Generalized disformal Horndeski theories:
cosmological perturbations and consistent matter coupling

In this subsection, we briefly review the invertible disformal transformation with second derivatives of the scalar field constructed in our previous study Takahashi et al. (2022a). We consider a generalized disformal transformation defined by

$$\bar{g}_{\mu\nu}[g,\phi]=F_{0}g_{\mu\nu}+F_{1}\phi_{\mu}\phi_{\nu}+2F_{2}\phi_{(\mu}X_{\nu)}+F_{3}X_{\mu}X_{\nu},$$

(3)

where $\phi_{\mu}\coloneqq\partial_{\mu}\phi$ and $X_{\mu}\coloneqq\partial_{\mu}X=2\phi_{\alpha}\phi^{\alpha}_{\mu}$ with $\phi_{\mu\nu}\coloneqq\nabla_{\mu}\nabla_{\nu}\phi$ and $\phi^{\alpha}_{\mu}=g^{\alpha\beta}\phi_{\beta\mu}$. Also, we have defined the symmetrization of a tensor $T_{\mu\nu}$ by $T_{({\mu\nu})}\coloneqq(T_{\mu\nu}+T_{\nu\mu})/2$. Here, $F_{i}$’s ($i=0,1,2,3$) are functions of $(\phi,X,Y,Z)$, with $Y$ and $Z$ defined as

$$Y\coloneqq\phi_{\mu}X^{\mu},\qquad Z\coloneqq X_{\mu}X^{\mu}.$$

(4)

We define the following quantities for later use:

$$\mathcal{F}\coloneqq F_{0}^{2}+F_{0}(XF_{1}+2YF_{2}+ZF_{3})+W(F_{2}^{2}-F_{1}F_{3}),\qquad W\coloneqq Y^{2}-XZ.$$

(5)

Note that the class of conventional disformal transformations of the form (1) is included in the class of generalized disformal transformations given by (3).

Let $\mathcal{D}$ denote the set of disformal transformations of the form (3). In order to discuss the inverse transformation, we consider a subset of $\mathcal{D}$ that forms groups under the following two binary operations:

• •

  Matrix product:^*3*3*3Strictly speaking, in order to guarantee the closedness under the matrix product, we need to enlarge the underlying set $\mathcal{D}$ to include the antisymmetric part of $\phi_{\mu}X_{\nu}$ Takahashi et al. (2022a). Nevertheless, the inverse element of an (invertible) element in $\mathcal{D}$ can also be found in $\mathcal{D}$, which allows us to construct the inverse disformal metric (8).

  $$\mathcal{D}^{2}\ni(\bar{g}_{\mu\nu},\hat{g}_{\mu\nu})\quad\mapsto\quad(\bar{g}~{}\begin{picture}(-1.0,1.0)(-1.0,-2.5)\circle*{3.0}\end{picture}~{}\;\hat{g})_{\mu\nu}\coloneqq\bar{g}_{\mu\alpha}g^{\alpha\beta}\hat{g}_{\beta\nu}.$$

  (6)

• •

  Functional composition:

  $$\mathcal{D}^{2}\ni(\bar{g}_{\mu\nu}[g,\phi],\hat{g}_{\mu\nu}[g,\phi])\quad\mapsto\quad(\bar{g}\circ\hat{g})_{\mu\nu}\coloneqq\bar{g}_{\mu\nu}[\hat{g},\phi].$$

  (7)

Note that both the operations are associative and have an identity element (given by $g_{\mu\nu}$ itself). The inverse element of $\bar{g}_{\mu\nu}$ associated with the matrix product (after raising the indices by $g^{\alpha\beta}$) gives the inverse metric $\bar{g}^{\mu\nu}$, while the one associated with the functional composition is nothing but the inverse transformation $g_{\mu\nu}[\bar{g},\phi]$. We shall clarify which subset of $\mathcal{D}$ can form groups under these operations.

Provided that $F_{0}\neq 0$ and $\mathcal{F}\neq 0$, it is straightforward to obtain the inverse metric with respect to $\bar{g}_{\mu\nu}$ Takahashi et al. (2022a):

$$\bar{g}^{\mu\nu}=\frac{1}{F_{0}}\bigg{[}g^{\mu\nu}-\frac{F_{0}F_{1}-Z(F_{2}^{2}-F_{1}F_{3})}{\mathcal{F}}\phi^{\mu}\phi^{\nu}-2\frac{F_{0}F_{2}+Y(F_{2}^{2}-F_{1}F_{3})}{\mathcal{F}}\phi^{(\mu}X^{\nu)}-\frac{F_{0}F_{3}-X(F_{2}^{2}-F_{1}F_{3})}{\mathcal{F}}X^{\mu}X^{\nu}\bigg{]}.$$

(8)

With the inverse metric, the kinetic term in the barred frame can be computed as

$$\bar{X}=\bar{g}^{\mu\nu}\phi_{\mu}\phi_{\nu}=\frac{XF_{0}-WF_{3}}{\mathcal{F}},$$

(9)

which is a function of $(\phi,X,Y,Z)$ in general. If $\bar{X}$ depends on $Y$ and/or $Z$ in a nontrivial manner, the quantities $\bar{Y}$ and $\bar{Z}$, which are respectively the barred counterparts of $Y$ and $Z$ defined in (4), yield unwanted higher derivatives like $\partial_{\mu}Y$ and $\partial_{\mu}Z$ through $\partial_{\mu}\bar{X}$. Therefore, we require

$$\bar{X}_{Y}=\bar{X}_{Z}=0,$$

(10)

where ${\bar{X}}_{Y}\coloneqq\partial{\bar{X}}/\partial Y$ and ${\bar{X}}_{Z}\coloneqq\partial{\bar{X}}/\partial Z$, so that $\bar{X}=\bar{X}(\phi,X)$. We also assume $\bar{X}_{X}\neq 0$ so that the relation $\bar{X}=\bar{X}(\phi,X)$ can be solved for $X$ to yield $X=X(\phi,\bar{X})$. Then, we have $\bar{X}_{\mu}\coloneqq\partial_{\mu}\bar{X}=\bar{X}_{X}X_{\mu}+\bar{X}_{\phi}\phi_{\mu}$, and hence

$$\begin{split}\bar{Y}&=\bar{g}^{\mu\nu}\phi_{\mu}\bar{X}_{\nu}=\bar{X}_{X}\frac{YF_{0}+WF_{2}}{\mathcal{F}}+\bar{X}_{\phi}\bar{X},\\ \bar{Z}&=\bar{g}^{\mu\nu}\bar{X}_{\mu}\bar{X}_{\nu}=\bar{X}_{X}^{2}\frac{ZF_{0}-WF_{1}}{\mathcal{F}}+2\bar{X}_{\phi}\bar{Y}-\bar{X}_{\phi}^{2}\bar{X}.\end{split}$$

(11)

We further require that these two equations can be solved for $Y$ and $Z$ to obtain $Y=Y(\phi,\bar{X},\bar{Y},\bar{Z})$ and $Z=Z(\phi,\bar{X},\bar{Y},\bar{Z})$, which is possible if the Jacobian determinant $|\partial(\bar{Y},\bar{Z})/\partial(Y,Z)|$ is nonvanishing. As a side note, the quantity $W$ is transformed as

$$\bar{W}\coloneqq\bar{Y}^{2}-\bar{X}\bar{Z}=\frac{\bar{X}_{X}^{2}}{\mathcal{F}}W.$$

(12)

The requirements discussed above provide the invertibility condition, namely Takahashi et al. (2022a),

$$F_{0}\neq 0,\qquad\mathcal{F}\neq 0,\qquad\bar{X}_{Y}=\bar{X}_{Z}=0,\qquad\bar{X}_{X}\neq 0,\qquad\left|\frac{\partial(\bar{Y},\bar{Z})}{\partial(Y,Z)}\right|\neq 0.$$

(13)

Under this condition, we can obtain the inverse transformation for (3) in the following form Takahashi et al. (2022a):

$$g_{\mu\nu}=\frac{1}{F_{0}}\left(\bar{g}_{\mu\nu}-\frac{\bar{X}_{X}^{2}F_{1}-2\bar{X}_{\phi}\bar{X}_{X}F_{2}+\bar{X}_{\phi}^{2}F_{3}}{\bar{X}_{X}^{2}}\phi_{\mu}\phi_{\nu}-2\frac{\bar{X}_{X}F_{2}-\bar{X}_{\phi}F_{3}}{\bar{X}_{X}^{2}}\phi_{(\mu}\bar{X}_{\nu)}-\frac{F_{3}}{\bar{X}_{X}^{2}}\bar{X}_{\mu}\bar{X}_{\nu}\right),$$

(14)

where the functions of $(\phi,X,Y,Z)$ in the right-hand side can be translated back into functions of $(\phi,\bar{X},\bar{Y},\bar{Z})$ by use of (9) and (11). In other words, the set of conditions (13) defines a subset of $\mathcal{D}$ that forms groups under the matrix product (6) and the functional composition (7).

Thus, we clarified the invertibility condition (13) for the generalized disformal transformation (3). Before closing this subsection, there are several things to note about the invertible generalized disformal transformation and its possible further extensions.

1. 1.

   The first is about the independent functional DOFs of invertible transformations. As we already mentioned in Takahashi et al. (2022a), one can regard $\bar{X}=\bar{X}(\phi,X)$ as a given function, which then fixes one of the coefficient functions, say $F_{3}$, in the generalized disformal transformation (3) through (9). Written explicitly,

   $$F_{3}=\frac{XF_{0}-\bar{X}(\phi,X)\left[F_{0}(F_{0}+XF_{1}+2YF_{2})+WF_{2}^{2}\right]}{W+\bar{X}(\phi,X)\left(ZF_{0}-WF_{1}\right)}.$$

   (15)

   Therefore, the invertible subclass of transformations is characterized by $\bar{X}(\phi,X)$ as well as the remaining three functions $F_{0}$, $F_{1}$, $F_{2}$ of $(\phi,X,Y,Z)$. Then, the third condition in (13) is trivial, and the independent functional DOFs should be chosen to satisfy the remaining four conditions.

2. 2.

   The invertibility condition (13) for generalized disformal transformations is precisely a generalization of (2) for conventional disformal transformations. Indeed, by choosing the independent functional DOFs of generalized disformal transformations as

   $$F_{0}=F_{0}(\phi,X),\qquad F_{1}=F_{1}(\phi,X),\qquad F_{2}=0,\qquad\bar{X}=\frac{X}{F_{0}+XF_{1}},$$

   (16)

   we have $F_{3}=0$ from (15), and hence the transformation law (3) reduces to the conventional one (1). Then, the last two conditions in (13) are degenerate, which yield $\bar{X}_{X}\propto F_{0}-XF_{0X}-X^{2}F_{1X}\neq 0$. Also, we now have $\mathcal{F}\propto F_{0}+XF_{1}\neq 0$. Therefore, we recover (2) from (13).

3. 3.

   The inverse transformation may not be unique in the following sense. We required the last two conditions in (13) so that equations (9) and (11) can be solved algebraically for $X$, $Y$, and $Z$. Since these equations are nonlinear algebraic equations in general, there could be multiple branches of (real) solutions. In this case, we have the inverse transformation for each branch of solution.

4. 4.

   It could happen that the set of conditions (13) is violated only at some spacetime point(s) for some particular configurations. If this happens for a solution in some scalar-tensor theory, then the configuration obtained as a result of the disformal transformation cannot be regarded as a solution in the disformal counterpart of the scalar-tensor theory in general. In other words, such a transformation could introduce a new branch of solution which is not connected to solutions in the original theory Jiroušek et al. (2022a, b).

5. 5.

   The transformation law (3) is not most general up to the second derivative of the scalar field, and there could be other types of invertible transformations (see Domènech et al. (2020) for an example constructed based on conformally/disformally invariant quantities). In principle, it would be possible to find a more general class of invertible disformal transformations with the second derivative of the scalar field, which we leave for future work.

6. 6.

   Finally, it is also possible to construct invertible disformal transformations with third or higher derivatives of the scalar field Takahashi et al. (2022a). Moreover, as we shall discuss in Appendix A, one can incorporate the curvature tensor in the transformation law.

Let us apply the generalized disformal transformation (3) satisfying the invertibility condition (13) to generate novel ghost-free theories beyond the quadratic/cubic DHOST class. For a given action $S_{\rm g}[g_{\mu\nu},\phi]$ of scalar-tensor theories, one can replace the metric by the generalized disformal metric (3) to obtain a new action, i.e.,

$\displaystyle S_{\rm g}[g_{\mu\nu},\phi]\quad\mapsto\quad\tilde{S}_{\rm g}[g_{\mu\nu},\phi]\coloneqq S_{\rm g}[\bar{g}_{\mu\nu},\phi].$

(17)

Suppose that the generalized disformal metric satisfies the invertibility condition (13). Since the two gravitational actions $S_{\rm g}[g_{\mu\nu},\phi]$ and $\tilde{S}_{\rm g}[g_{\mu\nu},\phi]$ are related to each other by invertible transformation, the two theories are mathematically equivalent in the absence of matter fields. However, the theory described by the action $\tilde{S}_{\rm g}[g_{\mu\nu},\phi]$ should be distinguished from the seed theory when matter fields are taken into account. Let us assume matter fields (which we denote collectively by $\Psi$) minimally coupled to gravity and consider the following two actions:

$$S[g_{\mu\nu},\phi,\Psi]\coloneqq S_{\rm g}[g_{\mu\nu},\phi]+S_{\rm m}[g_{\mu\nu},\Psi],\qquad\tilde{S}[g_{\mu\nu},\phi,\Psi]\coloneqq\tilde{S}_{\rm g}[g_{\mu\nu},\phi]+S_{\rm m}[g_{\mu\nu},\Psi].$$

(18)

Due to the existence of the matter fields, the two actions are no longer disformally related to each other. Indeed, performing the disformal transformation $(g_{\mu\nu},\phi)\mapsto(\bar{g}_{\mu\nu},\phi)$ on the first action, we obtain

$$S[g_{\mu\nu},\phi,\Psi]\quad\mapsto\quad S[\bar{g}_{\mu\nu},\phi,\Psi]=\tilde{S}_{\rm g}[g_{\mu\nu},\phi]+S_{\rm m}[\bar{g}_{\mu\nu},\Psi],$$

(19)

where we have used $S_{\rm g}[\bar{g}_{\mu\nu},\phi]=\tilde{S}_{\rm g}[g_{\mu\nu},\phi]$. This is different from $\tilde{S}[g_{\mu\nu},\phi,\Psi]$ because the matter fields are minimally coupled to $\bar{g}_{\mu\nu}$, not $g_{\mu\nu}$. In this sense, the two theories are no longer equivalent in the presence of matter fields.

In Takahashi et al. (2022a), we discussed the transformation of a general action via the generalized disformal transformation (3) and demonstrated that the transformation of a simplest scalar-tensor theory indeed generates a nontrivial higher-derivative theory. In what follows, we study how Horndeski theories, which form a large class of ghost-free scalar-tensor theories, are transformed under the generalized disformal transformation and write down the resultant action explicitly. As mentioned earlier, we are interested in the invertible subset of transformations, by which any ghost-free theory is mapped to another ghost-free theory. Indeed, this is the case when one performs the conventional disformal transformation on Horndeski theories, and we call the resultant class of theories the disformal Horndeski class (or the DH class for short). The DH class is (literally) a part of the DHOST class: It is a sum of the quadratic DHOST of class ²N-I and the cubic DHOST of class ³N-I Crisostomi et al. (2016); Ben Achour et al. (2016a). We will see that the generalized disformal transformation of Horndeski theories in general lies beyond the quadratic/cubic DHOST class. We call the resultant class of novel ghost-free scalar-tensor theories the generalized disformal Horndeski class (or the GDH class).

The action of Horndeski theories (in four spacetime dimensions) are given by

$\displaystyle S_{\rm H}[g_{\mu\nu},\phi]=\int{\rm d}^{4}x\sqrt{-g}\sum_{I=2}^{5}\mathcal{L}_{I}[g_{\mu\nu},\phi],$

(20)

with

$$\begin{split}\mathcal{L}_{2}&\coloneqq G_{2}(\phi,X),\\ \mathcal{L}_{3}&\coloneqq G_{3}(\phi,X)\Box\phi,\\ \mathcal{L}_{4}&\coloneqq G_{4}(\phi,X)R-4G_{4X}(\phi,X)g^{\alpha[\mu}g^{\nu]\beta}\phi_{\alpha\mu}\phi_{\beta\nu},\\ \mathcal{L}_{5}&\coloneqq G_{5}(\phi,X)G^{\mu\nu}\phi_{\mu\nu}+2G_{5X}(\phi,X)g^{\alpha[\mu|}g^{\beta|\nu|}g^{\gamma|\lambda]}\phi_{\alpha\mu}\phi_{\beta\nu}\phi_{\gamma\lambda}.\end{split}$$

(21)

Here, $G_{I}$’s are arbitrary functions of $(\phi,X)$, $G_{\mu\nu}$ denotes the Einstein tensor, and indices inside square brackets are anti-symmetrized (note that $\mu$, $\nu$, and $\lambda$ are anti-symmetrized in the second term of $\mathcal{L}_{5}$). The generalized disformal transformation of the action (20) is obtained by the replacement $g_{\mu\nu}\to\bar{g}_{\mu\nu}$, with $\bar{g}_{\mu\nu}$ being a functional of $g_{\mu\nu}$ and $\phi$ given by (3). By employing the transformation law for each building block of the action developed in Takahashi et al. (2022a), it is straightforward to compute the generalized disformal transformation of the action (20). Written explicitly, the action of GDH theories is given by

$$S_{\rm GDH}[g_{\mu\nu},\phi]\coloneqq S_{\rm H}[\bar{g}_{\mu\nu},\phi]=\int{\rm d}^{4}x\sqrt{-g}\,\mathcal{J}\sum_{I=2}^{5}\tilde{\mathcal{L}}_{I}[g_{\mu\nu},\phi],$$

(22)

where $\mathcal{J}\coloneqq F_{0}\mathcal{F}^{1/2}$ and

$$\begin{split}\tilde{\mathcal{L}}_{2}&\coloneqq G_{2},\\ \tilde{\mathcal{L}}_{3}&\coloneqq G_{3}\bar{g}^{\mu\nu}(\phi_{\mu\nu}-C^{\rho}{}_{\mu\nu}\phi_{\rho}),\\ \tilde{\mathcal{L}}_{4}&\coloneqq G_{4}\bar{g}^{\mu\nu}\left(R_{{\mu\nu}}-2C^{\alpha}{}_{\beta[\alpha}C^{\beta}{}_{\nu]\mu}\right)-2\left(G_{4\phi}\phi_{\alpha}+G_{4X}\bar{X}_{\alpha}\right)\bar{g}^{\mu[\nu}C^{\alpha]}{}_{{\mu\nu}}\\ &\quad\>-4G_{4X}\bar{g}^{\alpha[\mu}\bar{g}^{\nu]\beta}(\phi_{\alpha\mu}-C^{\sigma}{}_{\alpha\mu}\phi_{\sigma})(\phi_{\beta\nu}-C^{\rho}{}_{\beta\nu}\phi_{\rho}),\\ \tilde{\mathcal{L}}_{5}&\coloneqq G_{5}\left(\bar{g}^{\mu\lambda}\bar{g}^{\nu\sigma}-\frac{1}{2}\bar{g}^{{\mu\nu}}\bar{g}^{\lambda\sigma}\right)(\phi_{\mu\nu}-C^{\rho}{}_{\mu\nu}\phi_{\rho})\left(R_{\lambda\sigma}+2\nabla_{[\alpha}C^{\alpha}{}_{\sigma]\lambda}+2C^{\alpha}{}_{\beta[\alpha}C^{\beta}{}_{\sigma]\lambda}\right)\\ &\quad\>+2G_{5X}\bar{g}^{\alpha[\mu|}\bar{g}^{\beta|\nu|}\bar{g}^{\gamma|\lambda]}(\phi_{\alpha\mu}-C^{\sigma}{}_{\alpha\mu}\phi_{\sigma})(\phi_{\beta\nu}-C^{\rho}{}_{\beta\nu}\phi_{\rho})(\phi_{\gamma\lambda}-C^{\eta}{}_{\gamma\lambda}\phi_{\eta}),\end{split}$$

(23)

with all $G_{I}$’s and their derivatives evaluated at $(\phi,\bar{X})$. Here, we have defined the tensor $C$ as the change of the Christoffel symbol under the generalized disformal transformation:

$$C^{\lambda}{}_{\mu\nu}\coloneqq\bar{\Gamma}^{\lambda}_{\mu\nu}-\Gamma^{\lambda}_{\mu\nu}=\bar{g}^{\lambda\alpha}\left(\nabla_{(\mu}\bar{g}_{\nu)\alpha}-\frac{1}{2}\nabla_{\alpha}\bar{g}_{\mu\nu}\right),$$

(24)

which contains the third derivative of $\phi$ through $\nabla\bar{g}$ since $\bar{g}$ itself contains the second derivative of $\phi$ [see Eq. (3)]. Note that we have performed an integration by parts to remove a covariant derivative acting on the tensor $C$ in $\tilde{\mathcal{L}}_{4}$. There still remain terms containing $\nabla C$ and hence the fourth derivatives of $\phi$, but they can also be removed by integration by parts.^*4*4*4The authors would like to thank Hiroaki W. H. Tahara for useful discussion on this point. This means that the action (22) of GDH theories can be recast in a form where derivatives of the scalar field appear only up to the cubic order.

In Fig. 1, we illustrate the inclusion relation among ghost-free scalar-tensor theories. The GDH theory (22) is characterized by the functions $G_{I}(\phi,X)$ ($I=2,3,4,5$) associated with the seed Horndeski theory and the functions $\bar{X}(\phi,X)$ and $F_{i}(\phi,X,Y,Z)$ ($i=0,1,2$) associated with the invertible generalized disformal transformation. As mentioned above, the GDH class involves known ghost-free scalar-tensor theories as special subclasses. Clearly, it includes the Horndeski class as a subclass, which amounts to considering the identity transformation (i.e., $F_{0}=1$, $F_{1}=F_{2}=F_{3}=0$). For conventional disformal transformations characterized by $F_{0}(\phi,X)$ and $F_{1}(\phi,X)$ satisfying the invertibility condition (2), the action (22) reduces to DH theories, though the entire class of quadratic/cubic DHOST theories is not covered by GDH theories. It is known that the complement of the DH subclass in the quadratic/cubic DHOST class, which corresponds to the shaded region in Fig. 1, does not accommodate stable cosmological solution or otherwise the tensor perturbations are nondynamical Langlois et al. (2017). Hence, from a phenomenologically point of view, DH theories provide the only viable class within quadratic/cubic DHOST theories. With generic $F_{i}(\phi,X,Y,Z)$ satisfying the invertibility condition (13), the generalized disformal transformation allows us to go beyond quadratic/cubic DHOST theories and obtain a wider framework of GDH theories (22), which are free from Ostrogradsky ghosts and are expected to admit stable and viable cosmological solutions.

We first study how the Arnowitt-Deser-Misner (ADM) variables are transformed under the generalized disformal transformation. This is useful in the context of cosmology where the scalar field has a timelike profile and hence defines a preferred time direction. Note, however, that the results in this subsection apply to arbitrary spacetimes with a timelike scalar profile.

Let us introduce the ADM decomposition of the metric $g_{\mu\nu}$ as

$$g_{\mu\nu}{\rm d}x^{\mu}{\rm d}x^{\nu}=-\mathcal{N}^{2}{\rm d}t^{2}+\gamma_{ij}({\rm d}x^{i}+\mathcal{N}^{i}{\rm d}t)({\rm d}x^{j}+\mathcal{N}^{j}{\rm d}t),$$

(25)

where $\mathcal{N}$ is the lapse function, $\mathcal{N}^{i}$ is the shift vector, and $\gamma_{ij}$ is the induced metric. Likewise, the ADM decomposition of the disformal metric $\bar{g}_{\mu\nu}$ in (3) is written as

$$\bar{g}_{\mu\nu}{\rm d}x^{\mu}{\rm d}x^{\nu}=-\bar{\mathcal{N}}^{2}{\rm d}t^{2}+\bar{\gamma}_{ij}({\rm d}x^{i}+\bar{\mathcal{N}}^{i}{\rm d}t)({\rm d}x^{j}+\bar{\mathcal{N}}^{j}{\rm d}t).$$

(26)

Under the unitary gauge where the scalar field is spatially uniform [i.e., $\phi=\phi(t)$], we have

$$X=-\frac{\dot{\phi}^{2}}{\mathcal{N}^{2}},\qquad\bar{X}=-\frac{\dot{\phi}^{2}}{\bar{\mathcal{N}}^{2}},$$

(27)

with a dot denoting the derivative with respect to $t$.^*5*5*5In an epoch where the scalar field does not evolve monotonically in time, one cannot choose the unitary gauge. Such a situation typically happens if we identify the scalar field as the inflaton in the context of standard inflationary cosmology, where the inflaton oscillates in the reheating era. In this case, one should choose another appropriate gauge or perform a gauge-independent analysis. Nevertheless, it is straightforward to redo our analysis either in another gauge or in a gauge-independent way, and there would be no conceptual problem. In what follows, we study the relationship between $(\mathcal{N},\mathcal{N}_{i},\gamma_{ij})$ and $(\bar{\mathcal{N}},\bar{\mathcal{N}}_{i},\bar{\gamma}_{ij})$.

On performing the (generalized) disformal transformation, one could simultaneously redefine the time coordinate so that the new time-time component of the disformal metric coincides with the original one. It is natural to perform such a coordinate redefinition when one works in a coordinate system where the lapse function is set to be unity, as done, e.g., in Minamitsuji (2014); Motohashi and White (2016). However, we do not do so in the present paper in order to clarify how each component of the metric is changed under the disformal transformation. Hence, one should be careful about this issue when comparing the following results with those in earlier works.

For the transformation (3) to be invertible, we required that $\bar{X}$ should be written in the form $\bar{X}(\phi,X)$, under which we obtain

$$\bar{\mathcal{N}}^{2}=-\frac{\dot{\phi}^{2}}{\bar{X}(\phi,-\dot{\phi}^{2}/\mathcal{N}^{2})}.$$

(28)

Note that this equation can be solved for $\mathcal{N}$ algebraically as

$$\mathcal{N}^{2}=-\frac{\dot{\phi}^{2}}{X(\phi,-\dot{\phi}^{2}/\bar{\mathcal{N}}^{2})}.$$

(29)

Here, $X=X(\phi,\bar{X})$ is the inverse function of $\bar{X}=\bar{X}(\phi,X)$, which exists again due to the invertibility of the transformation (3) [see Eq. (13)]. It is also straightforward to obtain the barred counterpart of the shift vector and the induced metric as

$$\begin{split}\bar{\mathcal{N}}_{i}&=\bar{g}_{0i}=F_{0}\mathcal{N}_{i}+(F_{2}\dot{\phi}+F_{3}\dot{X}){\rm D}_{i}X,\\ \bar{\gamma}_{ij}&=\bar{g}_{ij}=F_{0}\gamma_{ij}+F_{3}{\rm D}_{i}X{\rm D}_{j}X,\end{split}$$

(30)

where ${\rm D}_{i}$ is the covariant derivative associated with $\gamma_{ij}$. The quantities $Y$ and $Z$ are written as

$$\begin{split}Y&=g^{0\mu}\dot{\phi}X_{\mu}=-\frac{\dot{\phi}}{\mathcal{N}^{2}}\left(\dot{X}-\mathcal{N}^{k}{\rm D}_{k}X\right),\\ Z&=g^{\mu\nu}X_{\mu}X_{\nu}={\rm D}_{k}X{\rm D}^{k}X+\frac{Y^{2}}{X},\end{split}$$

(31)

so that

$$W=Y^{2}-XZ=-X{\rm D}_{k}X{\rm D}^{k}X.$$

(32)

Note that spatial indices for unbarred quantities are raised by $\gamma^{ij}$. From this equation, we find that the quantity $W$ does not contain $\dot{X}$ (and hence $\dot{\mathcal{N}}$). Using the above relations, we obtain the inverse spatial metric in the barred frame as

$$\bar{\gamma}^{ij}=\frac{1}{F_{0}}\left(\gamma^{ij}-\frac{XF_{3}}{XF_{0}-WF_{3}}{\rm D}^{i}X{\rm D}^{j}X\right),$$

(33)

which is used to raise spatial indices for barred quantities. For instance,

$$\bar{\mathcal{N}}^{i}=\bar{\gamma}^{ij}\bar{\mathcal{N}}_{j}=\mathcal{N}^{i}+\dot{\phi}\frac{XF_{2}+YF_{3}}{XF_{0}-WF_{3}}{\rm D}^{i}X.$$

(34)

We stress that the invertibility of the generalized disformal transformation is crucial to obtain the transformation law for the lapse function in the simple form of (28). Then, as it should be, the map $(\mathcal{N},\mathcal{N}_{i},\gamma_{ij})\mapsto(\bar{\mathcal{N}},\bar{\mathcal{N}}_{i},\bar{\gamma}_{ij})$ is invertible. For noninvertible transformations that do not satisfy the condition (10), one can still obtain $\bar{\mathcal{N}}$ as a functional of unbarred variables by employing (9). However, the transformation law now contains $\dot{\mathcal{N}}$ and hence is no longer an algebraic equation for $\mathcal{N}$.

Hereafter, we restrict ourselves to the spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) background and perturbations about it. The background metric and scalar field are described by

$$g_{\mu\nu}{\rm d}x^{\mu}{\rm d}x^{\nu}=-N(t)^{2}{\rm d}t^{2}+a(t)^{2}\delta_{ij}{\rm d}x^{i}{\rm d}x^{j},\qquad\phi=\phi(t),$$

(35)

with $a(t)$ being the scale factor. Then, the scalar-field kinetic term $X$ also depends on $t$ only. From (27) and (31), the background value of the quantities $X$, $Y$, and $Z$ are given by

$$X_{\rm BG}\coloneqq-\frac{\dot{\phi}^{2}}{N^{2}},\qquad Y_{\rm BG}\coloneqq\frac{X_{\rm BG}\dot{X}_{\rm BG}}{\dot{\phi}},\qquad Z_{\rm BG}\coloneqq\frac{Y_{\rm BG}^{2}}{X_{\rm BG}},$$

(36)

respectively. Here and in what follows, quantities with the subscript “BG” indicate their background value (and we omit the subscript when unnecessary). Equation (36) means that the quantity $W=Y^{2}-XZ$ vanishes at the background level, which is an important property of the FLRW spacetime.

The barred-frame metric also takes the FLRW form,

$\displaystyle\bar{g}_{\mu\nu}{\rm d}x^{\mu}{\rm d}x^{\nu}$

$\displaystyle=-\bar{N}(t)^{2}{\rm d}t^{2}+\bar{a}(t)^{2}\delta_{ij}{\rm d}x^{i}{\rm d}x^{j},$

(37)

with $\bar{N}=\bar{N}(N)$ and $\bar{a}=\bar{a}[N,a]$ given by

$$\frac{\bar{N}^{2}}{N^{2}}=\frac{X_{\rm BG}}{\bar{X}(\phi,X_{\rm BG})}=\frac{\mathcal{F}_{\rm BG}}{F_{0,{\rm BG}}},\qquad\bar{a}=F_{0,{\rm BG}}^{1/2}\,a.$$

(38)

Hence, on the FLRW background, $F_{0}>0$ and $\mathcal{F}>0$ should be required in order to preserve the metric signature. Note that we have used (9) with $W=0$ and also $\bar{X}=\bar{X}(\phi,X)$, which follows from the invertibility condition (13). It should also be noted that (38) can be inverted to obtain $N=N(\bar{N})$ and $a=a[\bar{N},\bar{a}]$, which is consistent with the result of Babichev et al. (2022).

Let us now investigate perturbations about the FLRW background. In principle, the formulae in §III.1 provide transformation laws for nonlinear perturbations. Actually, the authors of Motohashi and White (2016) studied how nonlinear perturbations about the FLRW background are transformed under conventional disformal transformations. However, the extension of their discussion to generalized disformal transformations would be nontrivial due to the additional terms in the transformation laws, i.e., the second term of each equation in (30). In any case, the formulae in §III.1 are useful to study linear perturbations about the FLRW background, which we study in the following subsections.

We first consider tensor perturbations about the FLRW background. In the unbarred frame, the tensor perturbations $h_{ij}$ appear in the metric as

$$g_{\mu\nu}{\rm d}x^{\mu}{\rm d}x^{\nu}=-N^{2}{\rm d}t^{2}+a^{2}\left(\delta_{ij}+h_{ij}\right){\rm d}x^{i}{\rm d}x^{j},$$

(39)

with $h_{ii}=0=\partial_{j}h_{ij}$. Note that $X$ remains unperturbed under the tensor perturbations. Hence, performing the disformal transformation (3) on (39), we have

$$\bar{g}_{\mu\nu}{\rm d}x^{\mu}{\rm d}x^{\nu}=-\bar{N}^{2}{\rm d}t^{2}+\bar{a}^{2}\left(\delta_{ij}+h_{ij}\right){\rm d}x^{i}{\rm d}x^{j}.$$

(40)

This means that the tensor perturbations are disformally invariant, i.e.,

$$\bar{h}_{ij}=h_{ij},$$

(41)

which is consistent with the result in Minamitsuji (2021).

For Horndeski theories described by the action (20), the quadratic action for the tensor perturbations takes the form Kobayashi et al. (2011),

$\displaystyle S_{T}^{(2)}[h_{ij}]=\frac{1}{8}\int{\rm d}t{\rm d}^{3}x\,Na^{3}\left[\frac{\mathcal{G}_{T}}{N^{2}}\dot{h}_{ij}^{2}-\frac{\mathcal{F}_{T}}{a^{2}}(\partial_{k}h_{ij})^{2}\right],$

(42)

where we have defined

$$\begin{split}\mathcal{G}_{T}&\coloneqq 2\left[G_{4}-2XG_{4X}+\frac{X}{2}G_{5\phi}+H(-X)^{3/2}G_{5X}\right],\\ \mathcal{F}_{T}&\coloneqq 2\left(G_{4}-\frac{X}{2}G_{5\phi}+\frac{Y}{2}G_{5X}\right),\end{split}$$

(43)

with $H\coloneqq\dot{a}/(Na)$ being the Hubble parameter. So long as there is no tensor perturbation associated with the matter sector, one can use this quadratic action to study the linear dynamics of the tensor perturbations associated with the metric. Here and in what follows, coefficients in the quadratic action are evaluated at the background. Also, repeated spatial indices of perturbation variables are contracted by the Kronecker delta. From the above action, we find the squared sound speed for the tensor perturbations as

$$c_{T}^{2}=\frac{\mathcal{F}_{T}}{\mathcal{G}_{T}}.$$

(44)

We see that the tensor perturbations are free of ghost/gradient instabilities if

$$\mathcal{G}_{T}>0,\qquad\mathcal{F}_{T}>0.$$

(45)

The quadratic action (42) for Horndeski theories can be straightforwardly mapped to the one for GDH theories. More concretely, we first replace all the variables by barred ones and then translate it in the unbarred language by use of (38) and (41), i.e.,

	$\displaystyle\tilde{S}_{T}^{(2)}[h_{ij}]$	$\displaystyle=\frac{1}{8}\int{\rm d}t{\rm d}^{3}x\,\bar{N}\bar{a}^{3}\left[\frac{\bar{\mathcal{G}}_{T}}{\bar{N}^{2}}\dot{\bar{h}}_{ij}^{2}-\frac{\bar{\mathcal{F}}_{T}}{\bar{a}^{2}}(\partial_{k}\bar{h}_{ij})^{2}\right]$
		$\displaystyle=\frac{1}{8}\int{\rm d}t{\rm d}^{3}x\,Na^{3}\mathcal{J}\left[\frac{\tilde{\mathcal{G}}_{T}}{N^{2}}\dot{h}_{ij}^{2}-\frac{\tilde{\mathcal{F}}_{T}}{a^{2}}(\partial_{k}h_{ij})^{2}\right],$		(46)

where we have defined

$$\begin{split}\bar{\mathcal{G}}_{T}&\coloneqq 2\left[G_{4}-2\bar{X}G_{4X}+\frac{\bar{X}}{2}G_{5\phi}+\bar{H}(-\bar{X})^{3/2}G_{5X}\right],\\ \bar{\mathcal{F}}_{T}&\coloneqq 2\left(G_{4}-\frac{\bar{X}}{2}G_{5\phi}+\frac{\bar{Y}}{2}G_{5X}\right),\end{split}$$

(47)

with $\bar{H}\coloneqq\dot{\bar{a}}/(\bar{N}\bar{a})$ and

$$\tilde{\mathcal{G}}_{T}\coloneqq\frac{\bar{X}}{X}\bar{\mathcal{G}}_{T},\qquad\tilde{\mathcal{F}}_{T}\coloneqq\frac{\bar{\mathcal{F}}_{T}}{F_{0}}.$$

(48)

Note that the functions $G_{4}$ and $G_{5}$ as well as their derivatives are now evaluated at $(\phi,\bar{X})$. The squared sound speed is given by

$$\tilde{c}_{T}^{2}=\frac{\tilde{\mathcal{F}}_{T}}{\tilde{\mathcal{G}}_{T}}=\frac{X\bar{\mathcal{F}}_{T}}{\bar{X}F_{0}\bar{\mathcal{G}}_{T}}.$$

(49)

Also, the conditions for the absence of ghost/gradient instabilities are given by

$$\tilde{\mathcal{G}}_{T}>0,\qquad\tilde{\mathcal{F}}_{T}>0.$$

(50)

In what follows, we discuss issues on the speed of gravitational waves. First, it should be noted that we always assume the existence of matter field(s) which are minimally coupled, and hence light propagates at a unit speed in both the seed Horndeski and GDH theories. As mentioned earlier, the almost simultaneous detection of the gravitational-wave event GW170817 and the gamma-ray burst 170817A posed a tight constraint on the propagation speed of gravitational waves in the late-time universe, which motivates us to study theories where gravitational waves propagate at a unit speed. It is now well known that Horndeski theories with $G_{4X}=G_{5}=0$ satisfy $c_{T}^{2}=1$ regardless of the background evolution so long as we consider a homogeneous and isotropic background Creminelli and Vernizzi (2017); Ezquiaga and Zumalacárregui (2017); Baker et al. (2017). Indeed, we have $\mathcal{G}_{T}=\mathcal{F}_{T}=2G_{4}$ in this case, and hence $c_{T}^{2}=1$ is deduced from (44). A subclass of DHOST theories with $c_{T}^{2}=1$ is also isolated Langlois et al. (2018).

Here, we further enlarge the viable theory space by using the framework of GDH theories. Let us use Horndeski theories with $G_{5}=0$ (and not necessarily $G_{4X}=0$) as a seed, for which

$$\bar{\mathcal{G}}_{T}=2\left[G_{4}(\phi,\bar{X})-2\bar{X}G_{4X}(\phi,\bar{X})\right],\qquad\bar{\mathcal{F}}_{T}=2G_{4}(\phi,\bar{X}).$$

(51)

Note that both $\bar{\mathcal{G}}_{T}$ and $\bar{\mathcal{F}}_{T}$ can be translated into functions of $(\phi,X)$ since we are interested in the invertible subclass of the generalized disformal transformations for which $\bar{X}=\bar{X}(\phi,X)$. The reason why we set $G_{5}=0$ is to remove the term containing the scale factor $a$, whose time evolution depends on the matter sector. Then, the squared sound speed of tensor perturbations in (49) can be written as

$$\tilde{c}_{T}^{2}=\frac{XG_{4}(\phi,\bar{X})}{\bar{X}F_{0}\left[G_{4}(\phi,\bar{X})-2\bar{X}G_{4X}(\phi,\bar{X})\right]}.$$

(52)

Therefore, if we choose the conformal factor of the disformal transformation as^*6*6*6Since $W_{\rm BG}=Y_{\rm BG}^{2}-X_{\rm BG}Z_{\rm BG}=0$ on the FLRW background [see Eq. (36)], this requirement does not completely fix the functional form of $F_{0}(\phi,X,Y,Z)$. This ambiguity may be solved by studying a spatially inhomogeneous background (e.g., black hole solutions) where $W_{\rm BG}\neq 0$.

$$F_{0}=\frac{XG_{4}(\phi,\bar{X})}{\bar{X}\left[G_{4}(\phi,\bar{X})-2\bar{X}G_{4X}(\phi,\bar{X})\right]},$$

(53)

we can make $\tilde{c}_{T}^{2}=1$. This also means that one can map a Horndeski model with $c_{T}^{2}\neq 1$ to one with $\tilde{c}_{T}^{2}=1$ by generalized disformal transformation, which is a generalization of the results in Creminelli and Vernizzi (2017); Ezquiaga and Zumalacárregui (2017); Babichev et al. (2018). We note that the above expression for $F_{0}$ does not contain either $Y$ or $Z$ (though there could be an implicit dependence through $W$). As a result, $\mathcal{J}=F_{0}\mathcal{F}^{1/2}$ is also independent of $Y$ or $Z$ on the FLRW background, which can be verified by use of (38). As we shall see in §IV.1, this property leads to the consistency of matter coupling at the level of linear cosmological perturbations.