Luminal Scalar-Tensor theories for a not so dark Dark Energy

With the new era of multi-messenger astronomy, the initial impression was that a large class of Scalar-Tensor modifications of Einstein’s gravity is ruled out by the strict coincidence between the speed of Light ($c$) and Gravity ($c_{g}$). In particular, the almost simultaneous detection of the Gravitational Wave (GW) signal from the event GW170817 [1] and the gamma ray Burst GRB170817A [2, 3] placed the strong constraint,

$$\left|\frac{c_{g}}{c}-1\right|\leq 5\times 10^{-16}\,.$$

(1)

Nevertheless, it is clear that this constraint essentially indicates a relation between Gravity and Light. It, nevertheless, does not directly rule out modified gravity theories that could be relevant on cosmological scales, without assuming in first place something about light also on those scales. The standard approach is to take the following assumption:

1. (b)

   The Photon of Maxwell Electrodynamics (EM) remains minimally coupled even at the scales where General Relativity (GR) may need modification. Namely, $c=1$ even at the scales where the scalar of Dark Energy dominates the expansion of the universe,

thus, we see $c_{g}=c=1$. However, gravity couples universally to all matter, and in principle, one could also explore an alternative assumption to (b), where the scalar modification of gravity at cosmological scales shares this universal coupling property, e.g.:

1. (a)

   The scalar of Dark Energy (DE) couples to both, the Graviton and the Photon in a specific way, such that we see the luminality of GWs

$$\frac{c_{g}(t)}{c(t)}=1\,.$$

(2)

The assumption (b) conveniently fits EM at all scales, mainly to be consistent with laboratory experiments. However, it also quickly forbids additional input to more objectively constrain modified gravity theories.

The assumption (a) — which we take in this letter, and that clearly contains (b) as a particular case — opens a new set of observational possibilities: If DE is not really dark and also couples to the Photon, new types of laboratory and astrophysical tests are required [4].

Furthermore, (a) re-opens the path to non-minimally coupled theories for DE that were previously thought to be ruled out [5, 6, 7, 8, 9, 10, 11]. Interestingly, non-minimal couplings may be relevant in the wake of the recent DESI BAO data, favoring dynamical DE [12]. To that end a Horndeski theory with non-minimal couplings of the scalar to gravity would be necessary to safely cross the phantom divide [13, 14]. Although these possibilities are not conclusive [15], the theories shown in this letter — a broad generalization of the theories used in [13, 14]— open new opportunities for the cosmologist.

We consider the most general Degenerate Higher-Order Scalar-Tensor modifications of gravity (DHOST) that have been classified to date, which are by construction free of Ostragradky ghosts [16, 17, 18, 19, 20]. We deduce the DE–Photon couplings that are necessary for the observed luminality of GWs in these DHOST (2) — with Horndeski and Beyond Horndeski as particular cases. We find that only two types of DE–Photon couplings are necessary. One of them cannot be removed by a conformal/ disformal transformation of the metric. It is involved in a new Luminal Beyond Horndeski (BH) theory that we show below, for which the GW decay to DE is suppressed. Altogether passing the strong constraints on both the Luminality [6, 7, 8, 9, 10, 11] and non-decay of GW [21].

The model: In the usual parameterization, we consider 19 potentials depending on a scalar field $\pi$. They generalize the Einstein-Hilbert action in four dimensions (4D) with minimal and non-minimal couplings of $\pi$ to gravity. Let us denote the 19 scalar potentials as $a_{i},\,b_{j},\,f_{k},\,G_{k}$ with $i=1\,\dots 5,\,j=1\,\dots 10$ and $k=2,3$. In principle, we allow all of these potentials to be functions of a scalar field $\pi$ and $X=\pi_{\mu}\,\pi^{\mu}$, where $\pi_{\mu}=\nabla_{\mu}\pi$. However, some of these potentials are not free. There are relations among them in order to not propagate the Ostrogradsky ghost. These relations, known as degeneracy conditions, separate the theory space of DHOST into distinct classes. Thus, in all the theories we consider there are always less than 19 free scalar potentials of $\pi$ and $X$, with the specific number of free functions depending on the class. For instance in Horndeski theory there are up to $4$ free functions [22, 23]. A complete classification with the number of free functions, and properties is given in [17, 18, 19]. Below we only give the degeneracy conditions for the most physically relevant cases.

The Lagrangian is written as

L_DHOST_π=f_2 R+f_3 G_μν  π^μν+L_Linear+L_Quad+L_Cubic ,

(3)

where $R$ is the Ricci scalar, $G_{\mu\nu}$ is the Einstein tensor and $\nabla$ is the covariant derivative computed with the ambient metric of the $D$-dimensional manifold (of signature $-,+,+,+,\,\dots$), and $\mu=0,1,\,\dots\,D-1$. The main results in this letter will be in the usual $D=4$, however, as we explain latter on, they are most easily derived starting from $D=5$, as we will explicitly state when needed.

The last three terms in (3) contain Lorentz invariant combinations of, respectively, (up-to) linear, quadratic and cubic in $\nabla^{2}\pi$ terms. Explicitly,

$$\mathcal{L}_{\text{Linear}}=G_{2}+G_{3}\,\Box\pi$$

(4)

$$\mathcal{L}_{\text{Quad}}=\sum_{i=1}^{5}\,a_{i}(\pi,X)\,L_{i}^{(2)}\,,$$

(5)

where $L_{i}^{(2)}$ are of order $(\nabla^{2}\pi)^{2}$,

$L_{1}^{(2)}=(\pi_{\mu\nu})^{2}\,,$

$L_{2}^{(2)}=(\Box\pi)^{2}\,,$

$L_{3}^{(2)}=\Box\pi\,(\pi_{\mu\nu}\pi^{\mu}\pi^{\nu})\,,$

$$L_{4}^{(2)}=(\pi_{\mu\rho}\pi^{\mu})^{2}\,,\,\,L_{5}^{(2)}=(\pi_{\mu\nu}\pi^{\mu}\pi^{\nu})^{2}\,,$$

(6)

and

$$\mathcal{L}_{\text{Cubic}}=\sum_{j=1}^{10}\,b_{j}(\pi,X)\,L_{j}^{(3)}$$

(7)

where $L_{j}^{(3)}$ are of order $(\nabla^{2}\pi)^{3}$,

$L_{1}^{(3)}=(\Box\pi)^{3}\,,$

$L_{2}^{(3)}=\Box\pi(\pi_{\mu\nu})^{2}\,,$

$L_{3}^{(3)}=(\pi_{\mu\nu})^{3}\,,$

$L_{4}^{(3)}=(\Box\pi)^{2}(\pi_{\mu\nu}\pi^{\mu}\pi^{\nu})\,,$	$L_{5}^{(3)}=\Box\pi\,(\pi_{\mu\nu}\pi^{\mu})^{2}\,,$
$L_{6}^{(3)}=(\pi_{\rho\sigma})^{2}\,(\pi_{\mu\nu}\pi^{\mu}\pi^{\nu})\,,$	$L_{7}^{(3)}=\pi^{\mu\nu}\pi_{\nu\rho}\pi^{\rho\sigma}\pi_{\mu}\pi_{\sigma}\,,$
$L_{8}^{(3)}=(\pi^{\mu\nu}\pi_{\mu})^{2}(\pi^{\rho\sigma}\pi_{\rho}\pi_{\sigma})\,,$	$L_{9}^{(3)}=\Box\pi(\pi^{\rho\sigma}\pi_{\rho}\pi_{\sigma})^{2}\,,$

$$L_{10}^{(3)}=(\pi^{\rho\sigma}\pi_{\rho}\pi_{\sigma})^{3}$$

(8)

It was thought that a large set of DHOST theories – including Horndeski and Beyond Horndeski [24, 25] – are constrained to some extent in order to satisfy (2). As we noted, this belief assumed (b). In this letter, however, we work on the hypothesis (a), and thus, to $\mathcal{L}_{\text{DHOST}_{\pi}}$ we must add the precise Scalar of DE–Photon couplings such that we see the luminality of GWs (2). It was initially shown in [5] that a simple way to obtain them in 4D is to start from a 5 Dimensional (D) setup: thus, consider the action of DHOST only for a brief moment in 5D,

$$\int\sqrt{-{}^{(5)}g}\,d^{5}\text{x}\,\mathcal{L}_{\text{DHOST}_{\pi}}\,.$$

(9)

Writing the 5D metric ${}^{(5)}g$ as

$${}^{(5)}g_{B\,C}=\left(\begin{array}[]{cc}g_{\mu\,\nu}+\,A_{\mu}\,A_{\nu}&\,A_{\mu}\\ \,A_{\nu}&1\end{array}\right)\,,$$

(10)

where the latin indices are $B=0,\,\dots\,4$ and greek $\mu=0,\,\dots\,3$. Seen simply as a tool for our purpose in 4D, we compactify the 5th dimension with Kaluza’s cylinder condition [26], where we have assumed right away in Eqn. (10) a constant Dilaton, and such that the 4D fields $g$ and $A_{\mu}$ do not depend on the 5-th dimension. We further rescale the 4D fields to re-absorb the $\int d\text{x}^{4}$, and thus we rewrite (9) in terms of 4D fields only.

All in all, after compactification, the theory (9) takes the form of the usual DHOST plus a Scalar–Photon sector in 4D (11). As usual the $U(1)$ gauge invariance in the vector sector is inherited from diffeomorphisms in 5D.

It is clear that because the 4-vector $A_{\mu}$ and the 4D metric are just but components of the same metric in 5D, their speed in 4D is generally bound to be the same. The caveat is that we have broken isotropy in 5D by compactifying one spatial dimension and ignoring1 the dynamics of a Dilaton. Thus, there are special cases with unequal speeds which we single out below.

DHOST with Dark Energy–Photon couplings: From now on in 4D, the complete DHOST action with DE–Photon couplings reads,

$\displaystyle\int\sqrt{-g}\,d^{4}\text{x}\left(\mathcal{L}_{\text{DHOST}_{\pi}}+\mathcal{L}_{\text{DHOST}_{A}}\right)\,,$

(11)

with $\mathcal{L}_{\text{DHOST}_{\pi}}$ given in (3). The DE–Photon sector is,

	$\displaystyle\mathcal{L}_{\text{DHOST}_{A}}$	$\displaystyle=$	$\displaystyle\frac{f_{3}}{8}\left(4F_{\mu\nu}\nabla_{\rho}F^{\nu\rho}\pi^{\mu}+F^{2}\Box\pi-4F_{\mu}{}^{\nu}F^{\mu\rho}\pi_{\nu\rho}\right)$		(12)
		$\displaystyle-$	$\displaystyle\frac{f_{2}}{4}\,F^{2}+l_{\text{Quad}_{A}}+l_{\text{Cubic}_{A}}\,,$

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, and with obvious notation parallel to (5) and (7),

	$\displaystyle l_{\text{Quad}_{A}}$	$\displaystyle=$	$\displaystyle a_{1}(\pi,X)l_{1}^{(2)}$		(13)
	$\displaystyle l_{\text{Cubic}_{A}}$	$\displaystyle=$	$\displaystyle\sum_{j=\{2,3,6\}}b_{j}(\pi,X)l_{j}^{(3)}\,,$		(14)

where $l_{1}^{(2)}=l_{j}^{(3)}=\frac{1}{2}(F_{\mu\nu}\pi^{\mu})^{2}$ for $j=2,6$, and

$$l_{3}^{(3)}=\frac{3}{4}F_{\mu\nu}F_{\rho\sigma}\pi^{\mu\rho}\pi^{\nu}\pi^{\sigma}\,.$$

(15)

One identifies in principle three types of DE–Photon couplings $F^{2}\,\nabla^{2}\pi,\,F^{2}\,(\nabla\pi)^{2}$ and $F^{2}\,\nabla^{2}\pi\,(\nabla\pi)^{2}$. However, the latter — proportional to $b_{3}$ — will be removed below by the Luminality condition (2). It is essential to note that the DE–Photon couplings $f_{3}\,F^{2}\nabla^{2}\pi$ cannot be removed by a conformal/ disformal transformation of the metric that depends on up to first derivatives of $\pi$. Namely, one cannot obtain the $f_{3}$ DE–Photon couplings by such metric redefinition in the Maxwell term $-1/4\,F^{2}$.

In short, for each of the 6 contributions to $\mathcal{L}_{\text{DHOST}_{\pi}}$ labeled by the scalar potentials $f_{2},\,f_{3},\,a_{1},\,b_{2},\,b_{3},\,b_{6}$ there is a corresponding DE–Photon sector in Eqn. (12).

Luminal DHOST around the corner: As declared, not all Lagrangians in Eqn. (11) propagate gravitational and electromagnetic waves at the same speed. The approach below is to compute the Graviton and Photon speeds on a cosmological background for the action (11) and find the Lagrangians that can satisfy Eqn. (2). Note that previous cases suggest that the results below could also hold on (at least) spherically symmetric backgrounds [27, 28].

The scalar mode of DHOST is not modified by the new terms $\mathcal{L}_{\text{DHOST}_{A}}$ on the cosmological background. Thus we do not discuss any further the scalar sector in this letter. Furthermore, we will assume the DHOST classes that actually propagate a graviton [16, 17, 18, 19, 20].

We consider first order perturbations on a spatially flat FLRW background. With the perturbed metric $\text{d}s^{2}=(\eta_{\mu\nu}+\delta g_{\mu\nu})\text{d}x^{\mu}\text{d}x^{\nu}$ where $\eta_{\mu\nu}=-\text{d}t^{2}+a(t)^{2}\,\delta_{ij}\text{d}x^{i}\text{d}x^{j}$, we write only the symmetric, traceless and transverse tensor perturbation $h_{ij}$ and the two transverse vector perturbations $S_{i},\,F_{i}$ as,

$$\delta g=\left(2\,S_{i}\textrm{d}t\,\textrm{d}x^{i}+\left(\partial_{i}F_{j}+\partial_{j}F_{i}+2\,h_{ij}\right)\textrm{d}x^{i}\,\textrm{d}x^{j}\right)\,,$$

(16)

where we denote spatial indices with lowercase latin indices, $i=1,2,3$. The perturbed DHOST scalar $\pi(x^{\mu})$ is written as $\pi(t)+\chi(t,\vec{x})$ in the linearized expressions, within which $\pi(t)$ is the background scalar field. Finally, on the cosmological medium the photon amounts to the transverse perturbation $A_{i}(t,\vec{x})$, with vanishing background due to isotropy (and using background equations).

The quadratic action for the graviton reads,

$$\mathcal{S}_{Tensor}=\frac{1}{2}\int\text{d}t\,\text{d}^{3}x\,a^{3}\,\left(\mathcal{G}_{\tau}\,\dot{h}_{ij}^{2}-\frac{\mathcal{F}_{\tau}}{a^{2}}\,(\partial_{k}h_{ij})^{2}\right)\,,$$

(17)

with $f_{3,X}=\frac{\partial f_{3}}{\partial X}$ and so on,

	$\displaystyle\mathcal{G}_{\tau}$	$\displaystyle=$	$\displaystyle 2f_{2}+2\ddot{\pi}Xf_{3,X}-Xf_{3,\pi}-2Xa_{1}$
		$\displaystyle+$	$\displaystyle 2X(3\dot{\pi}H+\ddot{\pi})b_{2}+6\dot{\pi}XHb_{3}+2\ddot{\pi}X^{2}b_{6}\,,$
	$\displaystyle\mathcal{F}_{\tau}$	$\displaystyle=$	$\displaystyle 2f_{2}-2\ddot{\pi}Xf_{3,X}+Xf_{3,\pi}\,,$		(19)

while the action for the Photon is written as,

$$\mathcal{S}_{Vector}=\frac{1}{4}\int\text{d}t\,\text{d}^{3}x\,a\,\left(\mathcal{G}_{A}\,\dot{A}_{i}^{2}-\frac{\mathcal{F}_{A}}{a^{2}}\,(\partial_{k}A_{i})^{2}\right)\,,$$

(20)

where,

	$\displaystyle\mathcal{G}_{A}$	$\displaystyle=$	$\displaystyle\mathcal{G}_{\tau}-3\dot{\pi}XHb_{3}$		(21)
	$\displaystyle\mathcal{F}_{A}$	$\displaystyle=$	$\displaystyle\mathcal{F}_{\tau}\,.$		(22)

As expected, by construction, the coefficients in the quadratic actions are similar, e.g. as in (22). Now, with their speeds squared, respectively, $c_{g}^{2}=\frac{\mathcal{F}_{\tau}}{\mathcal{G}_{\tau}}$ and $c^{2}=\frac{\mathcal{F}_{A}}{\mathcal{G}_{A}}$, we find their ratio,

$$\frac{c_{g}^{2}}{c^{2}}=1-3\frac{\dot{\pi}XH\,b_{3}}{\mathcal{G}_{\tau}}\,.$$

(23)

Thus, in principle, the DHOST theories with

$$b_{3}=0\,,$$

(24)

would preserve the unit ratio of speeds (2). However, let us recall that depending on the degenerate class of DHOST being considered, the scalar potential $b_{3}$ may not be a free function but it may be fixed by the also crucial degeneracy conditions¹¹1If we had considered the Dilaton, with background $\Phi(t)$, then $\frac{c_{g}^{2}}{c^{2}}=1-3\frac{\dot{\pi}X(H-\frac{\dot{\Phi}}{\Phi})\Phi\,b_{3}}{\mathcal{G}_{\tau}}$. Restoring isotropy $\dot{\Phi}=\dot{a}$, we would see that (2) always holds. Although this choice is unphysical, this is at the very least a cross-check of our results. See [29] Section VC for a discussion. .

Degenerate and Luminal DHOST: thus, to apply the luminality condition (24) in DHOST, one is left with the task of establishing whether it is consistent with the degeneracy condition of the class. From the comprehensive classification in [18] Table 1 and [17] it is clear that there are many²²2A counterexample is the full mixed quadratic plus cubic BH [24, 25]. Let us see: the degeneracy condition is (28). While $b_{3}=0$ implies $F_{5}=-\frac{G_{5,X}}{3X}$. Assuming $F_{5},\,G_{5,X}\neq 0$ one finds from (28) a relation $F_{4}(G_{4},\,G_{4,X},\,G_{5,\pi})$ that sets $\mathcal{G}_{\tau}=\mathcal{G}_{A}=0$, which is a singular case with no Graviton and no Photon. However, note that the branch $F_{5}=G_{5,X}=0$ escapes the problem, because (28) is automatically satisfied with a totally free $F_{4}$. See the discussion below in the case (ii). Another counterexample is only cubic, full DHOST ${}^{3}\text{N-I}$, which contains cubic Horndeski and BH: as noted in [17] in this class $b_{3}=2b_{1}$, with $b_{1}$ free, up to the condition $b_{1}\neq 0$. Thus in this class (24) cannot be met. If we nevertheless take $b_{1}=0$, then we would be forced in another degenerate class, DHOST ${}^{3}\text{N-II}$ [17], which however has no graviton [18]. Scalar-Tensor theories with a graviton that can be made Luminal with Eqn. (24). We will focus, however, on the phenomenologically most relevant classes. The simplest successful case is:

1. (i)

   Every quadratic DHOST with the corresponding DE–Photon couplings, $\mathcal{L}_{\text{DHOST}_{A}}=-\frac{f_{2}}{4}\,F^{2}+\frac{a_{1}}{2}(F_{\mu\nu}\pi^{\mu})^{2}$, and with a graviton, satisfies Eqn. (2). Namely, the action (11) with $b_{i}=0$ and $f_{3}=0$ has luminal GWs. The degeneracy conditions on some of the functions $a_{i}$, for multiple classes of theories, are given for instance in [17] App. C.

In particular, (i) includes quadratic Horndeski and Beyond Horndeski theory ($\text{BH}_{4}$) as special cases. The latter is written with the action (11) and with the following degeneracy relations [16, 19],

$$f_{2}=G_{4},\,a_{1}=-a_{2}=2G_{4,X}+XF_{4},\,$$

$$a_{3}=-a_{4}=2F_{4},\,a_{5}=0\,.$$

(25)

From (12) and (25) the DE–Photon couplings that make luminal the quadratic BH theory are $\mathcal{L}_{BH_{4A}}$ in Eqn. (26), from which we recover $\mathcal{L}_{4A}$ given in [5] in the particular case $F_{4}=0$. The theory (i) also includes as a luminal class, for instance, the DHOST ${}^{2}\text{N-III/ IIa}$, which may still be phenomenologically relevant [18], yet disconnected from the Horndeski class.

Another successfully Luminal case is in the mixed quadratic plus cubic DHOST class:

1. (ii)

   The Quadratic Beyond Horndeski (BH) plus Cubic Horndeski theory, with $f_{3}=G_{5}(\pi)$ and with the corresponding DE-Photon couplings propagates Luminal GWs. Namely, the action (11) with the relations (25), and with $b_{j}=0$ with $j=1,\dots 10$. Explicitly,

   $$\int\text{d}^{4}x\big{(}\,\mathcal{L}_{BH_{4\pi}}+\mathcal{L}_{BH_{4A}}+\mathcal{L}_{H_{5\pi}}+\mathcal{L}_{H_{5A}}\big{)}\,,$$

   (26)

   with

   	$\displaystyle\mathcal{L}_{BH_{4\pi}}=G_{2}+G_{3}\Box\pi+G_{4}R-2G_{4,X}((\Box\pi)^{2}-\pi_{\mu\nu}^{2})$
   	$\displaystyle-F_{4}\Big{(}X(\Box\pi)^{2}-X\pi_{\mu\nu}^{2}+2(\pi_{\mu\nu}\pi^{\mu})^{2}-2\Box\pi\pi_{\mu\nu}\pi^{\mu}\pi^{\nu}\Big{)}$
   	$\displaystyle\mathcal{L}_{\text{BH}_{4A}}=-\frac{G_{4}}{4}\,F^{2}+\frac{2G_{4,X}+XF_{4}}{2}(F_{\mu\nu}\pi^{\mu})^{2}\,,$
   	$\displaystyle\mathcal{L}_{H_{5\pi}}=G_{5}\,G^{\mu\nu}\pi_{\mu\nu}$		(27)
   	$\displaystyle\mathcal{L}_{H_{5A}}=\frac{G_{5}}{8}\left(4F_{\mu\nu}\nabla_{\rho}F^{\nu\rho}\pi^{\mu}+F^{2}\Box\pi-4F_{\mu}{}^{\nu}F^{\mu\rho}\pi_{\nu\rho}\right)\,.$

   $G_{5}$ is a function of $\pi$ only, and we have taken $F_{5}=0$ (in the standard notation of BH [19]).

The theory (ii) is an apparently mild generalization of the quadratic plus cubic Luminal Horndeski theory that was shown in [5]. However, it is an essential one: namely, with this new theory it becomes possible to suppress the GWs decay to the scalar of DE, by fixing the newly free potential $F_{4}$. Let us see how: First note that $b_{3}=\frac{2}{3}(G_{5,X}+3XF_{5})=0$ is satisfied. Then Eqn. (2) follows; that is, in the theory (ii) the GWs are automatically Luminal without fixing any of the scalar potentials. Secondly, this theory is free of Ostrogradsky ghosts: the degeneracy condition³³3We take sign convention for $F_{4},\,F_{5}$ from [16, 19]. Note however, the opposite sign for $F_{4}$ taken in [10, 21, 30] in mixed quadratic plus cubic BH

$\displaystyle F_{4}G_{5,X}X=-3F_{5}\Big{(}G_{4}-2XG_{4,X}-\frac{X}{2}G_{5,\pi}\Big{)},\,$

(28)

is also automatically satisfied by $G_{5,X}=F_{5}=0$. And, contrary to the full Quadratic plus Cubic BH2 with $F_{5}=-\frac{G_{5,X}}{3X}\neq 0$ which also sets $b_{3}=0$, there are tensor and vector modes. Thirdly, it can be easily checked that $\mathcal{L}_{H_{5A}}$ is a vector-scalar Galileon term. Namely, it is of higher order in the Lagrangian but it has second order equations of motion. Again, no ghosts.

The essential aspect in the theory (ii) is that it has just the necessary amount of freedom, such that in a subclass within it, the GWs decay to DE may be suppressed: Indeed, in [21] it was shown that — in the case when $c_{g}(t)\neq 1$ — the following expression should be negligible (See Eqn. (87) in [21]), because if the GWs had considerably decayed to DE, we would have not observed them in first place:

F_4(4 G_4+X(2 G_4,X+3 G_5,π) )+X F_4,X(2 G_4 +X G_5,π )

(29)

+4 G_4,X^2+4 G_4 G_4,XX+G_5,π(4 G_4,X+2X G_4,XX+G_5,π)=0

(30)

where we have already used $G_{5,X}=F_{5}=0$ from the definition of (ii). Note that this constraint is independent of $H$ and $\ddot{\pi}$, thus also independent of the matter content.

As all scalar potentials — in particular $F_{4}(\pi,X)$ — remain free, there are theories in (ii) for which (30) is satisfied and the GWs decay to DE is suppressed. That is, since Eqn (30) is linear in $F_{4}$, it has solution [30]

$$F_{4}=\frac{1}{2X^{2}}\Big{(}2G_{4}-X(4G_{4,X}+G_{5,\pi})+\frac{4J_{4}(\pi)}{2G_{4}+XG_{5,\pi}}\Big{)},\,$$

(31)

where $J_{4}(\pi)$ is an integration "constant" (with respect to $X$). Notice two essential points to this conclusion: first, the DE–Photon couplings (i). They keep free the $F_{4}(\pi,X)$ function, while also keeping GWs luminal⁴⁴4Note this critical difference to Beyond Horndeski without the DE–Photon couplings $\mathcal{L}_{\text{BH}_{4A}},\,\mathcal{L}_{\text{H}_{5A}}$: in that case $F_{4}$ is not free to suppress the decay, because $F_{4}=\frac{-2G_{4,X}}{X}$ and $G_{5}=0$ are already fixed to preserve luminal GWs [6, 7, 8, 9, 10, 11].. Thus we can solve $F_{4}$ as (31). Secondly, the fact that the precise⁵⁵5Let us note that in [30] a similar looking Lagrangian to $\mathcal{L}_{\text{H}_{5A}}$ was considered with the aim to suppress the GWs decay while keeping their luminality, $\mathcal{L}_{SVT}^{(3)}\propto\bar{g}_{\alpha\beta}(g,\,\pi,\nabla\pi)\tilde{F}^{\mu\alpha}\tilde{F}^{\nu\beta}\pi_{\mu\nu}$, with $\tilde{F}$ the dual of $F$. However, $\mathcal{L}_{SVT}^{(3)}$ and $\mathcal{L}_{\text{H}_{5A}}$ are fundamentally different. Their quadratic Lagrangians and thus, their vector speeds are related in a matter dependent way, through combinations of $H,\,\ddot{\pi}$. Thus, in accordance with [30] it is not possible to find a matter independent solution to (30) and (2) with $\mathcal{L}_{SVT}^{(3)}$. $f_{3}F^{2}\nabla^{2}\pi$ couplings in $\mathcal{L}_{\text{H}_{5A}}$ cannot be removed by a conformal/ disformal transformation of the metric. This is significant: the disformal invariance of the decay, which was proven in [21], and used in the argument in [30], does not apply to this case.

Similarly, to the best of our knowledge, the GWs decay constraint [21] has only been computed for the BH theory. In particular, additional checks would be needed to rule out the full theory (i) shown above, which includes quadratic BH only as a particular case.

Conclusions: We have shown that 5 sets of self-consistent Dark Energy–Photon couplings are enough to render luminal the GWs in all DHOST theories (with a graviton) that first, are up to cubic in $\nabla^{2}\pi$, and second, whose degeneracy conditions are compatible with the sole condition $b_{3}(\pi,X)=0$.

For the cosmologist this means: The Scalar-Tensor theories with $b_{3}=0$ — such as the Beyond Horndeski theory (ii) Eqn. (26) or (i) — may be potentially used with minor consideration of the graviton speed, because DE—Photon couplings exist that can take care of the luminality of GWs and the experimental bound (1). Naturally, experimental constraints would be necessary on the DE—Photon couplings. Indeed, laboratory and astrophysical constraints have been already put on at least the disformal set of DE–Photon couplings shown in this letter [4] (See also [5, 30] for a discussion).

We showed at least one theory — a subclass of Luminal Beyond Horndeski — in which the decay of GWs to DE is suppressed on a cosmological background. This is relevant because such background is a good description in the bulk of the trajectory of GWs to Earth. We stressed that the essential type of DE–Photon coupling cannot be removed by conformal/ disformal transformation and thus the disformal invariance of the decay — which was proven in [21] — does not apply to this case.

We also showed some cases of BH and DHOST2 that remain ruled out by the bound (1), as they have no consistent DE–Photon coupling.

MVV is thankful to S. Ramazanov for valuable discussions. The work on this project has been supported by Russian Science Foundation grant № 24-72-10110,

https://rscf.ru/project/24-72-10110/.