IFT-UAM/CSIC-20-136

Scalar and tensor gravitational waves

Charles Dalang,1, ∗ Pierre Fleury,2, † and Lucas Lombriser1, ‡1Département de Physique Théorique, Université de Genève,

24 quai Ernest-Ansermet, 1211 Genève 4, Switzerland2Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid,

Cantoblanco, 28049 Madrid, Spain(Dated: April 7, 2021)

In dark-energy models where a scalar field is nonminimally coupled to the spacetime geometry, gravitationalwaves are expected to be supplemented with a scalar mode. Such scalar waves may interact with the standardtensor waves, thereby affecting their observed amplitude and polarization. Understanding the role of scalarwaves is thus essential in order to design reliable gravitational-wave probes of dark energy and gravity beyondgeneral relativity. In this article, we thoroughly investigate the propagation of scalar and tensor waves in thesubset of Horndeski theories in which tensor waves propagate at the speed of light. We work at linear orderin scalar and metric perturbations, in the eikonal regime, and for arbitrary scalar and spacetime backgrounds.We diagonalize the system of equations of motion and identify the physical tensor mode, which differs fromthe metric perturbation. We find that interactions between scalar and tensor waves generally depend on thescalar propagation speed. If the scalar waves are luminal or quasiluminal, then interactions are negligible. In thesubluminal case, scalar-tensor interactions are effectively suppressed due to the incoherence of the wave’s phases.

I. INTRODUCTION

After more than a hundred years of general relativity (here-after GR), deviations arising from its most popular alter-

natives remain elusive to observations [1]. While it is alwaysenough motivation to challenge the currently accepted theory,one should better know where deviations from appealing alter-natives may appear. Promising candidates have emerged in thepast two decades in light of cosmic acceleration [2, 3], which iswell described by GR with a cosmological constant but whichis poorly understood from a theoretical point of view [4].Of these viable alternatives, Horndeski theories form a

natural extension of GR, easy to study and featuring an extrascalar propagating degree of freedom, which will eventuallybe the main focus of this article. They form the most generalfour-dimensional Lorentz invariant set of scalar-tensor theoriesthat lead to second-order equations of motion [5], therebyavoiding Ostrogradski instabilities [6]. The idea of scalar fieldsthat would mix with gravitational degrees of freedom naturallyappears in the low-energy effective action of string theoriesor may emerge as a manifestation of Poincaré invariance inhigher dimensions [7, 8]. Alternatively, Horndeski theoriesmay be motivated as a generic phenomenological attempt tomodel cosmic acceleration periods such as inflation [9] ordark energy [10]. However, late-time deviations from GR areseverely constrained by local tests of gravity such as from lunarlaser ranging [11] and Shapiro time delay [12].In that context, an attractive feature of Horndeski theo-

ries is their ability to recover GR locally through screeningmechanisms. Well-known examples include the Vainshteinmechanism [13], the chameleon scenario [14], k-mouflage [15],and see also Refs. [16, 17] for more exotic scenarios.1 In

∗ charles.dalang@unige.ch† pierre.fleury@uam.es‡ lucas.lombriser@unige.ch1 SeeRef. [18] for recentmitigations concerning the effectiveness ofVainshtein

principle, this allows certain classes of Horndeski theoriesto escape local tests of gravity, but typically at the price ofno longer explaining the cosmic acceleration directly throughsufficiently large modified gravitational interactions [19].If the deviations from GR are screened in regions where

typical gravitation experiments are conducted, then the mostpromising area to look for them is the intergalactic medium.Gravitational waves (GWs) are a key candidate for that purpose,because they probe those inaccessible regions as they propagatethrough the Universe, as depicted in Fig. 1. The foremost exam-ple is the almost-simultaneous detection of GW170817 [20] andGRB 170817A [21], which has imposed stringent constraintson alternative theories of gravity. In practice, it has eliminatedall theories that predict a deviation from luminal propagationof GWs [22–24] for sources at reshift 𝑧 . 0.01. In particular, ithas posed severe challenges to a genuine explanation of cosmicacceleration frommodified gravity [25].2 Another consequenceof the interaction between GWs and dark energy is the possibledecay of the former into the latter, which practically rules outdegenerate higher-order scalar-tensor theories (DHOST) asviable explanations of dark energy, and sets an upper limit onthe kinetic-braiding parameter [28, 29].Besides the GW speed and decay rate, constraints may also

be obtained from the GW distance-redshift relation via theobservation of standard sirens [30, 31]. While the gravitationaland electromagnetic Hubble diagrams coincide in GR, theygenerally do not in alternative theories of gravity; in otherwords, 𝐷L (𝑧) ≠ 𝐷G (𝑧), where 𝐷L, 𝐷G respectively denote theluminosity distance (measured with electromagnetic signals)and the gravitational distance (measured with standard sirens).Such a discrepancy has been envisaged as a promising probe of

screening in light of a UV completion.2 As a caveat, let us mention that this argument may not directly be applicableto perturbations describing the large-scale structure of the Universe asthe energy scales involved in current GW experiments lie many orders ofmagnitude above cosmological scales [26, 27].

arX

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the cosmic evolution of the effective Planck mass [22, 32–35]and of the spatial clustering of dark energy [36].The difference between 𝐷G and 𝐷L is usually interpreted in

terms of an extra ‘friction’ which non-GR fields would exerton GWs. However, as initially suspected in Ref. [32, 37] anddemonstrated in Ref. [38], in the most popular models theratio 𝐷G/𝐷L only depends on local properties of gravity at theemission and reception of the GW. Thus, the prospects of anyprogram based on the difference between 𝐷G and 𝐷L shouldbe quite limited by screening.3 This point has been mostlyoverlooked in the literature dedicated to standard-siren tests ofgravity (see Ref. [42] for a recent proposition to exploit it).To be specific, Ref. [38] showed that, in Horndeski theories

for which GWs propagate at the speed of light, 𝐷G/𝐷L =

𝑀 (𝜑o)/𝑀 (𝜑s), where 𝑀 denotes the effective Planck massand 𝜑s, 𝜑o the dark-energy field at emission and observationof the GW. This turned out to be the only difference withGR. In particular, GWs still propagate along null geodesics ofthe background spacetime, and their polarization is parallel-transported.4Be that as it may, an important assumption in Ref. [38] was

to neglect scalar waves. It is thus natural to wonder whetherthe aforementioned results – distance formula, polarizationtransport – hold when scalar waves are properly accountedfor. Interactions between scalar and tensor waves are expectedto arise in a realistic, inhomogeneous Universe which doesnot enjoy the exact symmetries of the idealized Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime.5 Scalar-tensorinteractions may lead to energy transfers between the twosectors, thereby affecting, e.g., the observed 𝐷G. This questionis particularly relevant if GWs are lensed, or propagate throughnonlinear structures where screening may affect the scalarsector. Even if we could describe the background for GWs asan FLRW model with small perturbations, the accumulatedimpact of the latter over cosmic distances could still lead tosignificant effects. Any analysis of such setups that does notaccount for scalar waves is thus a priori inconsistent. Thisarticle aims to complete this gap, by proposing a joint analysisof both scalar and metric waves in Horndeski theories, withinan arbitrary scalar and spacetime background.The article is organized as follows. In Sec. II, after a brief

introduction to the reduced Horndeski models, we linearize itsequations of motion for both scalar and metric perturbationsaround an arbitrary background. In Sec. III, we focus ouranalysis on wavelike perturbations, we derive the dispersion

3 It should be noted that local experiments such as lunar laser ranging orShapiro time delay do not constrain directly the local effective Planck mass,but rather the effective Newton constant appearing in the Poisson equation,or the gravitational slip. These couplings may differ in general [39, 40].However, for known screening mechanisms in Horndeski gravity such as theVainshtein and chameleon scenarios, these couplings coincide in the deeplyscreened regime, leading to the absence of signature of the scalar field onthe gravitational luminosity distance with respect to GR [37, 41].

4 Note that Ref. [43] reached a different conclusion on the polarizationtransport; See Appendix E of Ref. [38] for an analysis of that discrepancy.

5 If scalar and tensor waves are treated as perturbations on an FLRW back-ground, the standard scalar-vector-tensor decomposition [44] implies thatthey involve independently at linear level; they do not at second order.

observer

GW source

screened

unscreen

ed

Figure 1. From its emission to its detection on Earth, a GW propagatesthrough unscreened regions of the Universe, where it may interactwith the scalar field that models dark energy. Note that this sketch isnot to scale.

relations, andwe identify the signatures of the three propagatingdegrees of freedom. In Sec. IV, we show that the interactionsbetween scalar and tensor GWs are negligible for luminal andquasiluminal scalar waves, defining a notion of scalar distancein the process. In the subluminal case, we discuss how scalar-tensor interactions may be suppressed due to the incoherenceof the phases of the scalar and tensor waves. Finally, wesummarize our results and conclude in Sec. V.

We adopt the Misner-Thorne-Wheeler conventions [45] forthe metric signature and the Riemann tensor. Greek indicesrun from 0 to 3, Latin indices from 1 to 3. A comma indicatesa partial derivative, 𝑍,` ≡ 𝜕`𝑍 , while a semicolon denotes acovariant derivative associated with the Levi-Civita connection,𝑍`;a ≡ ∇a𝑍`. Bold symbols represent Euclidean three-vectors;sans-serif symbols indicate matrices in the scalar-tensor fieldspace. Symmetrization and anti-symmetrization of indicesfollow 𝑍 (`a) ≡ 1

2 (𝑍`a + 𝑍a`) and 𝑍 [`a ] ≡ 12 (𝑍`a − 𝑍a`).

A bar indicates a background quantity, while a hat indicatesthe trace-reversed counterpart of a rank-two tensor, ��`a ≡𝑍`a − 1

2 (𝑔𝜌𝜎𝑍𝜌𝜎)𝑔`a . Units are such that 𝑐 = ℏ = 1.

II. LINEARIZED HORNDESKI MODELS IN THEEIKONAL REGIME

This section establishes the key equations governing linearperturbations of the gravitational field in Horndeski theories.The relevant action and equations of motion are given inSec. II A. Section II B is dedicated to the linearization of thelatter on an arbitrary background. In Sec. II C we identify thescalar and tensor modes by diagonalizing the kinetic term ofthe system of equations of motions.

3

A. Reduced Horndeski theories

In this article, we focus on the subset of Horndeski theoriesin which metric perturbations propagate at the speed of light.This class of models will be referred to as reduced Horndeskitheories throughout the article.

1. Action

Horndeski theories are an extension of GR featuring a scalarfield 𝜑, which can interact nonminimally with the spacetimemetric 𝑔`a . In the Jordan frame, Horndeski’s action reads

𝑆 = 𝑆m [𝜓m, 𝑔`a] + 𝑆g [𝜑, 𝑔`a] . (1)

The matter sector 𝑆m depends on the matter fields 𝜓m, assumedto be minimally coupled to the spacetime metric 𝑔`a .In reduced Horndeski theories, the gravitational sector reads

𝑆g [𝜑, 𝑔`a] =4∑𝑖=2

𝑆𝑖 [𝜑, 𝑔`a] =𝑀2P2

∫Md4𝑥

√−𝑔4∑𝑖=2

L𝑖 , (2)

where 𝑀P ≡ 1/√8𝜋𝐺 denotes the reduced Planck mass and

the three Lagrangian densities L𝑖 are

L2 = 𝐺2 (𝜑, 𝑋) , (3)L3 = 𝐺3 (𝜑, 𝑋)2𝜑 , (4)L4 = 𝐺4 (𝜑)𝑅 , (5)

where 𝑋 = − 12𝑔`a𝜑,`𝜑,a denotes the scalar field’s kinetic term,

2 ≡ 𝑔`a∇`∇a the d’Alembertian operator, and 𝑅 the Ricciscalar. Note that conditions (3)-(5) ensure that GWs propagateat light speed, as first formulated in Ref. [46].The functions 𝐺2...4 being mostly free, 𝑆g encapsulates a

wide class of scalar-tensor theories, notably GR itself, Jordan-Brans-Dicke theories [47], 𝑓 (𝑅) [48], quintessence [49], k-essence [50, 51] and the cubic covariant galileon [52]. Inprinciple, this leaves enough freedom to locally screen devia-tions from GR, e.g. to evade Solar-System constraints.

2. Equations of motion

Requiring the action (1) to be stationary with respect tovariations of the scalar field 𝜑, and variations of the inversemetric 𝑔`a , yields the equations of motion

E𝜑 = 0 , (6)E`a = 𝑀−2

P 𝑇`a , (7)

where

E𝜑 ≡2𝑀−2

P√−𝑔𝛿𝑆g

𝛿𝜑, (8)

E`a ≡2𝑀−2

P√−𝑔𝛿𝑆g

𝛿𝑔`a, (9)

𝑇`a ≡ − 2√−𝑔

𝛿𝑆m𝛿𝑔`a

. (10)

The scalar E𝜑 and tensor E`a may be split into three pieceseach, following the division of 𝑆g into 𝑆2, 𝑆3, 𝑆4:

E𝜑 =

4∑𝑖=2

E (𝑖)𝜑 , E (𝑖)

𝜑 ≡2𝑀−2

P√−𝑔𝛿𝑆𝑖

𝛿𝜑; (11)

E`a =

4∑𝑖=2

E (𝑖)`a , E (𝑖)

`a ≡2𝑀−2

P√−𝑔𝛿𝑆𝑖

𝛿𝑔`a. (12)

The expressions of the scalar terms are

E (2)𝜑 = 𝐺2,𝜑 + 𝐺2,𝑋2𝜑 − 2𝑋𝐺2,𝑋 𝜑 + 𝐺2,𝑋𝑋𝜑

,`𝑋,` (13)

E (3)𝜑 =

(2𝐺3,𝜑 − 2𝑋𝐺3,𝜑𝑋 + 𝐺3,𝑋𝑋𝜑

,`𝑋,` + 𝐺3,𝑋2𝜑)2𝜑

− 2𝑋𝐺3,𝜑𝜑 + 2𝐺3,𝜑𝑋𝜑,`𝑋,` + 𝐺3,𝑋𝑋𝑋,`𝑋,`

− 𝐺3,𝑋 (𝜑;`a𝜑;`a + 𝑅`a𝜑,`𝜑,a) (14)

E (4)𝜑 = 𝐺4,𝜑𝑅 , (15)

and those of the tensor terms are

E (2)`a = −1

2(𝐺2,𝑋𝜑,`𝜑,a + 𝐺2𝑔`a

)(16)

E (3)`a = −𝐺3,𝜑 (𝜑,`𝜑,a + 𝑋𝑔`a)

− 𝐺3,𝑋[𝜑, (`𝑋,a) +

122𝜑 𝜑,`𝜑,a −

12𝜑,𝜌𝑋,𝜌𝑔`a

](17)

E (4)`a = 𝐺4𝐸`a − 𝐺4;`a +2𝐺4 𝑔`a , (18)

where 𝐸`a denotes the Einstein tensor. From Eq. (6), it appearsthat the scalar equation of motion does not directly receivecontributions from 𝑆m. This happens because matter is notdirectly coupled to 𝜑 in the Jordan frame. However, sinceE (3)𝜑 , E (4)

𝜑 feature the Ricci tensor, which is itself sourced bymatter via Eq. (7), the scalar field is actually sourced by matterjust like the metric. This indirect coupling can be made explicitas follows. First, use Eq. (7) to express 𝑅`a and 𝑅 as

𝑅 = 𝐺−14

(32𝐺4 + E (2) + E (3) − 𝑀−2

P 𝑇)

(19)

𝑅`a = 𝐺−14

[(𝐺4;`a +

122𝐺4𝑔`a

)− E (2)

`a − E (3)`a + 𝑀−2

P 𝑇`a

],

(20)

with E (𝑖) ≡ 𝑔`aE (𝑖)`a , 𝑇 ≡ 𝑔`a𝑇`a , and where a hat indicates

the trace-reversed counterpart of a tensor, for instance

E (2)`a ≡ E (2)

`a − 12E (2)𝑔`a . (21)

Second, substitute the above in the expressions (14), (15) ofE (3)𝜑 and E (4)

𝜑 .Importantly, once the above operations are performed, the

scalar equation of motion does not contain second-order deriva-tives of 𝑔`a any longer – it expresses the dynamics of the scalarfield only. This does not happen with the tensor equation ofmotion (7), because E`a contains several terms with secondderivatives of the scalar field. We shall come back to this issueat the level of linear perturbations in Sec. II C.

4

B. Linear perturbations on an arbitrary background

Consider small perturbations over an arbitrary backgroundfor both the scalar field and the metric,

𝑔`a = ��`a + ℎ`a , | |ℎ`a | | � 1 , (22)𝜑 = �� + 𝛿𝜑 , |𝛿𝜑| � 1 , (23)

where | | . . . | | can be any reasonable notion of norm. The goalof this subsection is to expand the equations of motion (6), (7)at first order in both ℎ`a , 𝛿𝜑. In Ref. [38], we had neglectedscalar perturbations 𝛿𝜑 for simplicity, and considered metricperturbations only. We shall not make this approximation here,because we are precisely interested in how scalar and tensorperturbations may interact.By definition, ��, ��`a are solutions of the equations of motion

for some background energy-momentumdistribution𝑇`a . Thus,at first order in 𝛿𝜑, ℎ`a , we have

𝛿E𝜑 = 0 , (24)𝛿E`a = 𝑀−2

P 𝛿𝑇`a . (25)

From now on, we shall neglect GW sources,6 i.e., assume𝛿𝑇`a = 0. The equations of motion then simply become𝛿E𝜑 = 𝛿E`a = 0.The quantities 𝛿E𝜑 , 𝛿E`a depend on both 𝛿𝜑 and ℎ`a , but

also on their first and second derivatives. The many termsconstituting 𝛿E𝜑 , 𝛿E`a may thus be organized into groups,depending on the number of derivatives acting on 𝛿𝜑, ℎ`a .This is because we will eventually focus on rapidly oscillatingfields, in which case more derivatives means a larger term.An abstract but compact way of writing the outcome of thisclassification is(

𝛿E𝜑

𝛿E`a

)=

(K𝛼𝛽∇𝛼∇𝛽 + A𝛼∇𝛼 + M

) (𝛿𝜑

ℎ𝜌𝜎

)=

(00

), (26)

where sans-serif symbols indicate matrices in the field super-space spanned by 𝛿𝜑, ℎ`a . In practice, each symbol K𝛼𝛽 , A𝛼,Mmay be seen as an 11 × 11 matrix.7 Their respective role andinterpretation should be clearer under the form of block ma-trices, as shown hereafter. Note that, in Eq. (26), we chose toexpress the equations of motion in terms of the trace-reversed

6 The energy-momentum perturbation 𝛿𝑇`a actually has two distinct contribu-tions. On the one hand, any obvious addition to 𝑇`a , say a black hole binaryor a cosmic string, would be a physical source of GWs. We shall not considersuch contributions because we want to focus on the propagation of scalar andtensor waves; in particular, we assume that they are not externally sourced asthey propagate. On the other hand, as a GW propagate through a Universefilled with matter fields, it generates a contribution to 𝛿𝑇`a by perturbing themetric in ��`a . For standard forms of cosmological fluids, that contributionto 𝛿𝑇`a is a negligible O(𝜔0) term. Exceptions are relativistic fluids, forwhich that correction is O(𝜔1) , but which have negligible impact on thepropagation of GWs during matter or dark-energy dominated scenarios [53].

7 Why 11? Because the set (𝛿𝜑, ℎ`a) has 11 independent components: onefor 𝛿𝜑, and 10 for the independent components of the symmetric tensor ℎ`a .This counting does not account for the gauge freedom.

metric perturbation ℎ`a; this choice leads to slightly simplerexpressions for K𝛼𝛽 ,A𝛼.The kinetic matrix

K𝛼𝛽 =

(𝐾

𝜑𝛼𝛽𝜑 𝐾

𝜌𝜎𝛼𝛽𝜑

𝐾𝜑𝛼𝛽

`a 𝐾𝜌𝜎𝛼𝛽

`a

), (27)

whose explicit expressions of the blocks are given in Ap-pendix A 1 a, governs the second derivatives of the scalar andmetric perturbations. It is therefore the core of the dynamicalproperties of the equations of motion. For wavelike perturba-tions, K𝛼𝛽 governs the dispersion relations. In particular, thediagonal components 𝐾 𝜑𝛼𝛽

𝜑 (resp. 𝐾 𝜌𝜎𝛼𝛽`a ) would control

the dispersion relation of scalar (resp. tensor) waves in theabsence of tensor (resp. scalar) waves. The non-diagonal com-ponents encode kinetic mixing, i.e., how second derivatives of𝛿𝜑 contaminate the equation of motion of ℎ`a and vice-versa.Since these off-diagonal components are generally nonzero inreduced Horndeski theories, we conclude that 𝛿𝜑, ℎ`a are nottheir actual degrees of freedom. We shall explicitly addressthis issue in Sec. II C.The amplitude matrix

A𝛼 =

(𝐴

𝜑𝛼𝜑 𝐴

𝜌𝜎𝛼𝜑

𝐴𝜑𝛼

`a 𝐴𝜌𝜎𝛼

`a

)(28)

rules the first derivatives of 𝛿𝜑, ℎ`a . Forwavelike perturbations,A𝛼 controls the evolution of their amplitudes. The expressionsof the blocks are given in Appendix A 1 b. The diagonalterms are self-damping or self-amplification terms, while theoff-diagonal terms encode the interactions between scalar andtensor waves. In other words, the latter tell us how energy isexchanged between scalar and tensor waves as they propagate.Finally, the mass matrix

M =

(𝑀

𝜑𝜑 𝑀

𝜌𝜎𝜑

𝑀𝜑

`a 𝑀𝜌𝜎

`a

)(29)

contains the terms with no derivatives. Such terms would beinvolved in the dispersion relation of scalar and tensor waves atnext-to-next-to-leading order. We shall justify in Sec. III A thatthey can be neglected in the eikonal regime. Anticipating onthis simplification, we neglect all masslike terms from now on;thus, there is not need to explicitly compute their expressionsin the scope of this article.

Example: Let us illustrate the above in the simple case of GRwith minimally coupled quintessence, i.e. L2 = 𝑋,L3 =0,L4 = 𝑅. In that case, omitting masslike terms,

𝛿E𝜑 = 2𝛿𝜑 − ��,` ℎ`a;a (30)

𝛿E`a =12

[2ℎ ;𝜌

𝜌(` a) −2ℎ`a − ℎ𝜌𝜎;𝜌𝜎 ��`a], (31)

and hence

K𝛼𝛽 =

(��𝛼𝛽 00 𝐾

𝜌𝜎𝛼𝛽`a

), (32)

A𝛼 =

(0 −��,𝜌��𝜎𝛼

0 0

), (33)

5

with

𝐾𝜌𝜎𝛼𝛽

`a =12

[2��𝛼(𝜌𝛿𝜎)

(` 𝛿𝛽

a) − 𝛿(𝜌` 𝛿

𝜎)a ��𝛼𝛽

− ��𝛼(𝜌��𝜎)𝛽 ��`a

]. (34)

In this example, the kinetic matrix is diagonal. Thisreflects the minimal coupling between the scalar andtensor fields, which are the true degrees of freedom ofGR with quintessence.

C. Kinetic diagonalization and gauge fixing

Unlike the above simple example, 𝛿𝜑, ℎ`a are generallynot the actual propagating degrees of freedom of linearizedHorndeski models. Finding those requires to diagonalize thekinetic term of Eq. (26), which is the goal of this subsection.This can be done in two successive steps, which we sketch inSecs. II C 1 and II C 2. Further details on the actual operationscan be found in Appendix A 2 a. Additional simplifications ofthe resulting equations of motion are obtained by imposing ananalog of the harmonic gauge, as shown in Sec. II C 3.

1. Eliminating second derivatives of ℎ`a from 𝛿E𝜑

The first step of the diagonalization procedure consists inisolating an equation of motion for 𝛿𝜑 that does not contain anysecond derivatives of the metric perturbation. In other words,we aim here to remove the off-diagonal block 𝐾 𝜌𝜎𝛼𝛽

𝜑 .In fact, as already mentioned at the end of Sec. II A 2, this

can even be achieved nonperturbatively by substituting theRicci terms in E𝜑 by their expression obtained from E`a . Atthe linear-perturbation level, that substitution is equivalent tocombining 𝛿E𝜑 , 𝛿E`a as follows,

𝛿E𝜑 ↦→ 𝛿E𝜑 + 𝐶`a𝛿E`a , (35)𝛿E`a ↦→ 𝛿E`a , (36)

where we have introduced the tensor

𝐶`a ≡ ��−14

[��3,𝑋

(��,` ��,a + �� ��`a

)+ ��4,𝜑 ��`a

]. (37)

Thanks to the nontrivial identity (see Appendix A 2 a)

𝐾𝜌𝜎𝛼𝛽

𝜑 + 𝐶`a𝐾𝜌𝜎𝛼𝛽

`a = 0 , (38)

the scalar equation of motion resulting from Eq. (35) is freefrom second-derivatives of the metric perturbation. Note that aside effect is the change of all the other terms of 𝛿E𝜑 .From the matrix point of view, the transformations (35),

(36) are operations on the rows of K𝛼𝛽 ,A𝛼, because theyconsist in linear combinations of 𝛿E𝜑 , 𝛿E`a without mixingthe variables 𝛿𝜑, ℎ`a .

2. Eliminating second derivatives of 𝛿𝜑 from 𝛿E`a

The second step of the diagonalization process aims to getrid of the scalar kinetic terms in the tensor equation of motion,i.e. to remove the off-diagonal block 𝐾 𝜑𝛼𝛽

`a .Unlike the previous step, this operation cannot be achieved

with a mere combination of the equations of motion 𝛿E𝜑 , 𝛿E`a ,but instead requires to combine their variables 𝛿𝜑, ℎ`a to geteigenfunctions of the system. Specifically, we introduce theeigentensor perturbation

𝛾`a ≡ ℎ`a + ��`a𝛿𝜑 , (39)

where

��`a = ��−14

(��3,𝑋 ��,` ��,a − ��4,𝜑 ��`a

), (40)

is the trace-reversed counterpart8 of the tensor 𝐶`a definedin Eq. (37).9 The fact that the same tensor appears in bothdiagonalization operations surely is not a coincidence, but wecould not identify its fundamental origin.As shown in Appendix A 2 a, the transformation

𝛿𝜑 ↦→ 𝛿𝜑 , (41)ℎ`a ↦→ 𝛾`a , (42)

which may be seen as an operation on the columns of the matrixsystem, terminates the diagonalization procedure by removingsecond derivatives of the scalar perturbation from the tensorequation of motion. Its success is due to the relation

𝐾𝜑𝛼𝛽

`a − 𝐾 𝜌𝜎𝛼𝛽`a ��𝜌𝜎 = 0 , (43)

between the original blocks of the kinetic matrix K𝛼𝛽 . Just likein the first step, the transformation (42) modifies almost all theother blocks of K𝛼𝛽 ,A𝛼.In the end, our two diagonalization steps are equivalent to

the following operations on the kinetic and amplitude matrices:

8 Had we defined the matrices K𝛼𝛽 , A𝛼 , M in terms of ℎ`a instead of ℎ`a ,the first step would have featured ��`a instead of 𝐶`a .

9 Note that Eq. (39) agrees with the results of Ref. [54], which coincidentallywas submitted on the same day as the present paper.

6

K𝛼𝛽 ↦→ /K𝛼𝛽=

(1 𝐶`a

0 1

)K𝛼𝛽

(1 0

−��𝜌𝜎 1

)=

(/𝐾 𝜑𝛼𝛽𝜑 00 /𝐾 𝜌𝜎𝛼𝛽

`a

), (44)

A𝛼 ↦→ /A𝛼=

(1 𝐶`a

0 1

) [A𝛼 + 2K (𝛼𝛽) ∇𝛽

] (1 0

−��𝜌𝜎 1

)=

( /𝐴 𝜑𝛼𝜑

/𝐴 𝜌𝜎𝛼𝜑

/𝐴 𝜑𝛼`a

/𝐴 𝜌𝜎𝛼`a

), (45)

where a slash indicates a quantity obtained after the diago-nalization process. Because they are kinetically decoupled,the variables 𝛿𝜑, 𝛾`a must be considered the true degrees offreedom of the linear theory.

3. Harmonic gauge

The expressions of the matrices /K𝛼𝛽 and /A𝛼 can be furthersimplified by taking advantage of the theory’s gauge freedom,stemming from the diffeomorphism invariance of the action (1).Under any infinitesimal transformation 𝑥` ↦→ 𝑥` = 𝑥` −b`, where b` is an infinitesimal vector field, the functionalexpression of the background fields ��`a , �� are left unchanged ifthe following transformations are applied to the perturbations:

ℎ`a ↦→ ℎ`a = ℎ`a + 2b (`;a) (46)𝛿𝜑 ↦→ 𝛿�� = 𝛿𝜑 + ��,𝜌b𝜌 . (47)

It follows in particular that

𝛾 ;a`a ↦→ �� ;a

`a = 𝛾 ;a`a +2b` + ��`a ��,𝜌b

𝜌;a , (48)

up to negligible masslike terms. Since b` is arbitrary, we arefree to impose the generalized harmonic gauge

𝛾`a;a = 0 , (49)

because if 𝛾`a did not satisfy the above, then we could alwaysfind a gauge field that is a solution of the hyperbolic partialdifferential equation 2b` + ��`a ��,𝜌b

𝜌;a = −𝛾 ;a`a , so that the

gauge-transformed ��`a would.The main advantage of Eq. (49) is that it elegantly reduces

the kinetic term of 𝛾`a to

/𝐾 𝜌𝜎𝛼𝛽`a 𝛾𝜌𝜎;𝛼𝛽 = −1

2��42𝛾`a . (50)

The other blocks of the kinetic and amplitude matrices, afterdiagonalization and gauge fixing, can be found in Appen-dices A 2 b and A 2 c, respectively.

III. SCALAR AND TENSOR WAVES

Having identified the kinetically decoupled degrees of free-dom 𝛿𝜑, 𝛾`a , we shall now focus more specifically on the casewhere such perturbations are propagating waves. The waveansätze are presented in Sec. III A, where we also justify whymasslike terms were dropped in the previous section. Thedispersion relations of scalar and tensor waves are discussed inSec. III B, and their effect on matter in Sec. III C.

A. Wave ansätze and eikonal approximation

We consider scalar and tensor perturbations under the form

𝛿𝜑 = Φ ei𝑣 , (51)𝛾`a = Γ`a ei𝑤 , (52)

where Φ, Γ`a ∈ C represent the complex amplitudes of thewaves, 𝑣, 𝑤 ∈ R denote their respective phases.A key assumption in this article is that the waves satisfy the

eikonal (or WKB) approximation.10 This means that the typicalevolution scale of the waves’ phases 𝑣, 𝑤, be it temporal orspatial, is much shorter than any other characteristic length ortime scale of the system. In particular, the phases are varyingmuch quicker than the amplitudes

𝜕𝑣, 𝜕𝑤 � 𝜕 ln |Φ|, 𝜕 ln |Γ`a | , (53)

which is the traditional content of the eikonal regime; they alsovary much more quickly than the background fields

𝜕𝑣, 𝜕𝑤 � 𝜕 ln |��|, 𝜕 ln |��`a | . (54)

In that context, a useful book-keeping parameter is theangular frequency of the waves,11 𝜔 ∼ 𝜕𝑣, 𝜕𝑤. The manyterms involved in the equations of motion can thereby besorted depending on their power of 𝜔, i.e., depending on howmany derivatives are hitting the phases 𝑣, 𝑤. This implies thefollowing hierarchy in Eq. (26)

K𝛼𝛽∇𝛼∇𝛽︸ ︷︷ ︸O(𝜔2)

� A𝛼∇𝛼︸︷︷︸O(𝜔1)

� M︸︷︷︸O(𝜔0)

. (55)

In practice, we only keep the O(𝜔2) and O(𝜔1) terms, whichrule the dispersion relations and the amplitudes of the waves,

10 Although the eikonal approximation is frequently considered a synonym ofgeometric optics (including in our own previous work), they are in fact notequivalent. Geometric optics consists in neglecting wave-optics effects, suchas interference and diffraction. Yet such phenomena are usually studied inthe framework of the eikonal approximation (see e.g. Ref. [55] for a recentexample in gravitational-wave physics). Recent attempts to go beyond theeikonal approximation can be found in Refs. [56, 57].

11 The frequency of scalar and tensor waves could in principle be different.However, in practice we expect both types of waves to be emitted by the samekind of events, e.g., merging binaries of compact objects, and thereby withthe same frequency. When only one of these waves is emitted and decaysinto the other sector, both are also expected to have the same frequency.

7

respectively. This explains why we have chosen to drop themasslike terms right from the beginning.Note that derivatives of 𝛿𝜑, 𝛾`a actually contain terms with

different powers of 𝜔. From the ansätze (51), (52), we find

𝛿𝜑,𝛼 =(iΦ 𝑞𝛼 +Φ,𝛼

)ei𝑣 , (56)

𝛾`a;𝛼 =(iΓ`a𝑘𝛼 + Γ`a;𝛼

)ei𝑤 , (57)

where 𝑞𝛼 ≡ 𝑣,𝛼 and 𝑘𝛼 ≡ 𝑤,𝛼 are the wavevectors of thescalar and tensor waves, respectively. Any occurrence of𝑘𝛼, 𝑞𝛼 counting as a power of 𝜔, the above expressions containboth O(𝜔0) and O(𝜔1) terms. As for the second derivatives,

𝛿𝜑;𝛼𝛽 =[−Φ 𝑞𝛼𝑞𝛽 + 2iΦ, (𝛼𝑞𝛽) + iΦ 𝑞𝛼;𝛽+Φ;𝛼𝛽

]ei𝑣 , (58)

𝛾𝜌𝜎;𝛼𝛽 =[− Γ𝜌𝜎𝑘𝛼𝑘𝛽 + 2iΓ𝜌𝜎;(𝛼𝑘𝛽) + iΓ𝜌𝜎𝑘𝛼;𝛽

+ Γ𝜌𝜎;𝛼𝛽]ei𝑤 , (59)

both contain O(𝜔2),O(𝜔1), and O(𝜔0) terms.

B. Dispersion relations

Isolating the O(𝜔2) terms in the equations of motion, whichcan only come from the kinetic matrix, we find the dispersionrelations of the scalar and tensor waves,

/𝐾 𝜑𝛼𝛽𝜑 𝑞𝛼𝑞𝛽 = 0 ,

/𝐾 𝜌𝜎𝛼𝛽`a Γ𝜌𝜎𝑘𝛼𝑘𝛽 = 0 .

(60)

(61)

1. Tensor waves are luminal

Let us start with the easiest of the two dispersion relations,namely the tensor one. Due to the extremely simple form of/𝐾 𝜌𝜎𝛼𝛽`a ∝ 𝛿 (𝜌` 𝛿𝜎)

a ��𝛼𝛽 , Eq. (61) actually reduces to

𝑘𝛼𝑘𝛼 = 0 , (62)

which means that tensor waves propagate at the speed of light.It also implies that tensor waves follow null geodesics, because

0 = (𝑘𝛼𝑘𝛼);𝛽 = 2𝑘𝛼𝑤;𝛼𝛽 = 2𝑘𝛼𝑤;𝛽𝛼 = 2𝑘𝛼𝑘𝛽;𝛼 , (63)

which is the geodesic equation. These statements are, inparticular, independent of the polarization of the wave. Thiswould not happen if the indices of 𝐾 𝜌𝜎𝛼𝛽

`a were intertwinedin a more complicated way, as it is the case for more generalHorndeski theories, such as quartic or quintic Galileons [54].The polarization-independence of the tensor wave’s disper-

sion relation justifies, a posteriori, the fact that we considereda single phase factor ei𝑤 in the ansatz (36) for 𝛾`a . Indeed, ifthe dispersion relation depended on the polarization, then eachcomponent of 𝛾`a would generally propagate at its own speed,and hence should be equipped with its own phase 𝑤`a .

2. The scalar wave’s velocity is inhomogeneous and anisotropic

The dispersion relation of scalar waves (60) is phenomeno-logically richer. Although there is no polarization dependenceby definition, the speed of scalar waves generally depends ontheir location and on their direction of propagation. Let usbe more specific; since the tensor /𝐾 𝜑𝛼𝛽

𝜑 is symmetric andreal-valued, there exists a tetrad (𝑒𝛼0 , . . . , 𝑒

𝛼3 ) such that

/𝐾 𝜑𝛼𝛽𝜑 ∝ −𝑒𝛼0 𝑒

𝛽

0 + 𝑐21𝑒

𝛼1 𝑒

𝛽

1 + 𝑐22𝑒

𝛼2 𝑒

𝛽

2 + 𝑐23𝑒

𝛼3 𝑒

𝛽

3 , (64)

where 𝑐1, 𝑐2, 𝑐3 are the three main phase velocities of scalarwaves, as measured in the frame defined by 𝑒`0 . In Eq. (64), wehave assumed that /𝐾 𝜑𝛼𝛽

𝜑 does not depart too much from ��𝛼𝛽 ,in the sense that its nonstandard terms are not large enough tochange the causal structure of the scalar dynamics; if it werethe case the theory would suffer from severe instabilities. Let(𝑞𝛼) = (𝜔, 𝑞𝒖) be the tetrad components of the scalar wave’sfour-vector, with 𝒖 = (𝑢1, 𝑢2, 𝑢3) a Euclidean unit vector. Thenthe local scalar phase velocity reads

𝑐S ≡𝜔

𝑞=

√(𝑐1𝑢1)2 + (𝑐2𝑢2)2 + (𝑐3𝑢3)2 , (65)

in the rest frame defined by 𝑒`0 .From the expression of /𝐾 𝜑𝛼𝛽

𝜑 provided in Appendix A 2 b,we can see that its failure to be proportional to ��𝛼𝛽 , i.e.,the departure from 𝑐S = 1, is due to the coupling functions��2,𝑋𝑋 , ��3,𝑋 , ��3,𝑋 𝜑 , ��3,𝑋𝑋 . In the limit where such depar-tures are small, we have12

��2,𝑋 |𝑐S − 1| = O[��2,𝑋𝑋 (𝜕��)2, ��3,𝑋𝜕2��, ��3,𝜑𝑋 (𝜕��)2,

��3,𝑋𝑋 (𝜕��)2𝜕2��]. (66)

Conversely, if we exclude any background fine tuning, then

𝑐S = 1⇐⇒ 𝐺2,𝑋𝑋 = 𝐺3,𝑋 = 0 . (67)

In other words, scalar waves are luminal if and only if 𝜑 is aconformally-coupled quintessence field, in agreement with thefindings of Ref. [43].

C. The effect of scalar and tensor waves on matter

Consider the superposition of a scalar and a tensor wave.What is their effect on the matter through which they propagate,and how can they be detected?

1. Observables are curvature perturbations

In the action (1), matter is coupled to the spacetime geometryonly; in particular it is not directly coupled to the scalar

12 While Eq. (66) may also hold for superluminal scalar waves, we refrain fromconsidering that case because its interpretation would be unclear.

8

field. Therefore, observable effects of the scalar and tensorperturbations must be looked for in the spacetime curvature,i.e., in the Riemann tensor. The perturbation of the latteraround its background value ��`a𝜌𝜎 reads, at linear order [58],

𝛿𝑅`a𝜌𝜎 =12

(ℎ`𝜎;a𝜌 − ℎ`𝜌;a𝜎 − ℎa𝜎;`𝜌 + ℎa𝜌;`𝜎

), (68)

and thereby depends on the original metric perturbation

ℎ`a = ��`a − 𝐶`a𝛿𝜑 ≡ ℎT`a + ℎS`a . (69)

In other words, the curvature perturbation generally picks uptwo distinct contributions: a rather standard one from the tensorwave via ℎT`a = ��`a , but also one from the scalar wave viaℎS`a = −𝐶`a𝛿𝜑. This is how the fifth force associated withscalar waves arises in the Jordan frame. The above emphasizesthe importance of a clear identification of the theory’s degreesof freedom, and notably 𝛾`a ≠ ℎ`a .To be more explicit, by combining Eqs. (68) and (69) we see

that the curvature perturbation can be written as the superposi-tion of two waves, respectively linked to ℎT`a and ℎS`a ,

𝛿𝑅`a𝜌𝜎 = RT`a𝜌𝜎ei𝑤 + RS`a𝜌𝜎ei𝑣 (70)

and whose amplitudes read, at leading order in 𝜔,

RT`a𝜌𝜎 = 2𝑘 [a𝐻T`] [𝜌𝑘𝜎 ] = 2𝑘 [aΓ`] [𝜌𝑘𝜎 ] , (71)

RS`a𝜌𝜎 = 2𝑞 [a𝐻S`] [𝜌𝑞𝜎 ] = −2Φ𝑞 [a𝐶`] [𝜌𝑞𝜎 ] . (72)

2. The tensor contribution is standard

The contribution of the tensor wave to the metric (andcurvature) perturbation is what one usually refers to as a GW.Its properties were analyzed in details in our earlier work [38].Let us briefly summarize its main findings in the following.Because of the harmonic-gauge condition (49), which at

leading order imposes 𝑘aΓ`a = 0 on the amplitude of 𝛾`a , itcan be shown that its trace-reversed counterpart is decomposedinto a gauge mode and a transverse-traceless mode as

Γ`a ≡ 𝐻T`a = 𝐻G`a + 𝐻TT`a . (73)

The gauge mode, which takes the form 𝐻G`a = 2𝑘 (`𝐻a) where𝐻a is a vector field, is nonphysical: it does not contribute to thecurvature perturbation, it does not carry energy-momentum,and it can always be locally removed by a gauge transformation.The transverse-traceless mode 𝐻TT`a contains the physics of

the tensor wave. Its expression is conveniently written in termsof a null tetrad (𝑘`, 𝑛`, 𝑚`, 𝑚

∗`), whose vectors are all null,

and where a star denotes a complex conjugate; the only nonzeroinner products of the tetrad are ��`a𝑚`𝑚

∗a = ��`a𝑘`𝑛a = 1.

The vectors 𝑚`, 𝑚∗` can be seen as spanning a spatial screen

that is orthogonal to the wave’s direction of propagation (seeRef. [38] for details). The transverse-traceless mode then reads

𝐻TT`a = 𝐻𝑚`𝑚a + 𝐻�𝑚∗`𝑚

∗a , (74)

thereby defining the complex amplitudes 𝐻, 𝐻� of the left-handed and right-handed helicity modes. These are related tothe usual plus and cross polarizations 𝐻+, 𝐻× through

𝐻 = 𝐻+ − i𝐻× , (75)𝐻� = 𝐻+ + i𝐻×. (76)

In the rest frame of any observer, if the tensorwave propagatesin the 𝑧-direction, then the amplitude of the associated curvatureperturbation reads

(RT0𝑖0 𝑗 ) =𝜔2T2

©­«𝐻+ 𝐻× 0𝐻× −𝐻+ 00 0 0

ª®¬ , (77)

where 𝜔T is the observed cyclic frequency of the tensor wave.No force is produced in the direction of propagation.

3. Effect of a luminal scalar wave

The tidal forces provoked by scalar waves depend on theirpropagation speed. Let us start with the luminal case (𝑐S = 1).Following the discussion of Sec. III B 2, if we do not allow forfine-tuned setups, then the luminal condition imposes𝐺3,𝑋 = 0,so that 𝐶`a = ��−1

4 ��4,𝜑 ��`a . It is then straightforward to showthat, in the rest frame of any observer, the associated curvatureperturbation reads

(RS0𝑖0 𝑗 ) = −𝜔2S2��4,𝜑

��4

©­«Φ 0 00 Φ 00 0 0

ª®¬ , (78)

where, again, the 𝑧-direction corresponds to the local directionof propagation of the scalar wave, and 𝜔S is the observed cyclicfrequency of the scalar wave.Equation (78) represents the tidal forces that are intuitively

expected from a scalar wave. If a ring of freely-falling particleswere placed in the 𝑥𝑦-plane, then the ring’s radius wouldperiodically increase and decrease by an amount proportionalto Φ. It is, however, interesting to notice that the wave’s effectsremain transverse, in the sense that there are no tidal forcesalong the direction of propagation. If𝐺4,𝜑 = 0, the scalar wavedecouples and travels without interacting with interferometers.

4. Effect of a subluminal scalar wave

The phenomenology of subluminal scalar waves (𝑐S < 1) isricher. In any observer’s rest frame, the amplitude (72) of thecurvature perturbation that it causes reads

RS0𝑖0 𝑗 = −12Φ

[𝑞𝑖𝑞 𝑗𝐶00 − 2𝑞 (𝑖𝐶 𝑗)0 + 𝜔2S𝐶𝑖 𝑗

], (79)

which now features two unrelated directions: the wavevector 𝒒on the one hand, and the background scalar field’s gradient ∇��present in𝐶𝑖 𝑗 on the other hand. As a consequence, tidal forcesare generally triaxial; in particular, they are no longer transverse.

9

More explicitly, since RS0𝑖0 𝑗 is symmetric and real-valued, itmay be diagonalized in an orthonormal system (𝒆𝑎)𝑎=1,...,3 as

(RS0𝑎0𝑏) = −𝜔2S2

©­«C1Φ 0 00 C2Φ 00 0 C3Φ

ª®¬ , (80)

where C1, C2, C3 are three dimensionless shape parameters,which depend on ��3,𝑋/��4, 𝑐S, the derivatives of ��, and theangle between 𝒒 and ∇��.Note that the orthonormal basis (𝒆𝑎) used in Eq. (80) is

generally different from the orthonormal basis (𝒆𝑖) used inEqs. (77) and (78). In particular, the direction 𝑎 = 3 does notalways coincide with the direction of propagation of the wave.In a scenario of chameleon screening, we may expect the

derivatives 𝜕�� to be suppressed at the observer’s location. Thiswould imply 𝑐S ≈ 1 and 𝐶`a ∝ ��`a , thereby bringing us backto Sec. III C 3. Hence, screening would not remove the effectof a scalar wave, but rather reduce it to that of a luminal wavewhich would be delayed with respect to the tensor wave.

IV. (NON-)INTERACTION BETWEEN SCALAR ANDTENSOR WAVES

In the previous sections, we have shown that in reducedHorndeski theories, GWs consist of the superposition of atensor wave (two degrees of freedom) and a scalar wave (onedegree of freedom). While tensor waves propagate at the speedof light, scalar waves do not in general. In this section, weinvestigate the evolution of the amplitude of tensor and scalarwaves, and their mutual interactions.

A. Evolution of the amplitudes

The standard procedure to get evolution equations for thewave amplitudes consists in extracting the O(𝜔1) terms in theequations of motion. This is easily understood starting withthe tensor wave, whose kinetic term reads

/𝐾 𝜌𝜎𝛼𝛽`a 𝛾𝜌𝜎;𝛼𝛽 = −1

2��42𝛾`a (81)

=12��4

(Γ`a𝑘𝛼𝑘

𝛼 − iDΓ`a

)ei𝑤 , (82)

up to O(𝜔0) terms, and with the differential operator

D ≡ 2𝑘𝛼∇𝛼 + 𝑘𝛼;𝛼 . (83)

The O(𝜔1) term in the above contains DΓ`a , whose functionis to propagate Γ`a in the direction of 𝑘𝛼, i.e. along thenull geodesic followed by the tensor wave. The other O(𝜔1)contributions to 𝛿E`a come from the amplitude matrix, whichencodes both self interactions (diagonal terms), and interactionswith the scalar waves (off-diagonal terms). In GR, the resultwould simply read DΓ`a = 0, leading to the fact that the GW

amplitude essentially decreases as the area of its wavefront.Here, we have instead

0 = − ��42DΓ`a + /𝐴 𝜌𝜎𝛼

`a Γ𝜌𝜎𝑘𝛼 +(/𝐴 𝜑𝛼`a 𝑞𝛼Φ

)ei(𝑣−𝑤) .

(84)The same procedure, i.e. extracting O(𝜔1) terms, applied tothe scalar equation of motion yields

0 = DSΦ + /𝐴 𝜑𝛼𝜑 𝑞𝛼Φ +

(/𝐴 𝜌𝜎𝛼𝜑 Γ𝜌𝜎𝑘𝛼

)ei(𝑤−𝑣) (85)

with DS ≡ /𝐾 𝜑𝛼𝛽𝜑

(2𝑞𝛼∇𝛽 + 𝑞𝛼;𝛽

). (86)

Because scalar and tensor waves may propagate at differentspeeds, their phases 𝑣 and 𝑤 may differ. This makes theanalysis of the combined evolution of Φ, Γ`a more subtle thanthe scalar-wave-free setup studied in Ref. [38]. In the following,we shall successively consider the cases where scalar waves areluminal (𝑐S = 1), quasiluminal (𝑐S ≈ 1), and finally nonluminal(𝑐S ≠ 1).

B. Luminal scalar waves: no interactions

As discussed in Sec. III B 2, the condition 𝑐S = 1 imposes𝐺2,𝑋𝑋 = 𝐺3,𝑋 = 0, which drastically simplifies the problem.In particular, Eqs. (84) and (85) become propagation equationsfor the amplitudesΦ, Γ`a only. In what follows, we assume thatthe scalar and tensor waves propagate along the same geodesicand have the same frequency, so that 𝑘` = 𝑞`. This assumptionis motivated by the fact that we are primarily interested in theinteraction of scalar and tensor waves originating from thesame source. It is nevertheless straightforward to generalizeour results to other setups.

1. Tensor amplitude

First consider the evolution of the tensor amplitude, governedby Eq. (84), which for 𝑣 = 𝑤 becomes

0 = −12��4DΓ`a + /𝐴 𝜌𝜎𝛼

`a 𝑘𝛼Γ𝜌𝜎 + /𝐴 𝜑𝛼`a 𝑘𝛼Φ . (87)

The amplitude terms are obtained from Appendix A 2 by taking𝐺2,𝑋𝑋 = 𝐺3,𝑋 = 0,

/𝐴 𝜌𝜎𝛼`a 𝑘𝛼Γ𝜌𝜎 = ��4,𝜑

[2��,𝜌Γ𝜌(`𝑘a) − 𝑘𝜌 ��,𝜌Γ`a

], (88)

/𝐴 𝜑𝛼`a 𝑘𝛼 = −

(��2,𝑋 + 2��3,𝜑

) [�� (`𝑘 ,a) −

12𝑘𝜌 ��,𝜌��`a

]+ ��−1

4 ��24,𝜑

[��, (`𝑘 ,a) +

12𝑘𝜌 ��,𝜌��`a

].

(89)

As mentioned in Sec. III C 2, the tensor amplitude canbe decomposed into a nonphysical gauge mode and physical

10

transverse-traceless mode, which in turn consists of two helicitymodes with complex amplitudes

𝐻 = 𝑚∗`𝑚

∗a𝐻

`a

TT = 𝑚∗`𝑚

∗aΓ

`a = 𝑚∗`𝑚

∗aΓ

`a , (90)𝐻� = 𝑚`𝑚aΓ

`a . (91)Hence, the evolution of𝐻 and𝐻� (or equivalently of𝐻+, 𝐻×)can be obtained by projecting Eq. (87) on 𝑚∗

`𝑚∗a and 𝑚`𝑚a .

Since 𝑚`, 𝑚∗` are null and orthogonal to 𝑘`, all the projected

amplitude terms read𝑚`𝑚a /𝐴`a𝜌𝜎𝛼

𝑘𝛼Γ𝜌𝜎 = 𝑘𝜌��4,𝜌𝐻� (92)𝑚∗

`𝑚∗a/𝐴`a𝜌𝜎𝛼

𝑘𝛼Γ𝜌𝜎 = 𝑘𝜌��4,𝜌𝐻� (93)𝑚`𝑚a /𝐴`a𝜑𝛼

𝑘𝛼 = 𝑚∗`𝑚

∗a/𝐴`a𝜑𝛼

𝑘𝛼 = 0 . (94)Assuming without loss of generality that 𝑚` and 𝑚∗

` areparallel-transported along the worldline of the tensor wave, theprojections of Eq. (87) on 𝑚`𝑚` and 𝑚∗

`𝑚∗` then reduce to

D[√𝐺4 (��) 𝐻#

]= 0 , (95)

where 𝐻# stands for either of 𝐻, 𝐻�; but Eq. (95) wouldalso apply to 𝐻+, 𝐻× which are linear superpositions of thecircular modes. We conclude, in particular, that the physicaltensor modes are mutually independent and decoupled fromthe scalar wave.Equation (95) is identical to the main result of Ref. [38] in

which scalar waves were initially neglected. All the conclusionsof that reference thus hold in the presence of luminal scalarwaves. In particular: (i) the polarization of a tensor wave isparallel-transported along the wave’s worldline; (ii) as the wavepropagates, its amplitude changes as13

𝐻# ∝ 1√𝐺4 (��) 𝐷 (𝑧)

, (96)

where 𝑧 is the observed redshift and 𝐷 (𝑧) = (1 + 𝑧)𝐷A isthe corrected luminosity distance, with 𝐷A the observed an-gular diameter distance to the source. Equation (96) thusapplies regardless of the observer’s motion, lensing, integratedSachs-Wolfe effect, etc. Finally, (iii) since the gravitationaldistance 𝐷G is extracted from GW observations in such away that 𝐻# ∝ (1 + 𝑧)/𝐷G, that distance is related to theelectromagnetic luminosity distance 𝐷L = (1 + 𝑧)2𝐷A as14

𝐷G =

√𝐺4 (��o)𝐺4 (��s)

𝐷L , (97)

where ��o, ��s are the values of the background scalar field atthe observation and emission events. We stress that the word“background” does not necessarily refer to a homogeneous-isotropic cosmological setup, but rather designates the scalarfield’s state without scalar waves.

13 This step uses the fact that, in the operator D, the wavefront’s expansionrate reads 𝑘`

;` = 2d ln𝐷/d_, where _ is an affine parameter for the GW’sgeodesic. See Sec. III.C.3 of Ref. [38] for details.

14 Note that 𝐷G is, in fact, fundamentally related to the geometric angulardiameter distance 𝐷A. Equation (97) implicitly assumes the validity ofthe distance-duality law for electromagnetic signals, which requires theconservation of photon number.

2. Scalar amplitude

We then turn to the evolution of the scalar amplitude, which isdictated by Eq. (85). For luminal scalar waves, /𝐾 𝜑𝛼𝛽

𝜑 ∝ ��𝛼𝛽 ,and hence it is fully determined by its trace. Let us introduce

𝑁 ≡ 14��𝛼𝛽 /𝐾 𝜑𝛼𝛽

𝜑 = ��2,𝑋 + 2��3,𝜑 + 3��−14 ��

24,𝜑 , (98)

in terms of which DS = 𝑁D, so that Eq. (85) becomes

0 = 𝑁DΦ + /𝐴 𝜑𝛼𝜑 𝑘𝛼Φ + /𝐴 𝜌𝜎𝛼

𝜑 𝑘𝛼Γ𝜌𝜎 . (99)

The amplitude terms are obtained from Appendix A 2 c byimposing ��2,𝑋𝑋 = ��3,𝑋 = 0, which yields

/𝐴 𝜑𝛼𝜑 𝑘𝛼 = 𝑘𝜌𝑁,𝜌 , (100)

/𝐴 𝜌𝜎𝛼𝜑 𝑘𝛼Γ𝜌𝜎 = 0 , (101)

and hence Eq. (99) reduces to D(𝑁Φ) = 0. This confirms, inparticular, that there is no interaction between tensor wavesand luminal scalar waves.Following the same logic as in the tensor-wave case, we

conclude that the amplitude of a scalar wave evolves as

Φ ∝ 1𝑁 (��, ��) 𝐷

. (102)

We may define a notion of scalar distance similarly to how wedefined the gravitational distance for tensor waves, namely thequantity that governs the wave’s amplitude as Φ ∝ (1 + 𝑧)/𝐷S.Following that definition, the scalar distance would read

𝐷S =𝑁 (��o, ��o)𝑁 (��s, ��s)

𝐷L ≠ 𝐷G , (103)

with the function 𝑁 given in Eq. (98).

C. Quasiluminal scalar waves: negligible interactions

We have seen that there are no interactions between scalarand tensor waves if 𝑐S = 1. Does that conclusion hold when 𝑐Sis close enough to 1, so that Eqs. (84) and (85) can be used tostudy the evolution of Φ, Γ`a? In other words, is the problemcontinuous in the limit 𝑐S → 1?In order to address that question, we shall first determine the

condition on 𝑐S such that 𝑣 ≈ 𝑤. Consider for simplicity thata scalar wave and a tensor wave are emitted simultaneously,and that they are initially in phase.15 Let us determine thephase drift 𝑣 − 𝑤 along the tensor wave’s worldline, which weaffinely parametrize with _. The phase drift from emission (_s)

15 This does not restrict the generality of the discussion, since any initialrelative phase can always be absorbed in the complex amplitudes Φ, Γ`a .

11

to observation (_o) then reads

𝑣 − 𝑤 =

∫ _o

_s

d_(d𝑣d_

− d𝑤d_

)(104)

=

∫ _o

_s

d_(𝜕`𝑣 − 𝜕`𝑤

) d𝑥`d_

(105)

=

∫ _o

_s

d_ (𝑞` − 𝑘`)𝑘` (106)

=

∫ _o

_s

d_ 𝜔T𝜔S1 − 𝑐S𝑐S

, (107)

where we decomposed the wave four-vectors on a tetrad soas to exhibit their frequencies and phase velocities, 𝑘` =

𝜔T (1, 0, 0, 1) and 𝑞` = 𝜔S (1, 0, 0, 1/𝑐S); we neglected therelative deflection of scalar and tensor waves for simplicity. Inthe rest frame associated with the tetrad, the product dℓ = 𝜔Td_represents the physical distance over which the tensor wavetravels as the affine parameter changes by d_. Therefore,

𝑣 − 𝑤 =

∫ ℓs

0dℓ 𝜔S

1 − 𝑐S𝑐S

≡ ℓs⟨𝜔S1 − 𝑐S𝑐S

⟩. (108)

where ℓs is the affine-parameter based distance between thesource and the observer, and 〈. . .〉 is the ℓ-weighted average.We conclude that the difference between the phases 𝑣 and 𝑤

in Eqs. (84) and (85) is negligible if

ℓs

⟨𝜔S1 − 𝑐S𝑐S

⟩� 1 . (109)

This condition could have been guessed from intuitive argu-ments. Two waves can be considered to stay in phase if thedelay of the slowest wave with respect to the quickest remainsmuch smaller than their period. If the waves are emitted simul-taneously and travel over a distance ℓs, then the delay betweenthe receptions of scalar and tensor waves is on the order of(1 − 𝑐S)ℓs/𝑐S. The condition for that delay to remain muchsmaller than 𝑇S = 2𝜋/𝜔S thus matches Eq. (109).Relaxing the condition 𝑐S = 1, and hence allowing

𝐺2,𝑋𝑋 , 𝐺3,𝑋 ≠ 0, is comparable to opening Pandora’s boxand pouring its content into Eqs. (84) and (85). Let us re-spectively denote ΔΦ,Δ# the sum of these new terms in theevolution equations for the scalar and tensor amplitudes,

D[𝑁 (��, ��)Φ

]= ΔΦ , (110)

D[√𝐺4 (��) 𝐻#

]= Δ# . (111)

For instance, ΔΦ contains terms such as ��3,𝑋2��DΦ or��2,𝑋𝑋 ��

,𝛼𝑞𝛼Φ, and similarly for Δ#. In fact, careful ex-amination of the terms composing ΔΦ,Δ# reveals that they areall similar to the ones present in |𝑐S − 1|, as listed in Eq. (66).To be more specific, we have

ΔΦ,Δ# ∼ |𝑐S − 1| × (𝜔𝜕��) . (112)

But Eq. (108) shows that if 𝑣−𝑤 � 1, then |𝑐S−1| � O(𝜔−1),which implies that ΔΦ,Δ# are actually smaller than O(𝜔0),i.e., smaller than masslike terms within the eikonal hierarchy.

We conclude that, for quasiluminal scalar waves, scalar-tensor interactions can be safely neglected, so that the resultsof Sec. IVB hold. In other words, the problem is indeedcontinuous in the limit 𝑐S → 1.

D. Nonluminal scalar waves: incoherent interference

Let us finally tackle the case 𝑐S ≠ 1 beyond Sec. IVC.An important technical difficulty here is that the wavefrontsand worldlines of the scalar and tensor waves generally differ.Since Eqs. (84) and (85) only tell us about the local evolutionof Φ, Γ`a along their respective worldlines, it is impossibleto simultaneously solve for their evolution equations: theyconcern distinct lines in spacetime. Thus, we should in principleconsider the wavefronts of both waves and analyze their overlap.Fortunately, such an elaborate treatment is unnecessary if we

assume that the two waves are propagating in almost the samedirection. For concreteness, let us take the evolution of Γ`a (_)along the null geodesic of the tensor wave. Since Eq. (85) doesnot directly tell us aboutΦ(_), we may simply consider it as anunknown function, which varies much slower than the phasedrift 𝑣(_) − 𝑤(_). This is justified by16

d (𝑣 − 𝑤)d_

= 𝜔T𝜔S1 − 𝑐S𝑐S

= O(𝜔2) � dΦd_

= O(𝜔1) ,(113)

following the reasoning of Sec. IVC, which requires that thespatial directions of 𝑘` and 𝑞` coincide indeed. Note thatd(𝑣 − 𝑤)/d_ � dΦ/d_ only holds if 𝑐S is not too close from1, which is why we treated the quasiluminal case separately.Similarly to Sec. IVB1, we may project Eq. (84) on the

helicity basis; we notice that the projected tensor self-interactionterm is identical to the luminal case, and we find

2√𝐺4 (��)𝐷

d√𝐺4 (��)𝐷𝐻#

d_= /𝐴 𝜑𝛼

# 𝑞𝛼Φ ei(𝑣−𝑤)︸ ︷︷ ︸oscillates rapidly

, (114)

where # can stand for either or �, and 𝐷 = (1 + 𝑧)𝐷A.Integrating Eq. (114) between _s close to the source and a _then yields

𝐻# (_) =

√��4 (_s)��4 (_)

𝐷 (_s)𝐷 (_) 𝐻# (_s)

+ 12

∫ _

_s

d_′√��4 (_′)��4 (_)

𝐷 (_′)𝐷 (_)

/𝐴 𝜑𝛼

# 𝑞𝛼Φ ei(𝑣−𝑤) , (115)

where the first term is identical to the noninteracting case, whilethe second one is the average value of a rapidly oscillatingfunction, which thus is very small. We shall call incoherentinterference this phase-related suppression of interactions.

16 Recall that a derivative with respect to the affine parameter counts as onepower of 𝜔, since _ is connected to proper distance ℓ as d_ = 𝜔−1dℓ inthe frame of an observer who would measures 𝜔.

12

A similar reasoning could be applied to the scalar-amplitudecase, and would yield the same effective suppression of thescalar-tensor interaction due to the incoherence of the twowaves.Note however that the scalar self interaction generally differsfrom Sec. IVB2, because /𝐴 𝜑𝛼

𝜑 contains many additionalterms when 𝑐S ≠ 1. Since the emission of scalar waves isexpected to be extremely small, and since their conversion fromtensor waves is suppressed by incoherent interference, we didnot judge necessary to further push the present analysis, whichalready implies that scalar radiation is irrelevant to the tensormodes in reduced Horndeski theories.One possible loophole in the above, however, is the assump-

tion that scalar and tensor waves propagate (almost) in thesame direction. For example, this set-up does not cover theCherenkov-like scalar radiation that may be emitted by thepassage of a tensor wave. In that case indeed, the directionof constructive interference for scalar waves would typicallyform an angle \ = arccos(𝑐S) with the direction of propagationof the tensor wave. The treatment of such a case neverthelessrequires the use of tools that are beyond the scope of this article,and hence left for future work.

V. SUMMARY AND CONCLUSION

We have studied the propagation of GWs in the subset ofHorndeski theories where tensor perturbations propagate atlight speed (reduced Horndeski theories). Unlike our previouswork on the same topic [38], we now have accounted forpropagating scalar perturbations, i.e. scalar waves, which madethe present study technically much more involved.We derived for the first time the complete set of equations

governing the intertwined dynamics of scalar and tensor pertur-bations 𝛿𝜑, ℎ`a , for an arbitrary background scalar field andspacetime geometry ��, ��`a . Specifically, computations weremade at linear order in the perturbations, and in the limit wherethey vary on much smaller scales than the background.By diagonalizing the kinetic term of the resulting system of

equations of motion, we have shown that themetric perturbationℎ`a is not a fundamental degree of freedom of the linearizedtheory. Indeed, due to nonminimal couplings, ℎ`a actuallyencompasses some scalar information, which can be removedby considering 𝛾`a ≡ ℎ`a + ��`a𝛿𝜑 instead, 𝐶`a being definedin Eq. (37). The quantities 𝛿𝜑, 𝛾`a then represent the truescalar and tensor degrees of freedom of the linearized theory.

Considering wavelike ansätze for 𝛿𝜑, 𝛾`a , we confirmedthat tensor waves propagate at light speed, and carry two inde-pendent degrees of freedom which are the standard plus andcross modes of GWs. Scalar waves, however, are generallysubluminal, except if 𝐺2,𝑋𝑋 = 𝐺3,𝑋 = 0, i.e., for conformallycoupled quintessence. Scalar waves produce curvature pertur-bations and are thereby measurable in principle; the associatedtidal forces are transverse and circularly symmetric for luminalscalar waves, and generally triaxial in the subluminal case.We found that luminal and quasiluminal scalar waves do not

interact with tensor waves. In that case, scalar and tensor wavespropagate without seeing each others, just like the horizontaland vertical polarizations of light. Each wave, scalar or tensor,defines its own notion of distance, 𝐷S or 𝐷G, which quantifieshow its energy dilutes as the waves propagate. The expressionof 𝐷G, its connection with the electromagnetic luminositydistance, and the resulting discussions about standard sirens,remain unchanged compared to Ref. [38]. Because the emis-sion of scalar radiation is highly constrained by observations,we conclude that luminal and quasiluminal scalar waves areirrelevant to GW physics in reduced Horndeski theories.The case of frankly subluminal scalar waves is more subtle.

We argued that if scalar and tensor waves propagate in the samedirection, their interaction is effectively suppressed becausetheir phases are incoherent. However, the general case woulddeserve a dedicated study which is beyond the scope of thepresent work. In particular, tensor waves propagating withina subluminal scalar medium should generate scalar shockwaves similar to Cherenkov radiation. The analysis of thisphenomenon, together with its observational consequences,will be exposed in a subsequent article, for which the presentone constitutes a solid basis.

VI. ACKNOWLEDGMENTS

C.D. thanks Fabien Lacasa and Michele Oliosi for inspiringdiscussions. C.D. and P.F. warmly thank Miguel Zumalacár-regui and especially Jose María Ezquiaga for an enlighteningdiscussion which followed the first version of this article, andeventually led to the addition of Sec. IVD. C.D. and L.L. weresupported by a Swiss National Science Foundation (SNSF)Professorship grant (No. 170547). P.F. received the support ofa fellowship from “la Caixa” Foundation (ID 100010434). Thefellowship code is LCF/BQ/PI19/11690018.

Appendix A: Explicit expressions involved in the linearized equations of motion

This appendix provides the long explicit expressions involved in the linearized equations of motion for 𝛿𝜑, ℎ`a . Appendix A 1gives the result of a brute-force linearization of E𝜑 , E`a . Appendix A 2 is the result after diagonalization of the kinetic term andapplication of the generalized harmonic gauge (49).The formulas provided in this appendix have been independently derived and cross-checked by the first two authors of this

article, so as to mitigate the high risk of computational errors and typos.

13

1. Before diagonalization

Recall that we have classified the various terms arising from the equations of motion into three categories: (i) kinetic terms,with second derivatives of 𝛿𝜑, ℎ`a; (ii) amplitude terms, with first derivatives; and (iii) masslike terms, with no derivatives. Inpractice, this classification takes the following form:(

𝛿E𝜑

𝛿E`a

)=

[ (𝐾

𝜑𝛼𝛽𝜑 𝐾

𝜌𝜎𝛼𝛽𝜑

𝐾𝜑𝛼𝛽

`a 𝐾𝜌𝜎𝛼𝛽

`a

)︸ ︷︷ ︸

kinetic matrix K𝛼𝛽

∇𝛼∇𝛽 +(𝐴

𝜑𝛼𝜑 𝐴

𝜌𝜎𝛼𝜑

𝐴𝜑𝛼

`a 𝐴𝜌𝜎𝛼

`a

)︸ ︷︷ ︸amplitude matrix A𝛼

∇𝛼 +(𝑀

𝜑𝜑 𝑀

𝜌𝜎𝜑

𝑀𝜑

`a 𝑀𝜌𝜎

`a

)︸ ︷︷ ︸

mass matrix M

] (𝛿𝜑

ℎ𝜌𝜎

). (A1)

We give hereafter the explicit expressions of the kinetic and amplitude blocks, but not of the masslike terms which are neglected.Each block generically receives contributions from the three Lagrangians L2,L3,L4; hence, we shall write for instance

𝐾𝜑𝛼𝛽

𝜑 =

4∑𝑖=2

𝐾(𝑖) 𝜑𝛼𝛽

𝜑 , (A2)

where a superscript (𝑖) indicates that the associated term comes from L𝑖; the same terminology applies to the other blocks.When giving the expressions of the matrix blocks, we may choose to contract or not their indices, depending on what provides

the best readability.

a. Blocks of the kinetic matrix

Scalar kinetic terms in the scalar equation of motion: 𝐾 𝜑𝛼𝛽𝜑 𝛿𝜑;𝛼𝛽

𝐾(2) 𝜑𝛼𝛽

𝜑 = ��2,𝑋 ��𝛼𝛽 − ��2,𝑋𝑋 ��

,𝛼 ��,𝛽 (A3)

𝐾(3) 𝜑𝛼𝛽

𝜑 =(2��3,𝜑 − 2����3,𝜑𝑋 + ��3,𝑋𝑋 ��

,` ��,` + 2��3,𝑋2��)��𝛼𝛽

−(��3,𝑋𝑋 ��

,𝛼 ��,𝛽2�� + 2��3,𝜑𝑋 ��,𝛼 ��,𝛽 + 2��3,𝑋𝑋 ��, (𝛼 ��,𝛽) + 2��3,𝑋 ��;𝛼𝛽

)(A4)

𝐾(4) 𝜑𝛼𝛽

𝜑 = 0 (A5)

Metric kinetic terms in the scalar equation of motion: 𝐾 𝜌𝜎𝛼𝛽𝜑 ℎ𝜌𝜎;𝛼𝛽

𝐾(2) 𝜌𝜎𝛼𝛽

𝜑 ℎ𝜌𝜎;𝛼𝛽 = 0 (A6)

𝐾(3) 𝜌𝜎𝛼𝛽

𝜑 ℎ𝜌𝜎;𝛼𝛽 = −��3,𝑋 ��,𝜌 ��,𝜎𝛿𝑅𝜌𝜎 = −12��3,𝑋

[2��𝜌𝛼 ��,𝜎 ��,𝛽 −

(��,𝜌 ��,𝜎 + �� ��𝜌𝜎

)��𝛼𝛽

]ℎ𝜌𝜎;𝛼𝛽 (A7)

𝐾(4) 𝜌𝜎𝛼𝛽

𝜑 ℎ𝜌𝜎;𝛼𝛽 = ��4,𝜑𝛿𝑅 =12��4,𝜑

(2��𝜌𝛼��𝜎𝛽 + ��𝜌𝜎 ��𝛼𝛽

)ℎ𝜌𝜎;𝛼𝛽 (A8)

Scalar kinetic terms in the metric equation of motion: 𝐾 𝜑𝛼𝛽`a 𝛿𝜑;𝛼𝛽

𝐾(2) 𝜑𝛼𝛽

`a = 0 (A9)

𝐾(3) 𝜑𝛼𝛽

`a =12��3,𝑋

[2��, (` ��,𝛼𝛿

𝛽

a) − ��,` ��,a ��𝛼𝛽 − ��`a ��,𝛼 ��,𝛽

](A10)

𝐾(4) 𝜑𝛼𝛽

`a = ��4,𝜑

[��`a ��

𝛼𝛽 − 𝛿 (𝛼` 𝛿𝛽)a]

(A11)

Metric kinetic terms in the metric equation of motion: 𝐾 𝜌𝜎𝛼𝛽`a ℎ𝜌𝜎;𝛼𝛽

𝐾(2) 𝜌𝜎𝛼𝛽

`a ℎ𝜌𝜎;𝛼𝛽 = 0 (A12)

𝐾(3) 𝜌𝜎𝛼𝛽

`a ℎ𝜌𝜎;𝛼𝛽 = 0 (A13)

𝐾(4) 𝜌𝜎𝛼𝛽

`a ℎ𝜌𝜎;𝛼𝛽 = 𝐺4 𝛿𝐸`a =𝐺42

[2ℎ ;𝜌

𝜌(` a) −2ℎ`a − ℎ𝜌𝜎;𝜌𝜎 ��`a]

(A14)

14

b. Blocks of the amplitude matrix

Scalar amplitude terms in the scalar equation of motion: 𝐴 𝜑𝛼𝜑 𝛿𝜑,𝛼

𝐴(2) 𝜑𝛼

𝜑 =(−𝐺2,𝑋𝑋2�� + 𝐺2,𝑋 𝜑 + 2��𝐺2,𝑋𝑋𝜑 − 𝐺2,𝑋𝑋𝑋 ��

,` ��,`

)��,𝛼 + 2𝐺2,𝑋𝑋 ��

,𝛼 (A15)

𝐴(3) 𝜑𝛼

𝜑 =

[ (2����3,𝜑𝑋𝑋 − ��3,𝑋𝑋𝑋 ��,` ��

,` − ��3,𝑋𝑋2��)2�� + 2��3,𝜑𝜑 + 2����3,𝜑𝜑𝑋 − 2��3,𝜑𝑋𝑋 ��,` ��

,` − ��3,𝑋𝑋𝑋 ��,` ��,`

+ ��3,𝑋𝑋

(��;`a ��;`a + ��`a ��,` ��,a

) ]��,𝛼 +

[2��3,𝑋𝑋2�� + 4��3,𝜑𝑋

]�� ,𝛼 − 2��3,𝑋𝑋 ��,` ��

;`𝛼 − 2��3,𝑋 ��`𝛼 ��,` (A16)

𝐴(4) 𝜑𝛼

𝜑 = 0 (A17)

Metric amplitude terms in the scalar equation of motion: 𝐴 𝜌𝜎𝛼𝜑 ℎ𝜌𝜎;𝛼

𝐴(2) 𝜌𝜎𝛼

𝜑 = −��2,𝑋 ��,𝜌��𝜎𝛼 + 12��2,𝑋𝑋

(��,𝜌 ��,𝜎 + �� ��𝜌𝜎

)��,𝛼 (A18)

𝐴(3) 𝜌𝜎𝛼

𝜑 = −[2��3,𝜑 − 2����3,𝜑𝑋 + ��3,𝑋𝑋 ��

,` ��,` + 2��3,𝑋2��]��,𝜌��𝜎𝛼

+[(��3,𝜑𝑋 + 1

2��3,𝑋𝑋2��

)��,𝜎 + ��3,𝑋𝑋 ��

,𝛼

] (��,𝜌 ��,𝜎 + �� ��𝜌𝜎

)+ ��3,𝑋

[2��;𝛼(𝜌 ��,𝜎) − ��;𝛼_��,_��𝜌𝜎 − ��;𝜌𝜎 ��,𝛼 + 1

22����,𝛼��𝜌𝜎

](A19)

𝐴(4) 𝜌𝜎𝛼

𝜑 = 0 (A20)

Scalar amplitude terms in the metric equation of motion: 𝐴 𝜑𝛼`a 𝛿𝜑,𝛼

𝐴(2) 𝜑𝛼

`a =12

(��2,𝑋𝑋 ��,` ��,a + ��2,𝑋 ��`a

)��,𝛼 − ��2,𝑋 ��, (`𝛿𝛼a) (A21)

𝐴(3) 𝜑𝛼

`a = ��`a

[(��3,𝜑 + ����3,𝜑𝑋 − 1

2��3,𝑋𝑋 ��

,𝜌 ��,𝜌

)��,𝛼 + ��3,𝑋 �� ,𝛼

]+ ��,` ��,a

(��3,𝜑𝑋 + 1

2��3,𝑋𝑋2��

)��,𝛼

−[2��3,𝜑 ��, (` + ��3,𝑋 ��, (` + ��3,𝑋2����, (`

]𝛿𝛼,a) + ��3,𝑋𝑋 ��, (` ��,a) ��

,𝛼 + ��3,𝑋 ��, (` �� 𝛼;a) (A22)

𝐴(4) 𝜑𝛼

`a = 2��4,𝜑𝜑

[��`a ��

,𝛼 − �� (`𝛿𝛼a)

](A23)

Metric amplitude terms in the metric equation of motion: 𝐴 𝜌𝜎𝛼`a ℎ𝜌𝜎;𝛼

𝐴(2) 𝜌𝜎𝛼

`a = 0 (A24)

𝐴(3) 𝜌𝜎𝛼

`a =12��3,𝑋

[ (��,𝜌 ��,𝜎 + �� ��𝜌𝜎

) (12��`a ��

,𝛼 − ��, (`𝛿𝛼,a))+ ��,` ��,a ��, (𝜌��𝜎)𝛼

](A25)

𝐴(4) 𝜌𝜎𝛼

`a =12��4,𝜑

[(2��, (𝜌𝛿𝜎)

(` 𝛿𝛼a) − ��, (`𝛿

𝛼,a) ��

𝜌𝜎)−

(𝛿𝜌

(`𝛿𝜎a) −

12��`a ��

𝜌𝜎

)��,𝛼 − 2��`a ��, (𝜌��𝜎)𝛼

](A26)

The last two matrix elements being hard to read, we also provide their contracted counterparts:

𝐴(3) 𝜌𝜎𝛼

`a ℎ𝜌𝜎;𝛼 =12��3,𝑋

[ (��,𝜌 ��,𝜎 + �� ��𝜌𝜎

) (12��`a ��

,𝛼 ℎ𝜌𝜎;𝛼 − ℎ𝜌𝜎;(a ��,`))+ ��,` ��,a ��,𝜌 ℎ𝜌𝜎;𝜎

](A27)

𝐴(4) 𝜌𝜎𝛼

`a ℎ𝜌𝜎;𝛼 =12��4,𝜑 ��

,𝜌

(2ℎ𝜌(`;a) − ��𝜌(` ℎ,a) − ℎ`a;𝜌 +

12��`a ℎ,𝜌 − ��`a ℎ𝜌𝜎;𝜎

). (A28)

2. After diagonalization and harmonic-gauge fixing

a. Description of the operations

The kinetic term of the differential system (A1) can be diagonalized and simplified by applying the following operations:

15

1. Elimination of 𝐾 𝜌𝜎𝛼𝛽𝜑 . In this first step, we substitute the Ricci-curvature terms that appear in the scalar equation of

motion 𝛿E𝜑 using the tensor equation of motion 𝛿E`a . This operation is possible because 𝐾 𝜌𝜎𝛼𝛽`a ℎ`a;𝛼𝛽 ∝ 𝛿𝐸`a = 𝛿��`a .

In other words, the terms containing ℎ𝜌𝜎;𝛼𝛽 in the scalar equation of motion are linear combinations of the terms ℎ𝜌𝜎;𝛼𝛽 inthe tensor equation of motion. To be really specific, we have

𝐾𝜌𝜎𝛼𝛽

𝜑 ℎ𝜌𝜎;𝛼𝛽 = 𝐾(3) 𝜌𝜎𝛼𝛽

𝜑 ℎ𝜌𝜎;𝛼𝛽 + 𝐾(4) 𝜌𝜎𝛼𝛽

𝜑 ℎ𝜌𝜎;𝛼𝛽 (A29)= −��3,𝑋 ��,` ��,a𝛿𝑅`a + ��4,𝜑𝛿𝑅 (A30)= −

[��3,𝑋 (��,` ��,a + 𝑋��`a) + ��4,𝜑 ��`a

]𝛿𝐸`a (A31)

= −[��3,𝑋 (��,` ��,a + 𝑋��`a) + ��4,𝜑 ��`a

] (��−14 𝐾

𝜌𝜎𝛼𝛽`a ℎ𝜌𝜎;𝛼𝛽

)(A32)

= ��−14

[��3,𝑋 (��,` ��,a + 𝑋��`a) + ��4,𝜑 ��`a

]︸ ︷︷ ︸≡𝐶`a

(𝐾

𝜑𝛼𝛽`a 𝛿𝜑;𝛼𝛽 + 𝐴 𝜑𝛼

`a 𝛿𝜑,𝛼 + 𝐴 𝜌𝜎𝛼`a ℎ𝜌𝜎;𝛼

)(A33)

The tensor equation of motion, 𝛿E`a = 0, was used to go from the penultimate to the last line. When the above formulais substituted into the scalar equation of motion, the right-hand side contributes to the diagonal kinetic term and to theamplification matrix. Precisely, the following transformations apply:

𝐾𝜌𝜎𝛼𝛽

𝜑 ↦−→ 0 (A34)

𝐾𝜑𝛼𝛽

𝜑 ↦−→ 𝐾𝜑𝛼𝛽

𝜑 + 𝐶`a𝐾𝜑𝛼𝛽

`a (A35)𝐴

𝜑𝛼𝜑 ↦−→ 𝐴

𝜑𝛼𝜑 + 𝐶`a𝐴

𝜑𝛼`a (A36)

𝐴𝜌𝜎𝛼

𝜑 ↦−→ 𝐴𝜌𝜎𝛼

𝜑 + 𝐶`a𝐴𝜌𝜎𝛼

`a , (A37)

or, in matrix terms,

K𝛼𝛽 ↦−→(1 𝐶`a

0 1

)K𝛼𝛽 and A𝛼 ↦−→

(1 𝐶`a

0 1

)A𝛼 (A38)

which is an operation on the rows of the matrices.

We note that the additional terms to 𝐾 𝜑𝛼𝛽𝜑 , 𝐴

𝜑𝛼𝜑 , 𝐴

𝜌𝜎𝛼𝜑 are all quadratic in the coupling functions, i.e. they feature

pre-factors such as 𝐺3,𝑋𝐺4,𝜑𝜑 , or 𝐺23,𝑋 , etc. Because of that, they can no longer be associated with a specificLagrangian L2,L3,L4, and hence do not fit into the classification that we have used to far. All these hybrid terms willtherefore be gathered under the label (5) in the final result.

2. Elimination of 𝐾 𝜑𝛼𝛽`a . While the previous step consisted in combining the equations of motion, i.e. acting on the rows of

the system matrices, this second step will consist in mixing its variables, i.e. act on the columns of the system matrices.Namely, we introduce the new tensor variable

𝛾`a ≡ ℎ`a + ��`a𝛿𝜑 , ��`a = ��−14

(��3,𝑋 ��,` ��,a − ��4,𝜑 ��`a

)(A39)

being the trace-reversed counterpart of the tensor 𝐶`a that appeared in the previous step. Substituting, in the tensor equationof motion, ℎ𝜌𝜎;𝛼𝛽 by its expression in terms of 𝛾`a , 𝛿𝜑, eliminates the off-diagonal kinetic term 𝐾 𝜑𝛼𝛽

`a . This cancellationis due to the remarkable identity 𝐾 𝜌𝜎𝛼𝛽

`a ��𝜌𝜎 = 𝐾𝜑𝛼𝛽

`a , so that, up to masslike terms,

𝐾𝜌𝜎𝛼𝛽

`a ℎ𝜌𝜎;𝛼𝛽 + 𝐾 𝜑𝛼𝛽`a 𝛿𝜑;𝛼𝛽 = 𝐾

𝜌𝜎𝛼𝛽`a

(𝛾𝜌𝜎;𝛼𝛽 − ��𝜌𝜎𝛿𝜑;𝛼𝛽 − 2��𝜌𝜎;(𝛼𝛿𝜑,𝛽)

)+ 𝐾 𝜑𝛼𝛽

`a 𝛿𝜑;𝛼𝛽 (A40)

= 𝐾𝜌𝜎𝛼𝛽

`a 𝛾𝜌𝜎;𝛼𝛽 − 2𝐾 𝜌𝜎𝛼𝛽`a ��𝜌𝜎;(𝛼𝛿𝜑,𝛽) . (A41)

The change from ℎ`a to 𝛾`a also directly changes the amplitude matrix, since (again up to masslike terms)

𝐴𝜌𝜎𝛼

`a ℎ𝜌𝜎;𝛼 = 𝐴𝜌𝜎𝛼

`a

(𝛾𝜌𝜎;𝛼 − ��𝜌𝜎𝛿𝜑,𝛼

), (A42)

𝐴𝜌𝜎𝛼

𝜑 ℎ𝜌𝜎;𝛼 = 𝐴𝜑𝛼

`a

(𝛾𝜌𝜎;𝛼 − ��𝜌𝜎𝛿𝜑,𝛼

). (A43)

16

Summarizing, this second operation leads to the following transformations:

𝐾𝜑𝛼𝛽

`a ↦−→ 0 , (A44)

𝐴𝜑𝛼

`a ↦−→ 𝐴𝜑𝛼

`a − 2𝐾 𝜌𝜎 (𝛼𝛽)`a ��𝜌𝜎;𝛽 − 𝐴 𝜌𝜎𝛼

`a ��𝜌𝜎 , (A45)𝐴

𝜑𝛼𝜑 ↦−→ 𝐴

𝜑𝛼𝜑 − 𝐴 𝜌𝜎𝛼

𝜑 ��𝜌𝜎 , (A46)

where we must not forget to account for the modification of 𝐴 𝜌𝜎𝛼𝜑 that occurred in step 1. In matrix terms, that second

operation reads

K𝛼𝛽 ↦−→ K𝛼𝛽

(1 0

−��𝜌𝜎 1

)and A𝛼 ↦−→

[A𝛼 + 2K (𝛼𝛽) ∇𝛽

] (1 0

−��𝜌𝜎 1

). (A47)

In Eq. (A45), the correction coming from the kinetic matrix is linear in the coupling functions, and hence it directly changes𝐴

(3) 𝜑𝛼`a , 𝐴

(4) 𝜑𝛼`a . All the other corrections are quadratic in the coupling functions, and thus will be classified under the

label (5) as the modifications originating from the step 1.

3. Imposing the harmonic gauge. Impose the harmonic gauge 𝛾`a;a = 0, which removes a few terms in the equations ofmotion. In particular, this drastically simplifies the kinetic term for 𝛾`a .

The resulting system then reads[ (/𝐾 𝜑𝛼𝛽𝜑 00 /𝐾 𝜌𝜎𝛼𝛽

`a

)︸ ︷︷ ︸diagonal kinetic matrix /K𝛼𝛽

∇𝛼∇𝛽 +( /𝐴 𝜑𝛼

𝜑/𝐴 𝜌𝜎𝛼𝜑

/𝐴 𝜑𝛼`a

/𝐴 𝜌𝜎𝛼`a

)︸ ︷︷ ︸new amplitude matrix /A𝛼

∇𝛼

] (𝛿𝜑

𝛾𝜌𝜎

)=

(00

), (A48)

and the various terms generically decompose as, e.g.,

𝐾𝜑𝛼𝛽

𝜑 =

5∑𝑖=2

𝐾(𝑖) 𝜑𝛼𝛽

𝜑 , (A49)

where the terms with the label (5) are at least quadratic in the coupling functions.

b. Blocks of the kinetic matrix

Scalar kinetic terms in the scalar equation of motion: /𝐾 𝜑𝛼𝛽𝜑 𝛿𝜑;𝛼𝛽

/𝐾(2) 𝜑𝛼𝛽𝜑 = 𝐾

(2) 𝜑𝛼𝛽𝜑 = ��2,𝑋 ��

𝛼𝛽 − ��2,𝑋𝑋 ��,𝛼 ��,𝛽 (A50)

/𝐾(3) 𝜑𝛼𝛽𝜑 = 𝐾

(3) 𝜑𝛼𝛽𝜑 =

(2��3,𝜑 − 2����3,𝜑𝑋 + ��3,𝑋𝑋 ��

,` ��,` + 2��3,𝑋2��)��𝛼𝛽

−(��3,𝑋𝑋 ��

,𝛼 ��,𝛽2�� + 2��3,𝜑𝑋 ��,𝛼 ��,𝛽 + 2��3,𝑋𝑋 ��,𝛼 ��,𝛽 + 2��3,𝑋 ��;𝛼𝛽

)(A51)

/𝐾(4) 𝜑𝛼𝛽𝜑 = 𝐾

(4) 𝜑𝛼𝛽𝜑 = 0 (A52)

/𝐾(5) 𝜑𝛼𝛽𝜑 = 𝐶`a𝐾

𝜑𝛼𝛽`a = ��−1

4

[(3��24,𝜑 + 2����3,𝑋 ��4,𝜑 − ��2��23,𝑋

)��𝛼𝛽 − 2��3,𝑋

(��4,𝜑 + ����3,𝑋

)��, (𝛼 ��,𝛽)

](A53)

Metric kinetic terms in the metric equation of motion: /𝐾 𝜌𝜎𝛼𝛽`a 𝛾𝜌𝜎;𝛼𝛽

/𝐾(2) 𝜌𝜎𝛼𝛽`a = 𝐾

(2) 𝜌𝜎𝛼𝛽`a = 0 (A54)

/𝐾(3) 𝜌𝜎𝛼𝛽`a = 𝐾

(3) 𝜌𝜎𝛼𝛽`a = 0 (A55)

/𝐾(4) 𝜌𝜎𝛼𝛽`a = −1

2𝐺4𝛿

𝜌

, (`𝛿𝜎,a) ��

𝛼𝛽 i.e. /𝐾(4) 𝜌𝜎𝛼𝛽`a 𝛾𝜌𝜎;𝛼𝛽 = −1

2𝐺42𝛾`a (A56)

17

c. Blocks of the amplitude matrix

Scalar amplitude terms in the scalar equation of motion: /𝐴 𝜑𝛼𝜑 𝛿𝜑,𝛼

/𝐴(2) 𝜑𝛼𝜑 = 𝐴

(2) 𝜑𝛼𝜑 =

(−𝐺2,𝑋𝑋2�� + 𝐺2,𝑋 𝜑 + 2��𝐺2,𝑋𝑋𝜑 − 𝐺2,𝑋𝑋𝑋 ��

,` ��,`

)��,𝛼 + 2𝐺2,𝑋𝑋 ��

,𝛼 (A57)

/𝐴(3) 𝜑𝛼𝜑 = 𝐴

(3) 𝜑𝛼𝜑

=

[ (2����3,𝜑𝑋𝑋 − ��3,𝑋𝑋𝑋 ��,` ��

,` − ��3,𝑋𝑋2��)2�� + 2��3,𝜑𝜑 + 2����3,𝜑𝜑𝑋 − 2��3,𝜑𝑋𝑋 ��,` ��

,` − ��3,𝑋𝑋𝑋 ��,` ��,`

+ ��3,𝑋𝑋

(��;`a ��;`a + ��`a ��,` ��,a

) ]��,𝛼 +

[2��3,𝑋𝑋2�� + 4��3,𝜑𝑋

]�� ,𝛼 − 2��3,𝑋𝑋 ��,` ��

;`𝛼 − 2��3,𝑋 ��`𝛼 ��,` (A58)

/𝐴(4) 𝜑𝛼𝜑 = 𝐴

(4) 𝜑𝛼𝜑 = 0 (A59)

/𝐴(5) 𝜑𝛼𝜑 = 𝐶`a𝐴

𝜑𝛼`a − 𝐴 𝜌𝜎𝛼

𝜑 ��𝜌𝜎 − 𝐶`a𝐴𝜌𝜎𝛼

`a ��𝜌𝜎 (A60)

Metric amplitude terms in the scalar equation of motion: /𝐴 𝜌𝜎𝛼𝜑 𝛾𝜌𝜎;𝛼. The terms (2)-(4) slightly simplify because of the

harmonic gauge, which removes any contraction of 𝜌, 𝜎 with 𝛼.

/𝐴(2) 𝜌𝜎𝛼𝜑 =

12��2,𝑋𝑋

(��,𝜌 ��,𝜎 + �� ��𝜌𝜎

)��,𝛼 (A61)

/𝐴(3) 𝜌𝜎𝛼𝜑 =

[(��3,𝜑𝑋 + 1

2��3,𝑋𝑋2��

)��,𝛼 + ��3,𝑋𝑋 ��

,𝛼

] (��,𝜌 ��,𝜎 + �� ��𝜌𝜎

)+ ��3,𝑋

(2��;𝛼(𝜌 ��,𝜎) − ��;𝛼_��,_��𝜌𝜎 − ��;𝜌𝜎 ��,𝛼 + 1

22����,𝛼��𝜌𝜎

)(A62)

/𝐴(4) 𝜌𝜎𝛼𝜑 = 0 (A63)

/𝐴(5) 𝜌𝜎𝛼𝜑 = 𝐶`a𝐴

𝜌𝜎𝛼`a (A64)

Scalar amplitude terms in the metric equation of motion: /𝐴 𝜑𝛼`a 𝛿𝜑,𝛼. The terms (3), (4) change because of the kinetic term

that appears in the transformation (A45).

/𝐴(2) 𝜑𝛼`a = 𝐴

(2) 𝜑𝛼`a =

12

(��2,𝑋𝑋 ��,` ��,a + ��2,𝑋 ��`a

)��,𝛼 − ��2,𝑋 ��, (`𝛿𝛼a) (A65)

/𝐴(3) 𝜑𝛼`a =

[��3,𝜑 + ��3,𝑋2�� − ����3,𝜑𝑋 + 1

2��3,𝑋𝑋 ��

,𝜌 ��,𝜌

]��,𝛼��`a +

[��3,𝜑𝑋 ��

,𝛼 + ��3,𝑋𝑋 ��,𝛼 + 1

2��3,𝑋𝑋2����,𝛼

]��,` ��,a

+[−2��3,𝜑 − 2��3,𝑋2𝜑 + 2����3,𝑋 𝜑 − ��3,𝑋𝑋 ��

,𝜌 ��,𝜌]��, (`𝛿

𝛼,a) + 2��3,𝑋 ��, (` ��

𝛼;a) − ��3,𝑋 ��,𝛼 ��;`a (A66)

/𝐴(4) 𝜑𝛼`a = 𝐴

(4) 𝜑𝛼

`a = 0 (A67)

/𝐴(5) 𝜑𝛼`a = −𝐴 𝜌𝜎𝛼

`a ��𝜌𝜎 (A68)

Metric amplitude terms in the metric equation of motion: /𝐴 𝜌𝜎𝛼`a 𝛾𝜌𝜎;𝛼. The terms (3), (4) slightly simplify because of the

harmonic gauge, which removes any contraction of 𝜌, 𝜎 with 𝛼. The diagonalization process does not produce a (5) term.

/𝐴(2) 𝜌𝜎𝛼`a = 𝐴

(2) 𝜌𝜎𝛼`a = 0 (A69)

/𝐴(3) 𝜌𝜎𝛼`a =

12��3,𝑋

(��,𝜌 ��,𝜎 + �� ��𝜌𝜎

) (12��`a ��

,𝛼 − �� (`𝛿𝛼,a)

)(A70)

/𝐴(4) 𝜌𝜎𝛼`a =

12��4,𝜑

[(2��, (𝜌𝛿𝜎)

(` 𝛿𝛼a) − ��, (`𝛿

𝛼,a) ��

𝜌𝜎)−

(𝛿𝜌

(`𝛿𝜎a) −

12��`a ��

𝜌𝜎

)��,𝛼

](A71)

/𝐴(5) 𝜌𝜎𝛼`a = 0 (A72)

Again, since the (3) and (4) terms are hard to read, we provide their contracted counterparts:

/𝐴(3) 𝜌𝜎𝛼`a 𝛾𝜌𝜎;𝛼 =

12��3,𝑋

(��,𝜌 ��,𝜎 + �� ��𝜌𝜎

) (12��`a ��

,𝛼𝛾𝜌𝜎;𝛼 − 𝛾𝜌𝜎;(a ��,`))

(A73)

/𝐴(4) 𝜌𝜎𝛼`a 𝛾𝜌𝜎;𝛼 =

12��4,𝜑 ��

,𝜌

(2𝛾𝜌(`;a) − ��𝜌(`𝛾,a) − 𝛾`a;𝜌 +

12��`a𝛾,𝜌

). (A74)

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