisfy the metric (106) can be reorganized as In contrast that three scalars X, Y, Z can be constructed from φ µ and X µ , we can only construct two scalars X, W from φ µ and X µ because φ µ X µ = 0. Therefore, this special class is easier to handle because Y never appears in any manipulations (e.g., matrix product, functional composition) and only φ, X, W appears as independent variables. It is also easy to see that the second derivative along the time direction does not appear in this special class. We can rewrite X µ as This expression implies that X µ is proportional to ∇ α ∇ β φ projected perpendicular to the direction φ α ≡ ∇ α φ. Therefore, in the unitary gauge φ = t, the second derivative along the time direction does not appear as stated above. Before closing this section, let us examine the remaining invertibility conditions in Eq. (75) for the special class (109). For the restricted form (109), the conditions (75) are reduced to From the first two conditions, we obtain the following conditions on F U a : In writing down the remaining conditions, it is convenient to introduce the functions f U a (a = 0, 1, 2, 3) for the inverse metric through The functions f U a are expressed in terms of F U a as Using f U a , we can express X as The third and forth conditions in Eq. (111) imply that the combination above depends on X and not on W , respectively. Because X = X(φ, X), X µ can be computed as X µ = X φ φ µ + X X X µ . Here, the subscripts φ, X indicate the derivatives with respect to the corresponding variables. After some manipulation, we find and W can be expressed as The last condition in Eq. (111) implies that the combination above depends on W . Note that Y does not appear in the mapping (X, W ) ↔ (X, W ).
As an illustration, let us consider the case with F U 0 = 1, F U 1 = 0, F U 2 = 0 and F U 3 = 0 as a counterpart of the example in Ref. [19] (Section IIIA therein). In this case, X and W become Thus, the invertibility conditions (111) are solved bȳ for any function F U 3 (φ, X, W ) with F U 3 = P (φ, X) − W −1 for some function P (φ, X). We can also write it in the form (13) asḡ with As we have stated, the functions F a (a = 1, 2, 3) are related with each other to realize the cancellation between the higher-derivative terms.

VI. SUMMARY
In this work, we examined how a matter coupling changes the degeneracy for new theories of modified gravity generated by the metric transformation (7), which is a generalization of the disformally-transformed metric. Even when the Ostrogradsky mode does not exist in a theory only with the gravity action, it may revive once the matter coupling is introduced unless the matter coupling satisfies certain conditions. We derived such conditions to avoid re-appearance of the Ostrogradsky mode due to the matter coupling.
To derive the conditions to avoid the Ostrogradsky mode, we started from the Horndeski action with a matter coupling (15), in which a free scalar field σ is coupled to a metricḡ µν defined by Eq. (7). When the metric transformation (7) from g µν toḡ µν is invertible, this theory is equivalent to a new gravity theory which is generated from the Horndeski theory through the metric transformation (7) and has a minimal coupling to the matter scalar field σ. Thus, the action (12) incorporates the DHOST class I theories and their generalizations proposed in Ref. [19] with a minimally coupled matter scalar field. Working in the unitary gauge, we found that the degeneracy conditions of this system are exactly solvable and the matter metricḡ µν is restricted. We would like to note that this result would be nontrivial when we started from the action in the new gravity frame (i.e., the minimal coupling frame). In this frame, the matter action does not contain the higher derivatives nor affect the degeneracy of the kinetic matrix. To find the same restrictions, we will need to perform the full constraint analysis (see Appendix C).
We derived the conditions on the matter metric without and with assuming the invertibility in section III and in section IV, respectively. As explained in section V, the second-derivative terms of the gravity scalar field φ in the matter metric has a certain structure when both the degeneracy and invertibility conditions are imposed (see Eq. (109)): the second derivative ∇ µ ∇ ν φ only appears with the projection onto the orthogonal complement of ∇ µ φ. This manifestly shows the absence of the higher time-derivative of φ in the unitary gauge φ = t. This structure is similar to the so-called U-DHOST theories, which are degenerate in the unitary gauge but not in their covariant version [31]. Therefore, the resultant theory (12) with the matter metric (109) will have an instantaneous mode in the presence of the matter field. The instantaneous mode does not induce an instability as the Ostrogradsky mode but will lead to a peculiar phenomenology. It will be interesting to investigate how the phenomenology of the theory (12) is affected by the presence of the matter field in the environment.
Some directions of further studies are in order. First, as a solvable problem, we investigated the degeneracy conditions in the presence of a free scalar field as the matter. Further studies are necessary to see how much the matter metric is restricted for the realistic matter fields. Second, we limited our analysis within the unitary gauge. If we desire to eliminate the instantaneous mode, the degeneracy conditions should be imposed to the covariant version of the theory. It will be interesting to study how much the matter metric is further restricted in this case. Finally, we worked only on the metric transformation given by Eq. (7). In principal, this transformation can be generalized to incorporate more higher-derivative terms as done by Ref. [19]. Such an extension would be important to clarify the parameter space of the physically sensible theories without Ostrgradsky modes. where We show the relationship between the functions U a (a = 0, 1, 2, 3) in g µν (Eq. (16)) and the functions F a (a = 0, 1, 2, 3) in the matter metricḡ µν (Eq. (13)).
-Expression in the unitary gauge where -Expression of the components where In this appendix, we derive the expression of det(g µν ). The derivation is based on the matrix determinant lemma For the derivation, we define the following tensors: g µν (2) = U 0 g µν + U 1 n µ n ν + 2U 2 n (µ ∇ ν) N .

Appendix C: Structure of degeneracy in different frames
In the main text, we analyzed the degeneracy conditions for the action in the Horndeski frame (12) (Eq. (6)). When we instead use the action in the new gravity frame (Eq. (5)), the gravity action depends onN as well asṄ and we need constraints to remove them. In this Appendix, we see how the matter coupling affects the structure of degeneracy in the two different frames by using a simple example of a mechanical system: where the variables N , g, σ are not fields but the functions of t. We assume that the higher-derivative Lagrangian L G (N,Ṅ ,N , g,ġ) is generated from the Lagrangian L G (N, g,ġ) through the invertible transformation g →ḡ: L G (N,Ṅ ,N , g,ġ) = L G (N,ḡ,ġ) ;ḡ =ḡ(g, N,Ṅ ) .
Therefore, the action (C1) is equivalent to S[N,ḡ, σ] = dt L G (N,ḡ,ġ) + gσ 2 ; g = g(ḡ, N,Ṅ ) , where the relation g = g(ḡ, N,Ṅ ) is obtained by inverting the transformation g →ḡ. The action (C1) is a special class of the coupled mechanical systems analyzed in Refs. [33,34]. Following the analysis in Refs. [33,34], we show how the matter coupling gσ 2 makes us fail to remove the derivative terms of N from the equations of motion in the two frames (C1) and (C3).
The action (C3) corresponds to the Horndeski frame (12) (Eq. (6)). In this frame, the equations of motion are written as E N ≡ − d dt (gṄσ 2 ) + L G N + g Nσ 2 = −(gṄṄσ 2 )N + · · · , (C4) Eḡ ≡ − d dt (L Ġ g ) + L Ḡ g + gḡσ 2 = −L G NġṄ + gḡσ 2 + · · · , (C5) where only the highest-derivative term of N is shown in the r.h.s of each equation of motion. The subscripts represent the derivatives with respect to the corresponding variables. It is apparent that the matter coupling affects the degeneracy conditions to remove the highest derivativeN in the equations of motion. In the Hamiltonian analysis, this corresponds to the fact that the matter coupling affects the primary constraints. The action (C1) corresponds to the new gravity frame (Eq. (5)). In this frame, the equations of motion are written asĒ where only the highest-derivative term of N is shown in the r.h.s of each equation of motion. In this frame, the highest derivative in the equations of motion is .... N . We can show that the matter coupling does not affect the degeneracy condition for the highest derivative .... N . When the LagrangianL G is generated through the transformation as Eq. (C2), L G N andL Ġ g can be written in terms of L G :L Thus, the LagrangianL G satisfies the condition which also implies that the kinetic matrix is degenerate. The condition (C11) ensures that the highest derivative .... N can be removed by taking the following combination, Therefore, the matter coupling does not affect the degeneracy condition to remove the highest derivative .... N , i.e., the condition (C11). However, the matter coupling now appears in the terms withN andṄ through the term (vσ 2 ) · .
The matter coupling appears in the degeneracy conditions for the lower derivatives. In the Hamiltonian analysis, this corresponds to the fact that the matter coupling does not affect the primary constraints but does the higher-order (secondary, tertiary,...) constraints (see Ref. [34] and its reference for the relation of the Lagrangian analysis here to the Hamiltonian analysis).