# Gravitational wave energy momentum-tensor in reduced Horndeski theories

João C. Lobato  Universidade Federal do Rio de Janeiro, Instituto de Física
CEP 21941-972 Rio de Janeiro, RJ, Brazil  Maurício O. Calvão  Universidade
Federal do Rio de Janeiro, Instituto de Física
CEP 21941-972 Rio de Janeiro, RJ, Brazil

###### Abstract

We generalize, imposing the field equations only at dominant order, the
Isaacson formula for the gravitational wave (GW) energy-momentum tensor (EMT)
to the class of Horndeski theories in which the tensor modes travel at the
speed of light (reduced Horndeski theories) and scalar waves are present. We
discuss important particular cases such as: theories where scalar waves are
also luminal and theories in which the transverse-traceless gauge can be
achieved in an arbitrary open set. The vanishing of the trace of the
gravitational wave energy-momentum tensor is obtained for theories in which
all wave perturbations propagate at the speed of light. The trace is shown not
to vanish trivially in other cases. We obtain, as a particular case of our
general result, the GW EMTs, in a Brans-Dicke theory, both in the Einstein
frame, recovering previous results in the literature, and in the Jordan frame,
thereby showing the GW EMT is not conformally invariant. We further prove that
there exists a subclass of reduced Horndeski theories where, in contrast to
general relativity, the divergence of the GW EMT does not vanish even after
the imposition of the full equations of motion, assuming an eikonal solution.

##  I Introduction

The recent first detection of the stochastic gravitational wave background
(SGWB) by NANOGrav [1] has proven once more that there are still numerous new
wonders yet to be discovered about our universe. It has been responsible for
an additional exciting chapter in the continuous fresh series of discoveries
surrounding gravitational wave (GW) physics and its power to give us new
physics information will be enormous [2, 3]. The first space based
interferometer to be launched in the next decade, the Laser Interferometer
Space Antenna (LISA) [4, 5], will be able to provide another angle to the vast
GW landscape, improving our understanding of fundamental physics, astrophysics
and cosmology [6, 7, 8]. In particular, the SGWB coming from compact white
dwarf binaries in our own galaxy is expected to be detected by LISA [9].

The SGWB will serve as a new probe for alternative theories of gravity as
well. Short duration GWs need a number of interferometers in different parts
of the globe if the goal is to measure additional polarizations predicted by
modified theories of gravity. But, with the SGWB continuous signal, it is
possible to detect such polarizations with a single detector provided a
sufficiently long observation time is employed [10].

Another way SGWB can indicate deviations from GR is if direct corrections to
the detected signal exist. As pointed out by [11, 12], usually when discussing
the SGWB at a theoretical level, one pays attention to the spectral energy
density per unit solid angle but takes for granted the way this physical
quantity relates to the actually measurable SGWB signal. This relation between
what is directly measured and what is more physically relevant can change in
modified gravity. In [11], this relation is generalized to Brans-Dicke, Chern-
Simons and massive gravity theories. In [12], we obtained the aforementioned
relation, assuming there are no scalar waves, in the context of reduced
Horndeski theories, the most general scalar-tensor theory having second-order
equations of motion originating from a non-degenerate Lagrangian with tensor
waves traveling at the speed of light. The luminal property of tensorial
perturbations is observably justified, at least for low redshifts, by
multimessenger sources [13, 14].

The equation relating the directly measurable SGWB signal and the spectral
energy density of GWs has corrections for alternate theories of gravity
because, among other possible causes such as a change in GW propagation law,
the energy-momentum tensor (EMT) of GWs changes when compared to its simple
form in general relativity (GR). The GW EMT arises from the non-linear
character of the field equations. When assuming the presence of high-frequency
GW perturbations traveling through the background space-time, low frequency
contributions related with the GWs arise in the background part of the field
equations, a back-reaction mechanism that describes how GWs, while being a
result of the matter curved space-time, can themselves generate more
curvature. This process was first studied in [15, 16] in the GR context and
further mathematically developed by [19, 17, 18].

In [20], many crucial aspects regarding the computation of the GW EMT are
revised and generalized and the tensor is obtained for some alternate gravity
theories, such as Chern-Simons and scalar-tensor theories where the scalar
field has a canonical kinetic term in the action together with an interaction
term coupling the scalar field and scalar curvature invariants of second rank
or higher, that is, those constructed with the contraction of at least three
curvature related tensors. Such a family of theories does not include the
Lagrangian term $f(\varphi)R$, where $\varphi$ is the dynamical scalar field,
where $R$ is the Ricci scalar. Another particularity of the study made in [20]
is that only asymptotically flat space-times are taken into account. Thus,
important systems, such as cosmological ones, cannot be included in those
results.

In [21, 22] the GW EMT is obtained in $f(R)$ theories. In GR, the tensor has
always vanishing trace, being then interpreted as describing a radiation
fluid, while in $f(R)$ it can have negative values, which allows the resultant
fluid to be understood, in principle, as dark energy, responsible for the
present accelerated expansion stage of the universe.

The GW EMT for reduced Horndeski theories was originally obtained in [23]. But
in this work the presence of scalar waves was completely neglected. Further,
several results regarding scalar waves were derived in [24], but the GW EMT
was not generalized to contain scalar wave contributions.

In this present paper, our main goal is to obtain the off-shell GW EMT in the
context of reduced Horndeski theories assuming scalar waves. Several new
features must be considered: in this context, the transverse-traceless (TT)
gauge cannot always be achieved, scalar waves do not always travel at the
speed of light, the true degrees of freedom of GWs are mixed together inside
the tensor perturbations of the metric and some properties of the weak-limit
average used to obtain the tensor have to be reviewed.

We do not impose the full field equation in the most general context, but some
theories are explored in which, even after the field equations are used, the
divergence of the GW EMT is shown not to vanish. This implies that the duality
relation for the GW analogous cosmological distances (the luminosity and
angular diameter distances) is altered, which, as will be discussed further in
the text, can imply modifications on the spectral energy density of the SGWB.

Some particular subcases of interest are explored such as Horndeski theories
with luminal scalar waves and those in which the TT gauge can be achieved. In
the first case, the trace of the GW EMT is shown to vanish, as one should
expect of a radiation fluid, while in the other it is shown not to vanish in a
trivial way, which could potentially indicate a dark energy fluid, just as
suggested in [22] for $f(R)$ gravity.

In summary, the most important results of our study are (see Fig. (1)):

  * •

We obtain the GW EMT for a general reduced Horndeski theory, but only imposing
the equations of motion at dominant order (Eqs. (93), (94), (96)), (100).

  * •

We obtain the GW EMT for reduced Horndeski theories where scalar waves are
luminal and show that its trace vanishes (Eqs. (115) and (116)).

  * •

We obtain the GW EMT for reduced Horndeski theories where the transverse-
traceless gauge is achievable, showing that its trace does not vanish
trivially and thus may indicate an alternative for dark energy (Eqs.(122) and
(123)).

  * •

We reobtain the GW EMT for Brans-Dicke theory in the Einstein frame and
confirm previous results from literature (Eq. (128)).

  * •

We obtain the GW EMT for Brans-Dicke theory in the Jordan frame and show that
this object is not conformally invariant (Eqs. (139), (145) and (150)).

  * •

We give an example of a reduced Horndeski theory where, even after imposing
the complete field equations and assuming an eikonal solution, the divergence
of the GW EMT does not vanish, which impacts the GW duality relation and the
SGWB signal (Eqs.(156), (157) and (174)).

The results here can originate further investigation of possible modifications
of gravity that can be tested by several different ways, including the SGWB
signal analysis.

In Section II we present the theory we work on, define the GW EMT in a general
context, describe tensor and scalar wave propagation, discuss some limitations
on fixing gauges and present the weak-limit average. In Section III we expand
the action until second order in the amplitude of GWs. In Section IV we vary
the action and take the weak limit average to obtain the GW EMT. In Section V
we further explore particular cases of interest and reobtain the tensor in the
Brans-Dicke theory, confirming what was already known from literature. In
Section VI we calculate the divergence of the GW EMT in a particular subclass
of Horndeski and prove that, even after the imposition of the field equation,
it does not vanish for a general eikonal wave solution.

We use units in which $c=1$ and a Lorentzian metric with signature $+2$. The
Riemann will be defined by satisfying

$\displaystyle v_{\alpha;\beta\gamma}-v_{\alpha;\gamma\beta}=\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}{R^{\lambda}}_{\alpha\beta\gamma}v_{\lambda}$
(1)

where $v_{\alpha}$ is a generic covector and we define the Ricci tensor as:

$\displaystyle R_{\alpha\beta}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}={R^{\lambda}}_{\alpha\lambda\beta}.$
(2)

##  II Preliminaries

###  II.1 Reduced Horndeski theories

The reduced Horndeski theory gravitational action 111We will not consider
matter action in this study. is:

$\displaystyle S_{g}=\frac{1}{16\pi G}\int\sqrt{-g}[G_{2}(\varphi,X)$
$\displaystyle+G_{3}(\varphi,X)\Box\varphi$
$\displaystyle+G_{4}(\varphi)R]d^{4}x,$ (3)

where $\varphi$ is a scalar field, $\Box\varphi\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=g^{\mu\nu}\varphi_{;\mu\nu}$
and $X\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=-\varphi^{,\mu}\varphi_{,\mu}/2$
is the kinetic energy of the scalar field. It is the most general non-
degenerate scalar-tensor four-dimensional theory with second-order
differential field equations in which the tensorial modes of GWs travel at the
speed of light. Notice that no restriction is made for the speed of scalar
waves. GR is the special case in which $G_{2}=-2\Lambda$, $G_{3}=0$ and
$G_{4}=1$. 222 For a comprehensive review on Horndeski theory, see [25].

The family of scalar-tensor theories in which the GW EMT was obtained in [20]
has the gravitational action:

$\displaystyle S_{g}^{\prime}=S_{EH}+S_{int}+S_{\varphi},$ (4)

where

$\displaystyle S_{EH}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=\frac{1}{16\pi
G}\int\sqrt{-g}Rd^{4}x$ (5)

is the GR gravitational action,

$\displaystyle S_{\varphi}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=\beta\int\sqrt{-g}[X+V(\varphi)]d^{4}x$
(6)

is the canonical scalar term, with $\beta$ being a real constant and

$\displaystyle S_{int}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=\alpha\int\sqrt{-g}f(\varphi)\mathcal{R}d^{4}x,$
(7)

where $\alpha$ is another real constant and $\mathcal{R}$ is a scalar built
with curvature tensors (i.e. Riemann and Ricci tensors.), provided
$\mathcal{R}\neq R$. Actions of Eqs (3) and (4) do not coincide for at least
two reasons. First, in Eq. (4), $G_{2}=X+V(\varphi)$. Second, as stated
before, the coupling between the scalar field and the curvature, present in
$S_{int}$ is not of the form $G_{4}(\varphi)R$.

###  II.2 Definition of GW EMT in scalar-tensor theories

To obtain the GW EMT of a scalar-tensor theory, we first need to consider
solutions of the field equations having gravitational waves traveling on top
of a background spacetime with metric $\bar{g}_{\mu\nu}(x^{\sigma})$.

To that end, we will assume there is a coordinate system in which a two-
parameter family of metrics satisfying the field equations can be split as:

$\displaystyle
g_{\mu\nu}(x^{\sigma},\epsilon,\alpha)=\bar{g}_{\mu\nu}(x^{\sigma})+\alpha
h_{\mu\nu}(x^{\sigma},\epsilon)$ (8)

where $|\epsilon|,|\alpha|\ll 1$ are constant parameters (i.e. independent of
$x^{\sigma}$) related to, respectively, the GW wavelength and amplitude
($h_{\mu\nu}$ is assumed to vary substantially more and to be extremely weaker
than the background metric $\bar{g}_{\mu\nu}$). Henceforward, unless
explicitly stated otherwise, all covariant derivatives and the raising and
lowering of indices will be done with respect to the background metric. It is
important to emphasize that the background metric in this work is generic,
different from the treatment in [20], where $\bar{g}_{\mu\nu}$ is assumed to
be asymptotically flat, not comprising important cases, such as the isotropic
and homogeneous universes described by the Robertson-Walker metric.

In the same spirit, one splits the scalar field as:

$\displaystyle\varphi(x^{\mu},\epsilon,\alpha)=\bar{\varphi}(x^{\mu})+\alpha\delta\varphi(x^{\mu},\epsilon).$
(9)

Then, given the action of a theory, one can find the GW EMT by the following
procedure.

First, expand the action in powers of the GW amplitude:

$\displaystyle S=\bar{S}+\alpha
S^{(1)}(h_{\alpha\beta},\delta\varphi)+\alpha^{2}S^{(2)}(h_{\alpha\beta},\delta\varphi)+\mathcal{O}(\alpha^{3}),$
(10)

where $h_{\mu\nu}$ and $\delta\varphi$ still depend on $\epsilon$, but, of
course, $\bar{S}$ depends neither on $\epsilon$ nor on $\alpha$.

Second, vary the $\alpha$ dependent part of the action with respect to the
background metric and take the weak limit average (which will be described
further in the text, see Subsec. II.6):

$\displaystyle T^{(GW)}_{\mu\nu}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=-\frac{2}{\sqrt{-\bar{g}}}\bigg{\langle}\frac{\delta(S-\bar{S})}{\delta\bar{g}^{\mu\nu}}\bigg{\rangle}$
(11)

As a good first approximation, our study will only be concerned with
contributions coming from the linear and quadratic parts in Eq. (10).

The most general treatment one can give for the GW EMT is to assume $\alpha$
and $\epsilon$ as independent. But, to simplify our discussion, we will assume
from now on that these parameters are equal:

$\displaystyle\epsilon=\alpha.$ (12)

In principle, terms with different powers in $\epsilon$ can contribute to
$S^{(1)}$ and $S^{(2)}$, depending on the action we are interested, since Eq.
(10) is an expansion only in the $\alpha$ parameter.

Notice that, in Eq. (8), when we take the limit $\alpha\rightarrow 0$, the
wave vanishes and the total metric reduces to the background one. But when Eq.
(12) is assumed, it does not follow necessarily that

$\displaystyle\lim_{\epsilon\rightarrow 0}\epsilon
h_{\alpha\beta}(x^{\mu},\epsilon)=0.$ (13)

Here we will assume perturbations are such that Eq. (13) is valid. An example
of physical significance in which this is true is the eikonal function:

$\displaystyle
h_{\alpha\beta}(x^{\mu},\epsilon)=H_{\alpha\beta}(x^{\mu})e^{ik_{\nu}x^{\nu}/\epsilon}.$
(14)

Similar behavior is assumed for the scalar perturbation.

###  II.3 Gravitational waves in reduced Horndeski theories

The evolution equation for the tensorial and scalar parts of gravitational
wave can be compactly expressed as333For the complete expressions of each
matrix element in terms of the Galileon functions, see [24]:

$\displaystyle\bigg{[}\begin{pmatrix}K^{\alpha\beta}&K^{\rho\sigma\alpha\beta}\\\
{K^{\alpha\beta}}_{\mu\nu}&{K^{\rho\sigma\alpha\beta}}_{\mu\nu}\end{pmatrix}\nabla_{\alpha}\nabla_{\beta}+\begin{pmatrix}F^{\alpha}&F^{\rho\sigma\alpha}\\\
{F^{\alpha}}_{\mu\nu}&{F^{\rho\sigma\alpha}}_{\mu\nu}\end{pmatrix}\nabla_{\alpha}$
$\displaystyle\hskip 80.0pt+\begin{pmatrix}M&M^{\rho\sigma}\\\
M_{\mu\nu}&{M^{\rho\sigma}}_{\mu\nu}\end{pmatrix}\bigg{]}\begin{pmatrix}\delta\varphi\\\
\hat{h}_{\rho\sigma}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix},$ (15)

where $\hat{h}_{\alpha\beta}$ is the trace-reverse of $h_{\alpha\beta}$:

$\displaystyle\hat{h}_{\alpha\beta}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=h_{\alpha\beta}-\frac{h}{2}\bar{g}_{\alpha\beta}$
(16)

and the matrices acting upon the perturbation fields are called kinetic,
friction (or amplitude) and mass tensors.

In order to interpret the pair of perturbation fields in Eq. (15) as true
degrees of freedom of GWs, the kinetic tensor acting upon the double covariant
derivatives needs to be diagonal, as argued by [24]. In the most general case
of a reduced Horndeski theory, this does not occur for the pair
($\delta\varphi,\hat{h}_{\alpha\beta}$). Nevertheless, the diagonalization of
the kinetic term is still achievable if $\hat{h}_{\alpha\beta}$ is replaced by
the new tensor 444We choose here, differently to the notation adopted in [24],
to express $\hat{\gamma}_{\alpha\beta}$ with a hat since it is defined in
terms of hatted quantities.:

$\displaystyle\hat{\gamma}_{\alpha\beta}(x^{\mu},\epsilon)\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=\hat{h}_{\alpha\beta}(x^{\mu},\epsilon)+\hat{C}_{\alpha\beta}(x^{\mu})\delta\varphi(x^{\mu},\epsilon),$
(17)

where

$\displaystyle\hat{C}_{\mu\nu}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=\frac{1}{\bar{G}_{4}}(\bar{G}_{3,X}\bar{\varphi}_{,\mu}\bar{\varphi}_{,\nu}-\bar{G}_{4,\varphi}\bar{g}_{\mu\nu}),$
(18)

and $\bar{G}_{i}$ stands for $G_{i}(\bar{\varphi},\bar{X})$. The pair
$(\delta\varphi,\hat{\gamma}_{\alpha\beta})$ can then be thought as the true
degrees of freedom of GWs [24]. It is important to notice that the system of
equations governing the evolution of this new pair is still coupled, only the
kinetic term is diagonalized.

One can invert Eq. (16), to give:

$\displaystyle
h_{\alpha\beta}=\hat{h}_{\alpha\beta}-\bar{g}_{\alpha\beta}\frac{\hat{h}}{2}.$
(19)

Then, substituting Eq.(17) and its trace, one relates the initial perturbation
with the real tensorial degree of freedom of the system:

$\displaystyle
h_{\alpha\beta}=\hat{\gamma}_{\alpha\beta}-\frac{\bar{g}_{\alpha\beta}}{2}\hat{\gamma}-C_{\alpha\beta}\delta\varphi,$
(20)

where

$\displaystyle
C_{\mu\nu}=\frac{1}{\bar{G}_{4}}[\bar{G}_{3,X}(\bar{\varphi}_{,\mu}\bar{\varphi}_{,\nu}+\bar{X}\bar{g}_{\mu\nu})+\bar{G}_{4,\varphi}\bar{g}_{\mu\nu}]$
(21)

is the trace-reverse of $\hat{C}_{\mu\nu}$ 555Notice that
$\hat{\hat{A}}^{\alpha\beta}=A^{\alpha\beta}$, for any tensor
$A^{\alpha\beta}$..

###  II.4 Transverse-traceless gauge is not always achievable

A very powerful and standard way of representing GWs in general relativity is
by taking advantage of the gauge freedom the tensorial perturbation has to
reduce the number of non-zero independent components to two: one related with
the plus polarization and the other related with the cross polarization. The
resulting tensor has trace zero and travels along null geodesics transversely
with respect to the direction of propagation. The effect of GWs in free
particles is then found to be of a shearing nature, that is, it is an
anisotropic perturbation which preserves areas and volumes. The gauge in which
such scheme is possible is called the transverse-traceless (TT) gauge. Despite
its simplicity, the TT gauge is not always achievable in reduced Horndeski
theories.

Under the gauge change

$\displaystyle h_{\mu\nu}\rightarrow h_{\mu\nu}+2\xi_{(\mu;\nu)},$ (22)

where $\xi_{\mu}$ is the gauge vector, the divergence of
$\hat{\gamma}_{\mu\nu}$ transforms as

$\displaystyle{\hat{\gamma}_{\mu\nu}}^{;\nu}\rightarrow{\hat{\gamma}_{\mu\nu}}^{;\nu}+\Box\xi_{\mu}+{\bar{R}^{\mu}}_{\nu}\xi^{\nu}.$
(23)

It is always possible to find a $\xi_{\mu}$ such that the harmonic gauge
condition:

$\displaystyle{\hat{\gamma}_{\mu\nu}}^{;\nu}=0$ (24)

is satisfied.

Although the harmonic gauge condition is necessary for obtaining the more
restrictive TT gauge, it is not sufficient. As argued in [23], under the
geometrical optics regime, the TT gauge can only be achieved along the whole
GW null geodesic without violating Eq. (24) if $G_{4,\varphi}=0$ and
$G_{3,X}=0$. We will not restrict ourselves to this set of theories and will
not adopt the TT gauge in the discussion that follows, although the harmonic
gauge will be used throughout the study. This is a considerable difference
when comparing our results with the usual procedure for obtaining the GW EMT
in general relativity, since one expects, in the former case, the trace of the
tensorial degree of freedom to be present.

###  II.5 Unitary-harmonic gauge is not always achievable

In [26], a gauge in which Eq. (24) is valid together with $\delta\varphi=0$
simultaneously is presented. This is called the harmonic-unitary gauge in
[23]. In this gauge, all possible new contributions to the GW EMT we are
trying to investigate would be only a gauge effect, not having physical
consequences, since no scalar wave would result in real space-time curvature,
not generating geodesic deviation between freely falling particles.

But, as proven in [26], this gauge is only achievable when scalar waves are
luminal. And, even in this case, this is only a necessary condition. Since we
are dealing with a family of theories in which scalar waves are not
necessarily luminal and, even when they are luminal, this gauge cannot always
be achieved, the calculation of the GW EMT in reduced Horndeski theories can
still have physically relevant new contributions.

###  II.6 The weak limit average

Here we discuss some features regarding the weak limit average [16, 19, 27,
17, 20, 22] used to construct the GW EMT.

We will assume that although the perturbations have amplitudes of order
$\epsilon$ (remembering Eq. (12)), their derivatives can have greater order.
More specifically, we will impose that, for each derivative taken, the order
of the resulting quantity will drop by a power of $\epsilon$:

$\displaystyle\epsilon\hat{\gamma}_{\alpha\beta;\lambda_{1}\lambda_{2}...\lambda_{n}}=\mathcal{O}(\epsilon^{1-n}),$
(25)
$\displaystyle\epsilon\delta\varphi_{;\lambda_{1}\lambda_{2}...\lambda_{n}}=\mathcal{O}(\epsilon^{1-n}).$
(26)

This rule allows us to inspect how GWs can curve the background space-time by
contributing to the energy-momentum source of the background field equations.
If we assumed that derivatives were of the same order as the perturbations,
there would be no effect on the background space-time due to GWs. On the other
hand, if we assumed the drop in powers of $\epsilon$ for each derivative taken
to be greater than one, we would necessarily end up with divergent terms in
the GW EMT.

The weak limit average of a given tensor
$A_{\alpha_{1}\alpha_{2}...\alpha_{n}}(x^{\mu},\epsilon)$ is

$\displaystyle\langle
A_{\alpha_{1}\alpha_{2}...\alpha_{n}}(x^{\mu})\rangle\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=\lim_{\epsilon\rightarrow
0}\int\sqrt{-\bar{g}}$
$\displaystyle{f^{\beta_{1}...\beta_{n}}}_{\alpha_{1}\alpha_{2}...\alpha_{n}}(x^{\prime\mu},x^{\mu})$
$\displaystyle\times
A_{\beta_{1}...\beta_{n}}(x^{\prime\mu},\epsilon)d^{4}x^{\prime},$ (27)

where $f^{\alpha_{1}\alpha_{2}...\alpha_{n}}$ is a function that falls
smoothly to zero for $|x^{\mu}-x^{\prime\mu}|$ of the order of several
$\epsilon$ wavelengths but still small compared with the typical background
scale of variation. The limit is made to enforce the smallness of the
wavelength when compared with the scale in which the average is made.

Since $f^{\alpha_{1}\alpha_{2}...\alpha_{n}}$ is negligible for background
scales, background quantities, which do not depend on $\epsilon$ in any way,
can be considered constant under the region of integration:

$\displaystyle\langle\bar{B}_{\alpha_{1}\alpha_{2}...\alpha_{n}}\rangle=\bar{B}_{\alpha_{1}\alpha_{2}...\alpha_{n}}.$
(28)

Furthermore, we assume the limits

$\displaystyle\lim_{\epsilon\rightarrow
0}\hat{\gamma}_{\alpha\beta}(x^{\mu},\epsilon)=\hat{\gamma}_{\alpha\beta}(x^{\mu},0),$
(29) $\displaystyle\lim_{\epsilon\rightarrow
0}\delta\varphi(x^{\mu},\epsilon)=\delta\varphi(x^{\mu},0),$ (30)

to be well defined, which allow us to conclude that

$\displaystyle\langle\epsilon\hat{\gamma}_{\alpha\beta}\rangle=0,$ (31)
$\displaystyle\langle\epsilon\delta\varphi\rangle=0.$ (32)

Because of Eqs. (25) and (26), first derivatives of perturbations are of
background order. But one of the most important properties of the weak limit
is that, since $f^{\alpha_{1}\alpha_{2}...\alpha_{n}}$ does not depend on
$\epsilon$, such derivatives average to zero. In the case of the
$\hat{\gamma}_{\alpha\beta}$ perturbation:

$\displaystyle\langle\epsilon\hat{\gamma}_{\alpha\beta;\lambda}\rangle$
$\displaystyle=\lim_{\epsilon\rightarrow
0}\int\sqrt{-\bar{g}}{f^{\gamma\delta\phi}}_{\alpha\beta\lambda}\epsilon\hat{\gamma}_{\gamma\delta;\phi}d^{4}x^{\prime}$
$\displaystyle=-\lim_{\epsilon\rightarrow
0}\int\sqrt{-\bar{g}}{f^{\gamma\delta\phi}}_{\alpha\beta\lambda;\phi}\epsilon\hat{\gamma}_{\gamma\delta}d^{4}x^{\prime}=0.$
(33)

By induction, the n-th order derivative of the perturbations vanishes as well:

$\displaystyle\langle\epsilon\hat{\gamma}_{\alpha\beta;\lambda_{1}\lambda_{2}..\lambda_{n}}\rangle=0.$
(34)

Because of these properties, one can always use integration by parts in
products of derivatives of perturbations. For example:

$\displaystyle\langle\epsilon^{2}\hat{\gamma}_{\alpha\beta;\lambda}\delta\varphi_{;\delta}\rangle$
$\displaystyle=\langle\epsilon^{2}(\hat{\gamma}_{\alpha\beta;\lambda}\delta\varphi)_{;\delta}\rangle-\langle\epsilon^{2}\hat{\gamma}_{\alpha\beta;\lambda\delta}\delta\varphi\rangle$
$\displaystyle=-\langle\epsilon^{2}\hat{\gamma}_{\alpha\beta;\lambda\delta}\delta\varphi\rangle,$
(35)

since, analogously to Eq. (33), one shows that
$\langle\epsilon^{2}(\hat{\gamma}_{\alpha\beta;\lambda}\delta\varphi)_{;\delta}\rangle=0$.

Another important property is that covariant derivatives of perturbations
commute inside the average:

$\displaystyle\langle\epsilon\hat{\gamma}_{\alpha\beta;\mu\nu}\rangle$
$\displaystyle=\langle\epsilon\hat{\gamma}_{\alpha\beta;\nu\mu}\rangle+{\bar{R}^{\tau}}_{\alpha\mu\nu}\langle\epsilon\hat{\gamma}_{\tau\beta}\rangle+{\bar{R}^{\tau}}_{\beta\mu\nu}\langle\epsilon\hat{\gamma}_{\alpha\tau}\rangle$
$\displaystyle=\langle\epsilon\hat{\gamma}_{\alpha\beta;\nu\mu}\rangle.$ (36)

Using Eqs. (35) and (36), we get:

$\displaystyle\langle\epsilon^{2}\hat{\gamma}_{\alpha\beta;\lambda}\delta\varphi_{;\delta}\rangle=\langle\epsilon^{2}\hat{\gamma}_{\alpha\beta;\delta}\delta\varphi_{;\lambda}\rangle.$
(37)

Having in mind the properties derived in this subsection, one can simplify the
GW EMT defined in Eq. (38) [20]. Terms appearing in $S^{(1)}$ will necessarily
have a single perturbation or one of its derivatives. As a consequence of Eq.
(12) and the property illustrated in Eq. (34), these terms will necessarily
vanish under the average. This implies that we only need to compute $S^{(2)}$:

$\displaystyle
T^{(GW)}_{\mu\nu}\approx-\frac{2}{\sqrt{-\bar{g}}}\bigg{\langle}\epsilon^{2}\frac{\delta
S^{(2)}}{\delta\bar{g}^{\mu\nu}}\bigg{\rangle}.$ (38)

This is the first approximation of the quantity related to GWs that will serve
as an additional source to the background field equations of the theory.

###  II.7 Gravitational wave energy momentum tensor

In general relativity, one has the usual Isaacson formula:

$\displaystyle
T^{GW}_{\mu\nu}=\frac{1}{32}\langle\epsilon^{2}\hat{\gamma}_{\alpha\beta;\mu}{\hat{\gamma}^{\alpha\beta}}_{;\nu}\rangle=\frac{1}{32}\langle\epsilon^{2}\hat{h}_{\alpha\beta;\mu}{\hat{h}^{\alpha\beta}}_{;\nu}\rangle.$
(39)

where the last equality is valid because there is no scalar wave on the
theory.

The correction to this tensor in reduced Horndeski theories was already
obtained if we assumed scalar waves to be neglectible [23]:

$\displaystyle
T^{GW}_{\mu\nu}=\frac{\bar{G}_{4}}{32}\bigg{\langle}\epsilon^{2}\bigg{(}\hat{h}_{\alpha\beta;\mu}{\hat{h}^{\alpha\beta}}_{;\nu}-\frac{1}{2}\hat{h}_{,\mu}\hat{h}_{,\nu}\bigg{)}\bigg{\rangle},$
(40)

where we cannot assume the second term to vanish since the TT gauge may not be
achievable, as discussed previously.

Different corrections were obtained for the GW EMT when considering the
presence of scalar waves for theories like Brans-Dicke, Chern-Simmons and
massive gravity [11, 20], but no article handled the whole family of reduced
Horndeski theories with scalar waves.

In this last case, one expects corrections to these expressions involving
$\langle\delta\varphi_{,\mu}\delta\varphi_{,\nu}\rangle$,
$\langle\delta\varphi_{,\mu}\hat{\gamma}_{,\nu}\rangle$,
$\langle\hat{\gamma}_{,\mu}\hat{\gamma}_{,\nu}\rangle$ and $\langle
S^{\alpha\beta}\hat{\gamma}_{\alpha\beta,\mu}\delta\varphi_{,\nu}\rangle$,
with $S^{\alpha\beta}$ being a certain tensor to be deduced, where
$\hat{h}_{\alpha\beta}$ is exchanged by $\hat{\gamma}_{\alpha\beta}$ because
of the presence of the scalar waves. Since we do not assume a luminal nature
for scalar perturbations, we still can have contributions related with
$\Box{\delta\varphi}$ as well. Our aim is to express the GW EMT assuming all
these new elements from the theory.

##  III Expanding the action

We begin by expanding metric related quantities. Although the metric has only
linear perturbations, the inverse metric needs to have a second order term in
order to satisfy:

$\displaystyle g_{\alpha\beta}g^{\beta\gamma}={\delta^{\gamma}}_{\alpha},$
(41)

to second order. This implies in the following expression, correct to second
order:

$\displaystyle g^{\mu\nu}=\bar{g}^{\mu\nu}-\epsilon
h^{\mu\nu}+\epsilon^{2}{h^{\mu}}_{\alpha}h^{\alpha\nu}.$ (42)

Here, one might ask why haven’t we expanded the metric like

$\displaystyle g_{\mu\nu}=\bar{g}_{\mu\nu}+\epsilon
h_{\mu\nu}+\epsilon^{2}i_{\mu\nu}$ (43)

since we are interested in contributions in the action until second order. We
do not need to assume an $\epsilon^{2}$ term in the metric because it can only
appear on $S^{(2)}$ linearly, multiplying background terms. Because of Eq.
(34), they would vanish in the weak limit average.

We now expand the metric determinant. Taking the determinant of Eq. (9):

$\displaystyle g=\bar{g}\det(I+\epsilon H),$ (44)

where $g$ is the determinant of the total metric, $\bar{g}$ is the determinant
of the background metric and ${(I+\epsilon
H)^{\mu}}_{\nu}={\delta^{\mu}}_{\nu}+\epsilon{h^{\mu}}_{\nu}$. Defining the
logarithm of a matrix as:

$\displaystyle\log(A)\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=\sum_{k=1}^{\infty}(-1)^{k+1}\frac{(A-I)^{k}}{k},$
(45)

we may write, to second order in $\epsilon$,

$\displaystyle\det(I+\epsilon H)$ $\displaystyle=\exp[\log(\det(I+\epsilon
H))]$ $\displaystyle=\exp[Tr(\log(I+\epsilon H))]$
$\displaystyle=\exp\left(\epsilon
Tr(H)-\frac{\epsilon^{2}}{2}Tr(H^{2})\right)$ $\displaystyle=1+\epsilon
Tr(H)+\frac{\epsilon^{2}}{2}\\{[Tr(H)]^{2}-Tr(H^{2})\\},$ (46)

where in the second equality we have used a well known property (see [28], for
example) and $Tr$ is the trace of a matrix. Using this expansion in Eq. (44):

$\displaystyle\sqrt{-g}$
$\displaystyle=\sqrt{-\bar{g}}\left\\{1+\frac{\epsilon}{2}h+\frac{\epsilon^{2}}{8}\left[h^{2}-2{h^{\alpha}}_{\beta}{h^{\beta}}_{\alpha}\right]\right\\},$
(47)

where $h\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=Tr(H)=\bar{g}^{\alpha\beta}h_{\alpha\beta}$.

We next expand the scalar kinetic term $X$ using Eqs. (8) and (9):

$\displaystyle
X=-\frac{1}{2}g^{\alpha\beta}\varphi_{;\alpha}\varphi_{;\beta}=\bar{X}+\epsilon
X^{(1)}+\epsilon^{2}X^{(2)},$ (48)

where

$\displaystyle\bar{X}=-\frac{1}{2}\bar{g}^{\alpha\beta}\bar{\varphi}_{,\alpha}\bar{\varphi}_{,\beta},$
(49) $\displaystyle
X^{(1)}=\frac{h^{\alpha\beta}}{2}\bar{\varphi}_{,\alpha}\bar{\varphi}_{,\beta}-\bar{g}^{\alpha\beta}\bar{\varphi}_{,\alpha}\delta\varphi_{,\beta},$
(50) $\displaystyle
X^{(2)}=h^{\alpha\beta}\bar{\varphi}_{,\alpha}\delta\varphi_{,\beta}-\frac{\bar{g}^{\alpha\beta}}{2}\delta\varphi_{,\alpha}\delta\varphi_{,\beta}-\frac{1}{2}{h^{\alpha}}_{\mu}h^{\mu\beta}\bar{\varphi}_{,\alpha}\bar{\varphi}_{,\beta}.$
(51)

Notice that, in Eq. (48), we have used the full metric to contract the indices
instead of the background one, since we are interested in the action, which is
expressed in terms of the complete metric. With this, the Galileon functions
can be expanded:

$\displaystyle G_{i}(\varphi,X)$ $\displaystyle=\bar{G}_{i}+\epsilon
G_{i}^{(1)}+\epsilon^{2}G_{i}^{(2)}$ (52)

where

$\displaystyle
G_{i}^{(1)}=\bar{G}_{i,\varphi}\delta\varphi+\bar{G}_{i,X}X^{(1)}$
$\displaystyle
G_{i}^{(2)}=\bar{G}_{i,X}X^{(2)}+\bar{G}_{i,\varphi\varphi}\frac{\delta\varphi^{2}}{2}+\bar{G}_{i,\varphi
X}\delta\varphi X^{(1)}$ $\displaystyle\hskip
130.0pt+\bar{G}_{i,XX}\frac{{(X^{(1)})}^{2}}{2}.$ (53)

From the Ricci tensor expansion (see [28], for example), one obtains the Ricci
scalar to second order in $\epsilon$:

$\displaystyle R=g^{\alpha\beta}R_{\alpha\beta}=\bar{R}+\epsilon
R^{(1)}+\epsilon^{2}R^{(2)},$ (54)

where

$\displaystyle
R^{(1)}={h_{\beta\alpha;}}^{\beta\alpha}-\bar{\Box}h-\bar{R}_{\mu\nu}h^{\mu\nu}$
(55) $\displaystyle
R^{(2)}=\frac{1}{2}\bigg{[}\frac{3}{2}h_{\mu\alpha;\rho}{{h^{\mu\alpha}}_{;}}^{\rho}-h_{\mu\alpha;\rho}{{h^{\mu\rho}}_{;}}^{\alpha}$
$\displaystyle+\left(\frac{h_{;\alpha}}{2}-{h_{\alpha\rho;}}^{\rho}\right)(2{h^{\alpha\mu}}_{;\mu}-{h_{;}}^{\alpha})\bigg{]}+{h^{\mu}}_{\alpha}h^{\alpha\nu}\bar{R}_{\mu\nu}$
$\displaystyle\hskip
65.0pt+h^{\rho\alpha}(\bar{\Box}h_{\rho\alpha}+h_{;\rho\alpha}-{2{h^{\mu}}_{\rho}}_{;(\mu\alpha)}),$
(56)

and

$\displaystyle{{h^{\mu}}_{\rho}}_{;(\mu\alpha)}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=\frac{1}{2}({{h^{\mu}}_{\rho}}_{;\mu\alpha}+{{h^{\mu}}_{\rho}}_{;\alpha\mu}).$
(57)

To expand the D’alembertian in Eq. (3), we first need to expand the
Christoffel coefficients:

$\displaystyle\Gamma^{\alpha}_{\beta\gamma}=\bar{\Gamma}^{\alpha}_{\beta\gamma}+\epsilon\Gamma^{\alpha(1)}_{\beta\gamma}+\epsilon^{2}\Gamma^{\alpha(2)}_{\beta\gamma},$
(58)

where

$\displaystyle\Gamma^{\alpha(1)}_{\beta\gamma}=\frac{\bar{g}^{\alpha\lambda}}{2}(h_{\gamma\lambda;\beta}+h_{\beta\lambda;\gamma}-h_{\alpha\beta;\lambda})$
(59)

and

$\displaystyle\Gamma^{\alpha(2)}_{\beta\gamma}=-\frac{h^{\alpha\lambda}}{2}(h_{\gamma\lambda;\beta}+h_{\beta\lambda;\gamma}-h_{\alpha\beta;\lambda}).$
(60)

Then, if $\nabla_{\alpha}$ is the covariant derivative with respect to
$g_{\alpha\beta}$ we conclude

$\displaystyle\Box\varphi$
$\displaystyle=g^{\alpha\beta}\nabla_{\beta}\nabla_{\alpha}\varphi=g^{\alpha\beta}\nabla_{\beta}({\varphi_{,\alpha}})$
$\displaystyle=g^{\alpha\beta}[\varphi_{,\alpha\beta}-(\bar{\Gamma}^{\gamma}_{\alpha\beta}+\epsilon\Gamma^{\gamma(1)}_{\alpha\beta}+\epsilon^{2}\Gamma^{\gamma(2)}_{\alpha\beta})\varphi_{,\gamma}]$
$\displaystyle=\bar{\Box}\bar{\varphi}+\epsilon\bigg{[}\bar{\Box}\delta\varphi-\frac{\bar{g}^{\gamma\lambda}}{2}(2{h_{\mu\lambda}}^{;\mu}-h_{;\lambda})\bar{\varphi}_{,\gamma}-h^{\alpha\beta}\bar{\varphi}_{,\alpha\beta}\bigg{]}$
$\displaystyle\hskip
23.0pt+\epsilon^{2}\bigg{[}\bigg{(}{h_{\mu\lambda}}^{;\mu}-\frac{h_{;\lambda}}{2}\bigg{)}\bigg{(}h^{\gamma\lambda}\bar{\varphi}_{,\gamma}-\bar{g}^{\gamma\lambda}\delta\varphi_{,\gamma}\bigg{)}$
$\displaystyle\hskip
50.0pt+\frac{h^{\alpha\beta}\bar{g}^{\gamma\lambda}}{2}(2h_{\alpha\lambda;\beta}-h_{\alpha\beta;\lambda})\bar{\varphi}_{,\gamma}$
$\displaystyle\hskip
102.0pt-h^{\alpha\beta}\delta\varphi_{,\alpha\beta}+{h^{\alpha}}_{\mu}h^{\mu\beta}\bar{\varphi}_{,\alpha\beta}\bigg{]}.$
(61)

Using Eqs.(48), (52), (61) and (54), one finally finds

$\displaystyle S^{(2)}=\sum_{i=2}^{4}S^{(2)}_{i},$ (62)

where

$\displaystyle S^{(2)}_{2}=\frac{\sqrt{-\bar{g}}}{16\pi G}\bigg{\\{}$
$\displaystyle\frac{\bar{G}_{2}}{8}(h_{\alpha\beta}h^{\alpha\beta}-h^{2})+\frac{h}{2}\bar{G}_{2,\varphi}\delta\varphi+\frac{(\delta\varphi)^{2}}{2}\bar{G}_{2,\varphi\varphi}$
$\displaystyle+\bigg{[}\frac{1}{4}\bar{G}_{2,XX}[h_{\alpha\beta}\bar{\varphi}^{,\alpha}\bar{\varphi}^{,\beta}-2\bar{\varphi}^{,\alpha}\delta\varphi_{,\alpha}]+\delta\varphi\bar{G}_{2,\varphi
X}\bigg{]}\frac{1}{2}(h_{\alpha\beta}\bar{\varphi}^{,\alpha}\bar{\varphi}^{,\beta}-2\bar{\varphi}^{,\alpha}\delta\varphi_{,\alpha})$
$\displaystyle+\frac{1}{4}\bar{G}_{2,X}[-2\delta\varphi^{,\alpha}\delta\varphi_{,\alpha}-2h\delta\varphi^{,\alpha}\bar{\varphi}_{,\alpha}+\bar{\varphi}^{,\beta}(-2h^{\alpha\gamma}h_{\beta\gamma}\bar{\varphi}_{,\alpha}+h_{\alpha\beta}(4\delta\varphi^{,\alpha}+h\bar{\varphi}^{,\alpha}))]\bigg{\\}},$
(63) $\displaystyle S^{(2)}_{3}=\frac{\sqrt{-\bar{g}}}{16\pi G}\bigg{\\{}$
$\displaystyle\bar{G}_{3}\bigg{[}\frac{\bar{\varphi}^{,\alpha}}{4}[h(h_{,\alpha}-2{h_{\alpha\gamma}}^{;\gamma})-2h^{\beta\gamma}(h_{\beta\gamma;\alpha}-2h_{\alpha\beta;\gamma})-2h_{\alpha\beta}(h^{,\beta}-2{h^{\beta\gamma}}_{;\gamma})]$
$\displaystyle+\frac{1}{8}[4h_{,\alpha}\delta\varphi^{,\alpha}-8\delta\varphi_{,\alpha}{h^{\alpha\beta}}_{;\beta}-8h^{\alpha\beta}\delta\varphi_{;\alpha\beta}+4h\bar{\Box}\delta\varphi$
$\displaystyle+\bar{\varphi}_{;\gamma\beta}(8{h_{\alpha}}^{\gamma}h^{\alpha\beta}-4hh^{\gamma\beta})+\bar{\Box}\bar{\varphi}(h^{2}-2h_{\alpha\beta}h^{\alpha\beta})]\bigg{]}$
$\displaystyle+\bar{G}_{3,\varphi}\frac{\delta\varphi}{2}[\bar{\varphi}_{,\alpha}(h^{,\alpha}-2{h^{\alpha\beta}}_{;\beta})+2\bar{\Box}\delta\varphi-2h^{\alpha\beta}\bar{\varphi}_{;\alpha\beta}+h\bar{\Box}\bar{\varphi}]+\bar{G}_{3,\varphi\varphi}\frac{\delta\varphi^{2}}{2}\bar{\Box}\bar{\varphi}$
$\displaystyle+\bar{G}_{3,XX}\frac{\bar{\Box}\bar{\varphi}}{8}\bigg{[}h^{\alpha\beta}\bar{\varphi}_{,\alpha}\bar{\varphi}_{,\beta}-2\bar{\varphi}_{,\gamma}\delta\varphi^{,\gamma}\bigg{]}^{2}+\bar{G}_{3,\varphi
X}\bar{\Box}\bar{\varphi}\frac{\delta\varphi}{2}[h^{\alpha\beta}\bar{\varphi}_{,\alpha}\bar{\varphi}_{,\beta}-2\bar{\varphi}^{,\alpha}\delta\bar{\varphi}_{,\alpha}]$
$\displaystyle+\bar{G}_{3,X}\bigg{[}-\frac{\delta\varphi_{,\alpha}}{2}[(\delta\varphi^{,\alpha}-2h^{\alpha\beta}\bar{\varphi}_{,\beta})\bar{\Box}\bar{\varphi}+\bar{\varphi}^{,\alpha}(2\bar{\Box}\delta\varphi-2h^{\beta\gamma}\bar{\varphi}_{;\beta\gamma}+h\bar{\Box}\bar{\varphi})]$
$\displaystyle+\frac{1}{4}\bar{\varphi}_{,\alpha}\bar{\varphi}_{,\beta}[h^{\alpha\beta}(2\bar{\Box}\delta\varphi+\bar{\varphi}_{,\gamma}(h^{,\gamma}-2{h^{\gamma\delta}}_{;\delta})-2h^{\gamma\delta}\bar{\varphi}_{;\gamma\delta}+h\bar{\Box}\bar{\varphi})-2h^{\alpha\gamma}{h^{\beta}}_{\gamma}\bar{\Box}\bar{\varphi}$
$\displaystyle\hskip
230.0pt-2\delta\varphi^{,\alpha}(h^{,\beta}-2{h^{\beta\gamma}}_{;\gamma})]\bigg{]}\bigg{\\}},$
(64) $\displaystyle S^{(2)}_{4}=\frac{\sqrt{-\bar{g}}}{128\pi G}\bigg{\\{}$
$\displaystyle\bar{G}_{4}\bigg{[}8{h_{\alpha}}^{\gamma}h^{\alpha\beta}\bar{R}_{\beta\gamma}-4hh^{\beta\gamma}\bar{R}_{\beta\gamma}-2h^{\alpha\beta}h_{\alpha\beta}\bar{R}+\bar{R}h^{2}-2h_{,\beta}h^{,\beta}$
$\displaystyle-8({h^{\alpha\beta}}_{;\alpha}-h^{,\beta}){h_{\beta\gamma}}^{;\gamma}+8h^{\alpha\beta}(h_{;\alpha\beta}-{{h_{\alpha\gamma}}^{;\gamma}}_{\beta}-{{h_{\alpha\gamma}}_{;\beta}}^{\gamma}+\bar{\Box}h_{\alpha\beta})$
$\displaystyle+4h[{h^{\beta\gamma}}_{;\beta\gamma}-\bar{\Box}h]+2[3h_{\alpha\beta;\gamma}-2h_{\alpha\gamma;\beta}]h^{\alpha\beta;\gamma}\bigg{]}$
$\displaystyle+4\bar{G}_{4,\varphi}\delta\varphi[h\bar{R}-2h^{\alpha\beta}\bar{R}_{\alpha\beta}+2(h^{\alpha\beta}_{;\alpha\beta}-\bar{\Box}h)]+4\bar{G}_{4,\varphi\varphi}\bar{R}(\delta\varphi)^{2}\bigg{\\}}$
(65)

We can further simplify Eqs. (64) and (65) by using integration by parts to
express terms with two derivative acting upon a single perturbation as a
product of first derivatives of perturbations. As an example:

$\displaystyle 4\bar{G}_{4}h{h^{\beta\gamma}}_{;\beta\gamma}$
$\displaystyle=-4[\bar{G}_{4,\varphi}\bar{\varphi}_{,\gamma}h{h^{\beta\gamma}}_{;\beta}+\bar{G}_{4}h_{,\gamma}{h^{\beta\gamma}}_{;\beta}],$
(66)

where the the total derivative term vanished, since fields are assumed to be
zero at infinity, as usual. When considering the additional powers of
$\epsilon$ coming from the derivative of perturbations (as in Eq. (25)), the
first term in the above equation is of order $\epsilon$. It, then, vanishes in
the weak limit average (variation with respect to background metric does not
change the order of the term). This implies that one can make integration by
parts only paying attention to the perturbation factors of each term. In the
previous example:

$\displaystyle 4\bar{G}_{4}h{h^{\beta\gamma}}_{;\beta\gamma}$
$\displaystyle=-4\bar{G}_{4}h_{,\gamma}{h^{\beta\gamma}}_{;\beta}.$ (67)

With this result, one simplifies:

$\displaystyle S^{(2)}_{3}=\frac{\sqrt{-\bar{g}}}{16\pi G}\bigg{\\{}$
$\displaystyle\bar{G}_{3}\bigg{[}\frac{\bar{\varphi}^{,\alpha}}{4}[h(h_{,\alpha}-2{h_{\alpha\gamma}}^{;\gamma})-2h^{\beta\gamma}(h_{\beta\gamma;\alpha}-2h_{\alpha\beta;\gamma})-2h_{\alpha\beta}(h^{,\beta}-2{h^{\beta\gamma}}_{;\gamma})]$
$\displaystyle+\frac{1}{8}[\bar{\varphi}_{;\gamma\beta}(8{h_{\alpha}}^{\gamma}h^{\alpha\beta}-4hh^{\gamma\beta})+\bar{\Box}\bar{\varphi}(h^{2}-2h_{\alpha\beta}h^{\alpha\beta})]\bigg{]}$
$\displaystyle+\bar{G}_{3,\varphi}\frac{\delta\varphi}{2}[\bar{\varphi}_{,\alpha}(h^{,\alpha}-2{h^{\alpha\beta}}_{;\beta})+2\bar{\Box}\delta\varphi-2h^{\alpha\beta}\bar{\varphi}_{;\alpha\beta}+h\bar{\Box}\bar{\varphi}]+\bar{G}_{3,\varphi\varphi}\frac{\delta\varphi^{2}}{2}\bar{\Box}\bar{\varphi}$
$\displaystyle+\bar{G}_{3,XX}\frac{\bar{\Box}\bar{\varphi}}{8}\bigg{[}h^{\alpha\beta}\bar{\varphi}_{,\alpha}\bar{\varphi}_{,\beta}-2\bar{\varphi}_{,\gamma}\delta\varphi^{,\gamma}\bigg{]}^{2}+\bar{G}_{3,\varphi
X}\bar{\Box}\bar{\varphi}\frac{\delta\varphi}{2}[h^{\alpha\beta}\bar{\varphi}_{,\alpha}\bar{\varphi}_{,\beta}-2\bar{\varphi}^{,\alpha}\delta\varphi_{,\alpha}]$
$\displaystyle+\bar{G}_{3,X}\bigg{[}-\frac{\delta\varphi_{,\alpha}}{2}[(\delta\varphi^{,\alpha}-2h^{\alpha\beta}\bar{\varphi}_{,\beta})\bar{\Box}\bar{\varphi}+\bar{\varphi}^{,\alpha}(2\bar{\Box}\delta\varphi-2h^{\beta\gamma}\bar{\varphi}_{;\beta\gamma}+h\bar{\Box}\bar{\varphi})]$
$\displaystyle+\frac{1}{4}\bar{\varphi}_{,\alpha}\bar{\varphi}_{,\beta}[h^{\alpha\beta}(2\bar{\Box}\delta\varphi+\bar{\varphi}_{,\gamma}(h^{,\gamma}-2{h^{\gamma\delta}}_{;\delta})-2h^{\gamma\delta}\bar{\varphi}_{;\gamma\delta}+h\bar{\Box}\bar{\varphi})-2h^{\alpha\gamma}{h^{\beta}}_{\gamma}\bar{\Box}\bar{\varphi}$
$\displaystyle\hskip
230.0pt-2\delta\varphi^{,\alpha}(h^{,\beta}-2{h^{\beta\gamma}}_{;\gamma})]\bigg{]}\bigg{\\}},$
(68)

$\displaystyle S^{(2)}_{4}=\frac{\sqrt{-\bar{g}}}{128\pi G}\bigg{\\{}$
$\displaystyle\bar{G}_{4}\bigg{[}8{h_{\alpha}}^{\gamma}h^{\alpha\beta}\bar{R}_{\beta\gamma}-4hh^{\beta\gamma}\bar{R}_{\beta\gamma}-2h^{\alpha\beta}h_{\alpha\beta}\bar{R}+\bar{R}h^{2}$
$\displaystyle+2(h_{,\beta}h^{,\beta}-2h^{,\beta}{h_{\beta\gamma}}^{;\gamma}-h_{\alpha\beta;\gamma}h^{\alpha\beta;\gamma}+2h_{\alpha\gamma;\beta}h^{\alpha\beta;\gamma})\bigg{]}$
$\displaystyle+4\bar{G}_{4,\varphi}\delta\varphi[h\bar{R}-2h^{\alpha\beta}\bar{R}_{\alpha\beta}+2({h^{\alpha\beta}}_{;\alpha\beta}-\bar{\Box}h)]+4\bar{G}_{4,\varphi\varphi}\bar{R}(\delta\varphi)^{2}\bigg{\\}},$
(69)

where $S^{(2)}_{2}$ remains as in Eq. (63).

##  IV The GW EMT

###  IV.1 Basic variations

We assume as independent fields of the theory $\hat{\gamma}^{\mu\nu}$,
$\delta\varphi$, $\bar{\varphi}$, and $\bar{g}^{\alpha\beta}$. All other
quantities can be expressed in terms of them. Before varying the action
itself, we give some basic results needed.

First, we remember that (see [29], for example):

$\displaystyle\frac{\delta\bar{g}_{\mu\alpha}}{\delta\bar{g}^{\nu\lambda}}=-\frac{1}{2}(\bar{g}_{\lambda\alpha}\bar{g}_{\mu\nu}+\bar{g}_{\nu\alpha}\bar{g}_{\mu\lambda}),$
(70)
$\displaystyle\frac{\delta\bar{g}^{\alpha\beta}}{\delta\bar{g}^{\mu\nu}}=\frac{1}{2}(\delta^{\alpha}_{\mu}\delta^{\beta}_{\nu}+\delta^{\alpha}_{\nu}\delta^{\beta}_{\mu})$
(71)

and

$\displaystyle\frac{\delta\sqrt{-\bar{g}}}{\delta\bar{g}^{\alpha\beta}}=-\frac{\sqrt{-\bar{g}}}{2}\bar{g}_{\alpha\beta}.$
(72)

Then, one can vary all functions appearing on the Horndeski action. For
example, from Eq. (49):

$\displaystyle\frac{\delta\bar{X}}{\delta\bar{g}^{\alpha\beta}}=-\frac{1}{2}\bar{\varphi}_{,\alpha}\bar{\varphi}_{,\beta},$
(73)

which implies

$\displaystyle\frac{\delta\bar{G}_{i}}{\delta\bar{g}^{\mu\nu}}=-\frac{\bar{G}_{i,X}}{2}\bar{\varphi}_{,\mu}\bar{\varphi}_{,\nu}.$
(74)

The perturbation $h^{\alpha\beta}$ is not an independent field. It is
expressed in terms of the true GW degrees of freedom in Eq. (20). Varying such
relation, we find

$\displaystyle\frac{\delta
h^{\alpha\beta}}{\delta\bar{g}^{\mu\nu}}=\frac{1}{2}[\bar{g}^{\alpha\beta}\hat{\gamma}_{\mu\nu}-\hat{\gamma}\delta^{\alpha}_{(\mu}\delta^{\beta}_{\nu)}]-{\bar{A}^{\alpha\beta}}_{\mu\nu}\delta\varphi,$
(75)

where

$\displaystyle{\bar{A}^{\alpha\beta}}_{\mu\nu}$ $\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=\frac{\delta
C^{\alpha\beta}}{\delta\bar{g}^{\mu\nu}}$
$\displaystyle=\frac{1}{\bar{G}_{4}}\bigg{[}-\frac{1}{2}\bar{G}_{3,XX}\bar{\varphi}_{,\mu}\bar{\varphi}_{,\nu}(\bar{\varphi}^{,\alpha}\bar{\varphi}^{,\beta}+\bar{X}\bar{g}^{\alpha\beta})$
$\displaystyle+\bar{G}_{3,X}\bigg{(}2\bar{\varphi}^{(,\alpha}\delta^{\beta)}_{(\mu}\bar{\varphi}_{,\nu)}-\frac{\bar{g}^{\alpha\beta}}{2}\bar{\varphi}_{,\mu}\bar{\varphi}_{,\nu}+\bar{X}\delta^{\alpha}_{(\mu}\delta^{\beta}_{\nu)}\bigg{)}$
$\displaystyle\hskip
150.0pt+\bar{G}_{4,\varphi}\delta^{\alpha}_{(\mu}\delta^{\beta}_{\nu)}\bigg{]}.$
(76)

Taking the trace of Eq. (20) and varying once more:

$\displaystyle\frac{\delta
h}{\delta\bar{g}^{\mu\nu}}=-\hat{\gamma}_{\mu\nu}+\bar{B}_{\mu\nu}\delta\varphi,$
(77)

where

$\displaystyle\bar{B}_{\mu\nu}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=\frac{\delta\hat{C}}{\delta\bar{g}^{\mu\nu}}=\frac{\bar{\varphi}_{,\mu}\bar{\varphi}_{,\nu}}{\bar{G}_{4}}(\bar{G}_{3,X}+\bar{X}\bar{G}_{3,XX}).$
(78)

and $\hat{C}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=\bar{g}^{\alpha\beta}\hat{C}_{\alpha\beta}$.

###  IV.2 Variation of covariant derivatives

Several terms of the action upon which we wish to operate the variation with
respect to the background metric have covariant derivatives. Since these
derivatives have connection coefficients, one would need, in principle, to
handle the voluminous amount of variation terms arising from it. But one can
show that variations of connection components vanish under the weak limit
average when appearing in covariant derivatives of perturbation fields.
Although this result was already obtained in [20], we revisit it to point out
an important subtlety in our case.

A typical term we are interested here and that will be present after variation
of the action is:

$\displaystyle
T_{1}=\epsilon^{2}\int\sqrt{\bar{g}}{P^{\gamma}}_{\alpha\beta}\delta({h^{\alpha\beta}}_{;\gamma})d^{4}x,$
(79)

where $P^{\gamma}_{\alpha\beta}$ is a tensor that must be present so that the
overall quantity can become a scalar present in the action. Because of the
connections present in the covariant derivative, one can show that

$\displaystyle\delta({h^{\alpha\beta}}_{;\gamma})=(\delta{h^{\alpha\beta}})_{;\gamma}+\delta\bar{\Gamma}^{\alpha}_{\delta\gamma}h^{\delta\beta}+\delta\bar{\Gamma}^{\beta}_{\delta\gamma}h^{\alpha\delta}.$
(80)

The variation of the connection components are of the form

$\displaystyle\delta\bar{\Gamma}^{\alpha}_{\delta\gamma}\sim\bar{g}_{\lambda\delta}(\delta\bar{g}^{\lambda\alpha})_{;\gamma}$
(81)

So that they will appear in $T_{1}$ as

$\displaystyle T_{1}$
$\displaystyle\sim\epsilon^{2}\int\sqrt{\bar{g}}P^{\gamma}_{\alpha\beta}\bar{g}_{\lambda\delta}(\delta\bar{g}^{\lambda\alpha})_{;\gamma}h^{\delta\beta}d^{4}x$
$\displaystyle\sim-\epsilon^{2}\int\sqrt{\bar{g}}(P^{\gamma}_{\alpha\beta}{h_{\lambda}}^{\beta})_{;\gamma}\delta\bar{g}^{\lambda\alpha}d^{4}x.$
(82)

which, by Eq. (38), results in a contribution to the GW EMT that goes like

$\displaystyle
T^{GW}_{\alpha\lambda}\sim\langle\epsilon^{2}({P^{\gamma}}_{\alpha\beta}{h_{\lambda}}^{\beta})_{;\gamma}\rangle.$
(83)

By inspecting the terms in the action of the form of Eq. (79), one is able to
conclude that
$\mathcal{O}(\epsilon{P^{\gamma}}_{\alpha\beta})=\mathcal{O}(\epsilon)$ or
$\mathcal{O}(\epsilon^{0})$. Using an analogous argument as the one given in
Eq. (33), one can conclude that the above term must vanish. This implies that
we can discard connection variations when varying first covariant derivatives
of the tensorial perturbation. By a similar argument, one can conclude the
same property for the other perturbation derivatives, such as
$\delta\phi_{;\alpha\beta}$ and $h_{\alpha\beta;\mu\nu}$.

The subtlety present in this demonstration is that it is not valid if the
field being differentiated is of background nature. In theories where the
metric is the only degree of freedom, this is not a problem, since covariant
derivatives of the background metric are always zero. But in Horndeski
theories, we have the presence of the background scalar field. Consequently,
terms like

$\displaystyle
T_{2}=\epsilon^{2}\int\sqrt{-\bar{g}}L^{\alpha\beta}\delta(\bar{\varphi}_{;\alpha\beta})d^{4}x,$
(84)

where $L^{\alpha\beta}$ is some tensor necessary to form a scalar quantity in
the action, will appear after variation. Varying the correspondent connections
will give contributions to the GW EMT of the form:

$\displaystyle
T_{\phi\lambda}\sim\langle\epsilon^{2}({L^{\beta}}_{\phi}\bar{\varphi}_{,\lambda})_{;\beta}\rangle.$
(85)

But terms can be found in the action such that
$\mathcal{O}(\epsilon^{2}L^{\alpha\beta})=\mathcal{O}(\epsilon^{0})$. More
explicitly, they are the ones involving:

$\displaystyle
L^{\alpha\beta}=-\frac{1}{2}\bar{G}_{3,X}(\delta\varphi)_{,\lambda}(\delta\varphi)^{,\lambda}\bar{g}^{\alpha\beta}$
(86)

and

$\displaystyle
L^{\alpha\beta}=\frac{\bar{G}_{3,XX}}{2}(\bar{g}_{\mu\nu}\delta\varphi^{,\mu}\bar{\varphi}^{,\nu})^{2}\bar{g}^{\alpha\beta}.$
(87)

In these examples, we cannot neglect the variation of connection components.

###  IV.3 The energy momentum tensor

We are now finally able to obtain the GW EMT. We write the second order of the
action as:

$\displaystyle S^{(2)}=\frac{\sqrt{-\bar{g}}}{16\pi
G}\sum_{i=2}^{4}s^{(2)}_{i},$ (88)

where $s_{2}^{(2)},s_{3}^{(2)}$ and $s_{4}^{(2)}$ are related, respectively,
with Eqs. (63), (64) and (65). We remember that, because of Eqs. (25) and (26)
and the limit made in Eq. (27), the only terms surviving the averaging process
are those in which the number of derivatives acting on perturbation fields is
at least the number of perturbation factors appearing on it. Varying and using
Eq. (38), one can separate the GW EMT in two parts:

$\displaystyle
T^{GW}_{\mu\nu}=T^{GW(first)}_{\mu\nu}+T^{GW(second)}_{\mu\nu}.$ (89)

The first comes from the variation of $\sqrt{-\bar{g}}$ (Eq. (72)) and reads:

$\displaystyle T^{GW(first)}_{\mu\nu}$
$\displaystyle=\frac{\bar{g}_{\mu\nu}}{16\pi
G}\sum_{i=2}^{4}\langle\epsilon^{2}s^{(2)}_{i}\rangle=\frac{\bar{g}_{\mu\nu}}{16\pi
G}\bigg{\langle}\epsilon^{2}\bigg{\\{}\frac{1}{2}[\bar{G}_{2,XX}(\delta\varphi_{,\alpha}\bar{\varphi}^{,\alpha})^{2}-\bar{G}_{2,X}\delta\varphi_{,\alpha}\delta\varphi^{,\alpha}]+\bar{G}_{4,\varphi}\delta\varphi({h^{\alpha\beta}}_{;\alpha\beta}-\bar{\Box}h)$
$\displaystyle\hskip
130.0pt+\frac{\bar{G}_{4}}{4}\\{h_{,\beta}h^{,\beta}-2h^{,\beta}{h_{\beta\gamma}}^{;\gamma}+(2h_{\alpha\gamma;\beta}-h_{\alpha\beta;\gamma})h^{\alpha\beta;\gamma}\\}+\bar{G}_{3,\varphi}\delta\varphi\bar{\Box}\delta\varphi$
$\displaystyle\hskip
130.0pt+\bar{G}_{3,XX}\frac{\bar{\Box}\bar{\varphi}}{2}(\bar{\varphi}_{,\gamma}\delta\varphi^{,\gamma})^{2}+\frac{1}{2}\bar{G}_{3,X}\\{-\delta\varphi_{,\alpha}\delta\varphi^{,\alpha}\bar{\Box}\bar{\varphi}+\bar{\varphi}_{,\alpha}\bar{\varphi}_{,\beta}[h^{\alpha\beta}\bar{\Box}\delta\varphi$
$\displaystyle\hskip
130.0pt-\delta\varphi^{,\alpha}(h^{,\beta}-2{h^{\beta\gamma}}_{;\gamma})]-\delta\varphi_{,\alpha}\bar{\varphi}^{,\alpha}\bar{\Box}\delta\varphi\\}\bigg{\\}}\bigg{\rangle}.$
(90)

The second part comes from the variation of the $s_{i}^{(2)}$’s. Because
variation with respect to $\bar{g}^{\alpha\beta}$ does not change the order of
the quantity with respect to $\epsilon$,

$\displaystyle\bigg{\langle}\epsilon^{2}\frac{\delta
s^{(2)}_{i}}{\delta\bar{g}^{\mu\nu}}\bigg{\rangle}=\frac{\delta}{\delta\bar{g}^{\mu\nu}}\langle\epsilon^{2}s^{(2)}_{i}\rangle$
(91)

and we only need to vary the terms that survive the averaging process.

We conclude that

$\displaystyle T^{GW(second)}_{\mu\nu}=-\frac{1}{8\pi
G}\sum_{i=2}^{4}\langle\epsilon^{2}\delta
s_{i}^{(2)}\rangle=\sum_{i=2}^{4}T^{(second,i)}_{\mu\nu},$ (92)

where we present each $T^{(second,i)}_{\mu\nu}$ and details on how to obtain
them in Appendix A. Combining contributions of Eq. (90) and (92), we finally
find:

$\displaystyle T^{GW}_{\mu\nu}=\sum_{i=2}^{4}T^{GW(i)}_{\mu\nu},$ (93)

where

$\displaystyle T^{GW(2)}_{\mu\nu}=\frac{1}{16\pi
G}{\bar{W}^{\lambda\phi}}_{\mu\nu}\langle\epsilon^{2}\delta\varphi_{,\lambda}\delta\varphi_{,\phi}\rangle,$
(94)

with

$\displaystyle{\bar{W}^{\lambda\phi}}_{\mu\nu}=$
$\displaystyle\bar{G}_{2,X}\bigg{[}\delta^{\lambda}_{\mu}\delta^{\phi}_{\nu}-\frac{\bar{g}_{\mu\nu}}{2}\bar{g}^{\lambda\phi}\bigg{]}$
$\displaystyle-2\bar{G}_{2,XX}\bigg{[}\bar{\varphi}_{(,\mu}\delta^{\phi}_{,\nu)}\bar{\varphi}^{,\lambda}+\frac{\bar{\varphi}_{,\mu}\bar{\varphi}_{,\nu}}{4}\bar{g}^{\lambda\phi}$
$\displaystyle-\frac{\bar{g}_{\mu\nu}}{4}\bar{\varphi}^{,\lambda}\bar{\varphi}^{,\phi}\bigg{]}+\bar{G}_{2,XXX}\frac{\bar{\varphi}_{,\mu}\bar{\varphi}_{,\nu}}{2}\bar{\varphi}^{,\lambda}\bar{\varphi}^{,\phi},$
(95)
$\displaystyle T^{GW(4)}_{\mu\nu}=\frac{1}{16\pi G}\bigg{\\{}$
$\displaystyle\frac{\bar{G}_{4}}{2}\bigg{[}\langle\epsilon^{2}\hat{\gamma}_{\alpha\beta;\mu}{\hat{\gamma}^{\alpha\beta}}_{;\nu}\rangle-\frac{1}{2}\langle\epsilon^{2}\hat{\gamma}_{,\mu}\hat{\gamma}_{,\nu}\rangle\bigg{]}$
$\displaystyle+{\bar{O}^{\lambda\phi}}_{\mu\nu}\langle\delta\varphi_{,\lambda}\delta\varphi_{,\phi}\rangle+\bar{G}_{4,\varphi}\langle\hat{\gamma}_{,\mu}\delta\varphi_{,\nu}\rangle$
$\displaystyle+\langle
N_{\mu\nu}\bar{\Box}\delta\varphi\rangle+{\bar{Q}^{\alpha\beta\lambda\phi}}_{\mu\nu}\langle\hat{\gamma}_{\alpha\beta,\lambda}\delta\varphi_{,\phi}\rangle\bigg{\\}},$
(96)

with

$\displaystyle{\bar{O}^{\alpha\beta}}_{\mu\nu}=\frac{\bar{G}_{4}}{2}$
$\displaystyle[(C_{\gamma\lambda}C^{\gamma\lambda}-C^{2})\delta^{\alpha}_{\mu}\delta^{\beta}_{\nu}-4{\bar{A}^{\alpha\gamma}}_{\mu\nu}{C_{\gamma}}^{\beta}$
$\displaystyle\hskip
9.0pt-(C\bar{g}_{\mu\nu}+2\bar{B}_{\mu\nu})C^{\alpha\beta}+2C{\bar{A}^{\alpha\beta}}_{\mu\nu}$
$\displaystyle\hskip
25.0pt+(2\bar{g}_{\gamma\mu}\bar{g}_{\delta\nu}+\bar{g}_{\mu\nu}\bar{g}_{\gamma\delta})C^{\gamma\alpha}C^{\delta\beta}]$
$\displaystyle+2\bar{G}_{4,\varphi}\bigg{[}C\delta_{\mu}^{\alpha}\delta_{\nu}^{\beta}+\frac{\bar{g}_{\mu\nu}}{2}C^{\alpha\beta}-{\bar{A}^{\alpha\beta}}_{\mu\nu}\bigg{]},$
(97)

$\displaystyle
N_{\mu\nu}=\frac{\bar{G}_{4}}{2}\bigg{[}2h\bar{B}_{\mu\nu}+2h_{\alpha\beta}{\bar{A}^{\alpha\beta}}_{\mu\nu}$
$\displaystyle-2\bar{C}_{\mu\beta}{h_{\nu}}^{\beta}$
$\displaystyle+\frac{\bar{g}_{\mu\nu}}{2}(hC-
h_{\alpha\beta}\bar{C}^{\alpha\beta})\bigg{]}$
$\displaystyle+2\bar{G}_{4,\varphi}\delta\varphi\bigg{[}\bar{B}_{\mu\nu}+\frac{\bar{g}_{\mu\nu}}{2}\bar{C}\bigg{]},$
(98)
$\displaystyle{\bar{Q}^{\alpha\beta\lambda\phi}}_{\mu\nu}=\frac{\bar{G}_{4}}{2}[3C^{\lambda\phi}\delta^{\alpha}_{\mu}\delta^{\beta}_{\nu}-2C^{\alpha\beta}\delta^{\lambda}_{\mu}\delta^{\phi}_{\nu}]$
(99)

and, finally,

$\displaystyle T^{GW(3)}_{\mu\nu}=\frac{1}{16\pi G}\bigg{\\{}$
$\displaystyle{\bar{E}^{\lambda\phi}}_{\mu\nu}\langle\epsilon^{2}\delta\varphi_{,\lambda}\delta\varphi_{,\phi}\rangle+\langle\epsilon^{2}D_{\mu\nu}\bar{\Box}\delta\varphi\rangle$
$\displaystyle\hskip
35.0pt-\frac{\bar{G}_{3,X}}{2}\bar{\varphi}_{,\gamma}\bar{\varphi}^{,\gamma}\langle\epsilon^{2}\hat{\gamma}_{,\mu}\delta\varphi_{,\nu}\rangle$
$\displaystyle\hskip
15.0pt+{\bar{I}^{\alpha\beta\lambda\phi}}_{\mu\nu}\langle\epsilon^{2}\hat{\gamma}_{\alpha\beta,\lambda}\delta\varphi_{,\phi}\rangle+\bar{J}_{\mu\nu}$
$\displaystyle\hskip
40.0pt+\bar{G}_{3,X}\bar{\varphi}^{\alpha}\langle\epsilon^{2}\delta\varphi_{,\alpha}\delta\varphi_{;\mu\nu}\rangle\bigg{\\}},$
(100)

with

$\displaystyle D_{\mu\nu}=$
$\displaystyle\bar{G}_{3,\varphi}\bar{g}_{\mu\nu}\delta\varphi+\bar{G}_{3,\varphi
X}\bar{\varphi}_{,\mu}\bar{\varphi}_{,\nu}\delta\varphi$
$\displaystyle-\bar{G}_{3,X}\bigg{[}\bar{\varphi}_{,\alpha}\bar{\varphi}_{,\beta}\frac{\delta
h^{\alpha\beta}}{\delta\bar{g}^{\mu\nu}}+\bar{\varphi}_{;\mu\nu}\delta\varphi-\bar{\varphi}_{(,\mu}\delta\varphi_{,\nu)}$
$\displaystyle\hskip
45.0pt-\frac{\bar{g}_{\mu\nu}}{2}(\delta\varphi\bar{\Box}\bar{\varphi}+\bar{\varphi}_{,\alpha}\bar{\varphi}_{,\beta}h^{\alpha\beta}-\delta\varphi_{,\alpha}\bar{\varphi}^{,\alpha})\bigg{]}$
$\displaystyle-\bar{G}_{3,XX}\bigg{[}\frac{\bar{\varphi}_{,\mu}\bar{\varphi}_{,\nu}}{2}(\delta\varphi_{,\alpha}\bar{\varphi}^{,\alpha}-\delta\varphi\bar{\Box}\bar{\varphi}-\bar{\varphi}_{,\alpha}\bar{\varphi}_{,\beta}h^{\alpha\beta})\bigg{]}$
$\displaystyle\hskip
90.0pt+\bar{G}_{3,XXX}\frac{\bar{\varphi}_{,\mu}\bar{\varphi}_{,\nu}}{2}(\bar{\varphi}_{,\gamma}\delta\varphi^{,\gamma})^{2},$
(101) $\displaystyle{\bar{E}^{\lambda\phi}}_{\mu\nu}=$ $\displaystyle
2\bar{G}_{3,\varphi}\delta^{\lambda}_{\mu}\delta^{\phi}_{\nu}$
$\displaystyle+\bar{G}_{3,X}\bigg{\\{}[\bar{\Box}\bar{\varphi}-\bar{C}^{\alpha\beta}\bar{\varphi}_{\alpha}\bar{\varphi}_{\beta}]\delta^{\lambda}_{\mu}\delta^{\phi}_{\nu}$
$\displaystyle+2\bar{\varphi}_{,\beta}\bar{\varphi}_{(,\mu}\delta^{\lambda}_{\nu)}\bar{C}^{\beta\phi}+\frac{\bar{g}_{\mu\nu}}{2}\bar{\varphi}^{,\lambda}[\bar{C}\bar{\varphi}^{,\phi}-2\bar{C}^{\beta\phi}\bar{\varphi}_{,\beta}]$
$\displaystyle\hskip
20.0pt-\bar{\varphi}^{,\lambda}[2\bar{C}\delta^{\phi}_{(\mu}\bar{\varphi}_{,\nu)}-\bar{B}_{\mu\nu}\bar{\varphi}^{,\phi}-2{\bar{A}^{\beta\phi}}_{\mu\nu}\bar{\varphi}_{,\beta}]\bigg{\\}}$
$\displaystyle-2\bar{G}_{3,XX}\bar{\varphi}^{,\lambda}\bigg{\\{}\bar{\varphi}_{(,\mu}\delta^{\phi}_{\nu)}\bar{\Box}\bar{\varphi}-\frac{1}{2}\bar{\varphi}^{,\phi}\bar{\varphi}_{;\mu\nu}$
$\displaystyle\hskip
20.0pt-\frac{\bar{g}_{\mu\nu}}{4}\bar{\varphi}^{\phi}\bar{\Box}\bar{\varphi}-\frac{\bar{\varphi}_{,\mu}\bar{\varphi}_{,\nu}}{4}[C\bar{\varphi}^{,\phi}-2\bar{C}^{\beta\phi}\bar{\varphi}_{,\beta}]\bigg{\\}},$
(102) $\displaystyle{\bar{I}^{\alpha\beta\lambda\phi}}_{\mu\nu}=$
$\displaystyle\bar{G}_{3,X}(\bar{\varphi}^{,\alpha}\bar{\varphi}^{,\beta}\delta^{\lambda}_{\mu}\delta^{\phi}_{\nu}-2\bar{\varphi}^{,\lambda}\bar{\varphi}^{,\phi}\delta^{\alpha}_{\mu}\delta^{\beta}_{\nu}),$
(103) $\displaystyle\bar{J}_{\mu\nu}=$
$\displaystyle\bigg{(}\delta^{\alpha}_{(\mu}\delta^{\beta}_{\nu)}-\bar{g}^{\alpha\beta}\frac{\bar{g}_{\mu\nu}}{2}\bigg{)}\langle[(\bar{G}_{3,XX}(\bar{\varphi}_{,\gamma}\delta\varphi^{,\gamma})^{2}$
$\displaystyle\hskip
86.0pt-\bar{G}_{3,X}\delta\varphi_{,\gamma}\delta\varphi^{,\gamma})\bar{\varphi}_{,\alpha}]_{;\beta}\rangle$
(104)

Because the quantities $\bar{C}_{\mu\nu}$, $\bar{B}_{\mu\nu}$ and
${\bar{A}^{\alpha\beta}}_{\mu\nu}$ are totally symmetric in their indices and
because of the property exemplified by Eq. (37), almost all terms appearing in
the GW EMT are symmetric at a glance. The only term not manifestly symmetric
is
$-\langle\bar{G}_{4}\bar{C}_{\mu\beta}{h_{\nu}}^{\beta}\bar{\Box}\delta\varphi\rangle$,
present in Eq. (98). It is simple to notice, using the properties derived in
Subsec. (II.6), that:

$\displaystyle\langle\bar{G}_{4}\bar{C}_{\mu\beta}{h_{\nu}}^{\beta}\bar{\Box}\delta\varphi\rangle=\langle\bar{G}_{4}{h_{\nu}}^{\beta}\bar{\Box}(\bar{C}_{\mu\beta}\delta\varphi)\rangle$
(105)

since the difference between the sides of the equation vanishes in the weak
limit. Furthermore, since we are assuming luminal tensorial modes, Eq. (182),
one may add a vanishing quantity to obtain:

$\displaystyle\langle\bar{G}_{4}\bar{C}_{\mu\beta}{h_{\nu}}^{\beta}\bar{\Box}\delta\varphi\rangle$
$\displaystyle=-\langle\bar{G}_{4}{h_{\nu}}^{\beta}\bar{\Box}(\hat{\gamma}_{\mu\beta}-\bar{C}_{\mu\beta}\delta\varphi)\rangle$
$\displaystyle=-\bigg{\langle}\bar{G}_{4}{h_{\nu}}^{\beta}\bar{\Box}h_{\mu\beta}+\frac{1}{2}h_{\mu\nu}\bar{\Box}\hat{\gamma}\bigg{\rangle},$
(106)

where in the last equality we have used Eq. (20). While the second term is
evidently symmetric, the first one is shown to be as well by noticing that:

$\displaystyle\langle\bar{G}_{4}{h_{\nu}}^{\beta}\bar{\Box}h_{\mu\beta}\rangle=\langle\bar{G}_{4}h_{\mu\beta}\bar{\Box}{h_{\nu}}^{\beta}\rangle,$
(107)

a result of the property exemplified by Eq. (35). We then can conclude that

$\displaystyle T^{(GW)}_{\mu\nu}=T^{(GW)}_{\nu\mu},$ (108)

a property expected either from physical reasons or simply from the definition
of GW EMT, Eq. (11).

Another important aspect we would like to investigate is the trace of the
tensor. In GR, GWs behave like a radiation fluid, since the trace of its EMT
vanishes. But in more general theories of gravity, the tensor can have non-
vanishing trace. In particular, if such quantity is negative, as studies in
$f(R)$ theories already have suggested to be the case [22], it can effectively
act as a dark energy candidate in the context of cosmological space-times.

We can inquire if the trace of the obtained GW EMT vanishes in general or if
there is the possibility of the correspondent fluid to have distinct behaviors
depending on the subclass of theories one is interested in. If one wants to
prove that such trace does not vanish in the general reduced Horndeski scheme,
it is only necessary to show that the trace of a particular coefficient of the
$\bar{G}_{i}$ functions does not vanish, since they are independent from each
other.

Assuming, for example, that $\bar{G}_{2,X}\neq 0$ but $\bar{G}_{2,XX}=0$:

$\displaystyle T^{GW(2)}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}={{T^{GW(2)}}_{\mu}}^{\mu}$
$\displaystyle=-\frac{1}{16\pi
G}\langle\epsilon^{2}\delta\varphi_{,\mu}\delta\varphi^{,\mu}\rangle$
$\displaystyle=\frac{1}{16\pi
G}\langle\epsilon^{2}\delta\varphi\bar{\Box}\delta\varphi\rangle,$ (109)

which is not necessarily zero for theories where scalar waves are non-luminal.
We then conclude that there are theories within the reduce Horndeski family
that have the GW EMT with trace different from zero. It would be interesting
to investigate, for example, in what subclass of theories this tensor has
negative trace and, thus, can behave as dark energy.

We additionally point out that, although the weak limit average discards all
contributions of order $\epsilon^{n}$, with $n>0$, it amplifies terms having
negative powers in this perturbation factor. They are terms in which the
number of derivatives acting on perturbations exceeds the number of
perturbation factors. In the tensor calculated, there are terms of this kind
only in $T^{GW(3)}_{\mu\nu}$, more precisely, in $D_{\mu\nu}$ and in the last
term of Eq. (100). The $\bar{J}_{\mu\nu}$ contribution, although having more
derivatives than perturbations, is not problematic since one of the
derivatives is a total derivative and, by the property established in Eq. 34,
it has order $\epsilon^{0}$. The terms with negative powers in $\epsilon$ have
divergent behavior under the weak limit and, if present, make the GW EMT loose
its physical meaning.

We emphasize that the complete equations of motion were not imposed on the GW
EMT and that they may be helpful when dealing with such divergent terms. For
example, it is easy to see that all divergent terms vanish when we deal with
luminal scalar-waves, since the dominant order in powers of $\epsilon$ of the
equation of motion make D’alembertian terms vanish in the EMT and
$\bar{G}_{3,X}=0$. Furthermore, theories in which $\bar{G}_{3,X}=0$ (a
necessary but not sufficient condition for luminal scalar-waves) do not have
any divergent term. On the other hand, we believe that, in theories in which
such terms do not vanish after imposing the equations of motion, the procedure
adopted here to obtain the energy-momentum information of GWs is incomplete
and another formalism must be developed in order to treat it. A possible
solution to this stalemate is to maintain the amplitude and wavelength
perturbative parameters ($\alpha$ and $\epsilon$) initially introduced in Eq.
(9) as independent variables (instead of assuming them to be of same order, as
in Eq.(12)) and to make a limit procedure in each one, imposing a hierarchy of
smallness between them.

##  V Special cases

We now investigate particular important cases of the GW EMT. We take the
opportunity to reobtain results already derived in literature regarding the
particular case of a Brans-Dicke theory. The special cases are summarized in
Fig.(1).

Figure 1: Special cases studied in this work, with the equations that give the
corresponding GW EMT. Dots stand for particular theories and where they fit in
the diagram.

###  V.1 Luminal scalar waves

Scalar waves become luminal when the diagonalized version of the kinetic
operator present in Eq. (15) is proportional to the metric which,
consequently, implies that the second order differential operator acting on
the scalar perturbation is just the D’alembertian. As pointed out in [24],
this occurs in theories where:

$\displaystyle\bar{G}_{2,XX}=\bar{G}_{3,X}=0.$ (110)

Inserting this equation in Eqs. (21), (76) and (78), we find:

$\displaystyle{\bar{A}^{\alpha\beta}}_{\mu\nu}$
$\displaystyle=\frac{\bar{G}_{4,\varphi}}{\bar{G}_{4}}\delta^{\alpha}_{(\mu}\delta^{\beta}_{\nu)},$
(111) $\displaystyle\bar{B}_{\mu\nu}$ $\displaystyle=0,$ (112)
$\displaystyle\bar{C}_{\mu\nu}$
$\displaystyle=\frac{\bar{G}_{4,\varphi}}{\bar{G}_{4}}\bar{g}_{\mu\nu}.$ (113)

Analogous to Eq. (182), the dominant term of the equation of motion in orders
of $\epsilon$ for scalar waves imply, then:

$\displaystyle\langle\epsilon^{2}W\bar{\Box}\delta\varphi\rangle=0,$ (114)

where $W=\mathcal{O}(1)$.

With these simplifications, the GW EMT becomes (remembering the integration by
parts property exemplified in Eq. (35)):

$\displaystyle T^{GW}_{\mu\nu}=\frac{1}{16\pi G}\bigg{\\{}$
$\displaystyle\bigg{(}\bar{G}_{2,X}+2\bar{G}_{3,\varphi}+3\frac{\bar{G}_{4,\varphi}^{2}}{\bar{G}_{4}}\bigg{)}\langle\epsilon^{2}\delta\varphi_{,\mu}\delta\varphi_{,\nu}\rangle$
$\displaystyle+\frac{\bar{G}_{4}}{2}\bigg{[}\langle\epsilon^{2}\hat{\gamma}_{\alpha\beta;\mu}{\hat{\gamma}^{\alpha\beta}}_{;\nu}\rangle-\frac{1}{2}\langle\epsilon^{2}\hat{\gamma}_{,\mu}\hat{\gamma}_{,\nu}\rangle\bigg{]}\bigg{\\}}.$
(115)

Since in this subclass the D’alembertian of every perturbation vanishes, the
trace of the GW EMT continue to be zero:

$\displaystyle T^{GW}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=T^{GW\mu}_{\mu}=0.$
(116)

which is expected since here all fields travel at the speed of light and the
correspondent fluid should be of radiation nature as a consequence.

###  V.2 Theories with TT gauge

As mentioned previously, although the harmonic gauge can always be achieved,
this is not true to the more restrict TT gauge. By the arguments given in
[24], the TT gauge can only be obtained in an open set if 666The condition
here should be $G_{3,X}=G_{4,\varphi}=0$. Expanding these in power of
$\epsilon$ and demanding that the $\epsilon^{0}$ contributions must vanish
alone, gives the correspondent statements for the background quantities.:

$\displaystyle\bar{G}_{3,X}=\bar{G}_{4,\varphi}=0.$ (117)

In this case:

$\displaystyle\bar{C}_{\mu\nu}=\bar{B}_{\mu\nu}=0,$ (118)
$\displaystyle{\bar{A}^{\alpha\beta}}_{\mu\nu}=0.$ (119)

Since $C_{\mu\nu}=0$,

$\displaystyle\hat{\gamma}_{\mu\nu}=\hat{h}_{\mu\nu}.$ (120)

Since the TT gauge is achievable, we demand that:

$\displaystyle\hat{h}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}={\hat{h}^{\alpha}}_{\alpha}=0.$
(121)

The GW EMT gives:

$\displaystyle T^{GW}_{\mu\nu}=T^{GW(2)}_{\mu\nu}+$
$\displaystyle\frac{1}{16\pi
G}\bigg{\\{}\frac{\bar{G}_{4}}{2}\langle\epsilon^{2}{\hat{\gamma}_{\alpha\beta}}_{;\mu}{\hat{\gamma}^{\alpha\beta}}_{;\nu}\rangle$
$\displaystyle+\bar{G}_{3,\varphi}[2\delta^{\lambda}_{\mu}\delta^{\phi}_{\nu}-\bar{g}_{\mu\nu}\bar{g}^{\lambda\phi}]\langle\epsilon^{2}\delta\varphi_{,\lambda}\delta\varphi_{,\phi}\rangle\bigg{\\}}.$
(122)

In this case, the trace is not trivially zero:

$\displaystyle T^{GW}=$
$\displaystyle[-\bar{G}_{2,X}\bar{g}^{\lambda\phi}-\bar{X}(\bar{G}_{2,XXX}\bar{\varphi}^{,\lambda}\bar{\varphi}^{,\phi}-\bar{G}_{2,XX}\bar{g}^{\lambda\phi})$
$\displaystyle\hskip
97.0pt-2\bar{G}_{3,\varphi}\bar{g}^{\lambda\phi}]\langle\epsilon^{2}\delta\varphi_{,\lambda}\delta\varphi_{,\phi}\rangle.$
(123)

###  V.3 Brans-Dicke theory

For the Brans-Dicke theory, the GW EMT and the consequent signal for the
stochastic gravitational-wave background were already obtained in [11], for
the Einstein frame. Here we reobtain it as a particular case of our result and
obtain the correspondent Jordan frame tensor as well, comparing both.

The action of Brans-Dicke theory in the Jordan frame is:

$\displaystyle S^{(BD)}=\frac{1}{16\pi G}\int\sqrt{-g}\bigg{[}\varphi
R-\frac{\omega_{BD}}{\varphi}g^{\alpha\beta}\varphi_{,\alpha}\varphi_{,\beta}\bigg{]}d^{4}x,$
(124)

where $\omega_{BD}$ is a real number. This is a special case of reduced
Horndeski theory with

$\displaystyle G_{2}(\varphi,X)$
$\displaystyle=2\frac{\omega_{BD}}{\varphi}X,$ (125) $\displaystyle
G_{4}(\varphi,X)$ $\displaystyle=\varphi,$ (126) $\displaystyle
G_{3}(\varphi,X)$ $\displaystyle=G_{5}(\varphi,X)=0.$ (127)

It is then a theory in which scalar waves are luminal ($G_{3,X}=G_{2,XX}=0$)
but the TT gauge cannot be achieved ($G_{4,\varphi}\neq 0$). From Eq. (115),
then, the GW EMT in the Jordan frame is:

$\displaystyle 16\pi
GT_{\mu\nu}^{(BD}=\frac{(2\omega_{BD}+3)}{\bar{\varphi}}\langle\epsilon^{2}\delta\varphi_{,\mu}\delta\varphi_{,\nu}\rangle$
$\displaystyle-\frac{\bar{\varphi}}{4}\langle\epsilon^{2}\hat{\gamma}_{;\mu}\hat{\gamma}_{;\nu}\rangle$
$\displaystyle+\frac{\bar{\varphi}}{2}\langle\epsilon^{2}\hat{\gamma}_{\alpha\beta;\mu}{\hat{\gamma}^{\alpha\beta}}_{;\nu}\rangle.$
(128)

One usually states the theory in terms of another action, that is more similar
to the GR one:

$\displaystyle\undertilde{S}^{(BD)}=\frac{1}{16\pi
G}\int\sqrt{-\undertilde{g}}[\undertilde{R}-2\undertilde{g}^{\alpha\beta}\undertilde{\varphi}_{,\alpha}\undertilde{\varphi}_{,\beta}]d^{4}x,$
(129)

where the new metric and the new scalar field are defined as:

$\displaystyle\undertilde{g}_{\mu\nu}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=\frac{\varphi}{\bar{\varphi}}g_{\mu\nu},$
(130) $\displaystyle\undertilde{\varphi}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=\undertilde{\bar{\varphi}}+\epsilon\undertilde{\delta\varphi}.$
(131)

with

$\displaystyle\frac{\varphi}{\bar{\varphi}}$ $\displaystyle=\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}e^{-2\frac{\epsilon\undertilde{\delta\varphi}}{\sqrt{3+2\omega_{BD}}}}.$
(132)

When such transformation is made, we say we have changed from the Jordan to
the Einstein frame. Undertilde quantities indicate that the Einstein frame
fields are being used.

Expanding both sides of Eq. (130), one finds:

$\displaystyle\undertilde{\bar{g}}_{\mu\nu}$ $\displaystyle=\bar{g}_{\mu\nu},$
(133) $\displaystyle\undertilde{h}_{\mu\nu}$
$\displaystyle=h_{\mu\nu}+\frac{\delta\varphi}{\bar{\varphi}}\bar{g}_{\mu\nu}.$
(134)

Because of Eq. (133), one does not need to distinguish between contractions or
covariant derivatives with one or the other background metric. Expanding Eq.
(132) to linear order in $\epsilon$:

$\displaystyle\undertilde{\delta\varphi}=-\frac{\delta\varphi}{\bar{\varphi}}\frac{\sqrt{3+2\omega_{BD}}}{2}.$
(135)

Notice that, assuming the Einstein frame fields to be the fundamental ones,
the Horndeski functions would be:

$\displaystyle\undertilde{G}_{2}(\undertilde{\varphi},\undertilde{X})$
$\displaystyle=4\undertilde{X},$ (136)
$\displaystyle\undertilde{G}_{4}(\undertilde{\varphi},\undertilde{X})$
$\displaystyle=1,$ (137)
$\displaystyle\undertilde{G}_{3}(\undertilde{\varphi},\undertilde{X})$
$\displaystyle=\undertilde{G}_{5}(\undertilde{\varphi},\undertilde{X})=0.$
(138)

We see that in this case, scalar waves continue to be luminal, but now the TT
gauge can be achieved, since
$\undertilde{G}_{4,\undertilde{\phi}}=0=\undertilde{G}_{3,\undertilde{X}}$.

The Einstein frame GW EMT is, then:

$\displaystyle 16\pi
G\undertilde{T}^{(BD)}_{\mu\nu}=4\langle\epsilon^{2}\undertilde{\delta\varphi}_{,\mu}\undertilde{\delta\varphi}_{,\nu}\rangle$
$\displaystyle+\frac{1}{2}\langle\epsilon^{2}\undertilde{\hat{h}}_{\alpha\beta;\mu}{\undertilde{\hat{h}}^{\alpha\beta}}_{;\nu}\rangle$
$\displaystyle-\frac{1}{4}\langle\epsilon^{2}\undertilde{\hat{h}}_{,\mu}\undertilde{\hat{h}}_{,\nu}\rangle,$
(139)

where we have used that Eq. (120) is valid on the Einstein frame but have not
imposed the TT gauge 777Notice that no TT gauge is necessary to obtain Eq.
(120). It is only necessary for the theory to have the right Horndeski
functions so that the TT gauge is achievable. The gauge we are using in this
section is the given by Eq. (142). so that the comparison with literature can
be done. The next step is to use Eqs. (134) and (135) to express everything in
terms of the Jordan fields. We have:

$\displaystyle
4\langle\delta\undertilde{\varphi}_{,\mu}\delta\undertilde{\varphi}_{,\nu}\rangle=\frac{3+2\omega_{BD}}{\bar{\varphi}^{2}}\langle\delta\varphi_{,\mu}\delta\varphi_{,\nu}\rangle$
(140)

and

$\displaystyle\undertilde{\hat{h}}^{\alpha\beta}$ $\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=\undertilde{h}^{\alpha\beta}-\bar{g}^{\alpha\beta}\frac{\undertilde{h}}{2}$
$\displaystyle=h^{\alpha\beta}-\frac{\delta\varphi}{\bar{\varphi}}\bar{g}^{\alpha\beta}-\bar{g}^{\alpha\beta}\frac{h}{2}.$
(141)

The GW EMT for Brans-Dicke was obtained in [11]. The gauge used there is such
that:

$\displaystyle h=-2\frac{\delta\varphi}{\bar{\varphi}}.$ (142)

Which implies, then, by Eq. (141),

$\displaystyle\undertilde{\hat{h}}^{\alpha\beta}=h^{\alpha\beta},$ (143)

so that the Einstein frame EMT is expressed by the Jordan frame fields as:

$\displaystyle 16\pi
G\undertilde{T}^{(BD)}_{\mu\nu}=\frac{3+2\omega_{BD}}{\bar{\varphi}^{2}}\langle\epsilon^{2}\delta\varphi_{,\mu}\delta\varphi_{,\nu}\rangle$
$\displaystyle+\frac{1}{2}\langle\epsilon^{2}h_{\alpha\beta;\mu}{h^{\alpha\beta}}_{;\nu}\rangle$
$\displaystyle\hskip
20.0pt-\frac{1}{4}\langle\epsilon^{2}h_{,\mu}h_{,\nu}\rangle.$ (144)

Finally, using Eq. (142) again,

$\displaystyle 16\pi
G\undertilde{T}^{(BD)}_{\mu\nu}=\frac{2(1+\omega_{BD})}{\bar{\varphi}^{2}}\langle\epsilon^{2}\delta\varphi_{,\mu}\delta\varphi_{,\nu}\rangle$
$\displaystyle\hskip
120.0pt+\frac{1}{2}\langle\epsilon^{2}h_{\alpha\beta;\mu}{h^{\alpha\beta}}_{;\nu}\rangle.$
(145)

Which coincides with the result obtained in [11].

Since we have computed the GW EMT in both frames as special cases of our
result, we can compare them. To do so, one must express the Einstein EMT in
terms of $\hat{\gamma}_{\alpha\beta}$. With Eqs. (125-127), Eq. (20) becomes:

$\displaystyle
h_{\alpha\beta}=\hat{\gamma}_{\alpha\beta}-\frac{\bar{g}_{\alpha\beta}}{2}\hat{\gamma}-\frac{\bar{g}_{\alpha\beta}}{\bar{\varphi}}\delta\varphi$
(146)

Using the gauge condition of Eq. (142) in the trace of the above equation:

$\displaystyle\hat{\gamma}=-2\frac{\delta\varphi}{\bar{\varphi}},$ (147)

which implies in

$\displaystyle h_{\alpha\beta}=\hat{\gamma}_{\alpha\beta}.$ (148)

Since Eq. (120) is valid in the Einstein frame and Eq. (143) is also valid, we
also conclude that

$\displaystyle\undertilde{\hat{\gamma}}_{\alpha\beta}=\hat{\gamma}_{\alpha\beta},$
(149)

that is, although the $h_{\alpha\beta}$ in both frames are not equal to each
other, the true tensorial degrees of freedom $\hat{\gamma}_{\alpha\beta}$ are.

Using the gauge of Eq. (147) in the Jordan frame GW EMT given by Eq. (128) and
comparing the result with the Einstein frame EMT given by Eq. 145) with Eq.
(148) replaced on it:

$\displaystyle
T^{GW(BD)}_{\mu\nu}=\bar{\varphi}\undertilde{T}^{GW(BD)}_{\mu\nu}.$ (150)

This is not the same relation the matter EMT $T_{\mu\nu}^{(m)}$ would have
under conformal transformation [31]:

$\displaystyle
T_{\mu\nu}^{(m)}=\frac{\varphi}{\bar{\varphi}}\undertilde{T}_{\mu\nu}^{(m)}$
(151)

Requiring further that four-velocities must have norm equal to $-1$, they
relate as:

$\displaystyle
u^{\alpha}=\sqrt{\frac{\varphi}{\bar{\varphi}}}\undertilde{u}^{\alpha}.$ (152)

The energy density of GWs in both frames relate, then, as:

$\displaystyle\rho^{GW}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=u^{\alpha}u^{\beta}T^{GW}_{\alpha\beta}=\varphi\undertilde{u}^{\alpha}\undertilde{u}^{\beta}\undertilde{T}^{GW}_{\alpha\beta}=\varphi\undertilde{\rho}^{GW}$
(153)

which implies that, in principle, different from what is sustained in [11],
the GW energy density is not conformally invariant.

##  VI Divergence of the GW EMT

As we have demonstrated, some features of the GW EMT that are valid in GR, are
not valid in the general context of reduced Horndeski theories, such as the
trace being zero. We know that in GR, the divergence of this tensor vanishes,
which implies in conservation of the number of gravitons. We want now to
investigate if this is still true in our scope.

In this section, we will work with solutions of the eikonal form. That is:

$\displaystyle\hat{\gamma}_{\alpha\beta}=\Gamma_{\alpha\beta}(x^{\mu})e^{iw(x^{\mu})/\epsilon},$
(154) $\displaystyle\delta\varphi=\Phi(x^{\mu})e^{iv(x^{\mu})/\epsilon},$
(155)

where $\Gamma_{\alpha\beta},\Phi$ are complex functions and $w,v$ are real.

We choose a particular theory given by conditions

$\displaystyle\bar{G}_{2}=\bar{G}_{3,X}=\bar{G}_{4,\varphi}=\bar{G}_{4,X}=0.$
(156)

This theory has luminal scalar waves and admits the TT gauge. Its GW EMT
reads:

$\displaystyle T^{GW}_{\mu\nu}=$ $\displaystyle\frac{1}{16\pi
G}\bigg{\\{}\frac{\bar{G}_{4}}{2}\langle\epsilon^{2}{\hat{\gamma}_{\alpha\beta}}_{;\mu}({\hat{\gamma}^{\alpha\beta}}_{;\nu})^{*}\rangle$
$\displaystyle\hskip
50.0pt+2\bar{G}_{3,\varphi}\langle\epsilon^{2}\delta\varphi_{,\mu}(\delta\varphi_{,\nu})^{*}\rangle\bigg{\\}}.$
(157)

where the star stands for complex conjugate. It is present on the above
equation so that the final tensor remains real.

Taking the divergence of first term and replacing Eq.(154), one finds:

$\displaystyle\bigg{[}\frac{\bar{G}_{4}}{2}\langle\epsilon^{2}\hat{\gamma}_{\alpha\beta;\mu}({\hat{\gamma}^{\alpha\beta}}_{;\nu})^{*}\rangle\bigg{]}^{;\nu}=\frac{\bar{G}_{4}}{2}(k_{\mu}k_{\nu}\Gamma^{2})^{;\nu},$
(158)

where all other contributions vanish under the weak limit average and

$\displaystyle\Gamma^{2}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=\Gamma_{\alpha\beta}(\Gamma^{\alpha\beta})^{*}.$
(159)

and

$\displaystyle k_{\mu}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=w_{,\mu}.$
(160)

We now use the equations of motion. The $\epsilon^{-1}$ contribution of the
diagonalized version of Eq. (15) gives, for the tensorial part, in any reduced
Horndeski theory [24]:

$\displaystyle k_{\mu}k^{\mu}=0.$ (161)

Meaning that the tensorial waves travel through null curves. Using the
definition of $k_{\mu}$, the above equation implies that this curve is also a
geodesic, namely:

$\displaystyle k^{\nu}k_{\mu;\nu}=0.$ (162)

Furthermore, the terms of order 1 in $\epsilon$ of the equation of motion
combine to give, in the particular theory where Eq.(156) is valid, [24]

$\displaystyle\frac{\bar{G}_{4}}{2}(2k^{\alpha}\Gamma_{\mu\nu;\alpha}+{k^{\alpha}}_{;\alpha}\Gamma_{\mu\nu})=\bar{G}_{3,\varphi}\Phi
e^{i(v-w)}($ $\displaystyle\bar{\varphi}^{,\gamma}q_{\gamma}\bar{g}_{\mu\nu}$
$\displaystyle-2\bar{\varphi}_{(,\mu}q_{\nu)}).$ (163)

where

$\displaystyle q_{\mu}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=v_{,\mu}.$
(164)

Contracting the equation with $(\Gamma^{\mu\nu})^{*}$ and summing with its
complex conjugate contracted with $\Gamma^{\mu\nu}$, one is able to write

$\displaystyle\frac{\bar{G}_{4}}{2}(k^{\alpha}\Gamma^{2})_{;\alpha}=-2\bar{G}_{3,\varphi}\bar{\varphi}_{,\mu}q_{\nu}\text{Re}(\Phi
e^{i(v-w)}(\Gamma^{\mu\nu})^{*})$ (165)

where Re(…) indicates the real part of a function and we have used that, under
the TT gauge we are assuming, $\Gamma_{\mu\nu}$ has vanishing trace.

Using Eqs. (162) and (165) on Eq. (158), one ends up with:

$\displaystyle\bigg{[}\frac{\bar{G}_{4}}{2}\langle\epsilon^{2}\hat{\gamma}_{\alpha\beta;\mu}({\hat{\gamma}^{\alpha\beta}}_{;\nu})^{*}\rangle\bigg{]}^{;\nu}=$
$\displaystyle-2\bar{G}_{3,\varphi}\bar{\varphi}_{,\alpha}q_{\beta}k_{\mu}\times$
$\displaystyle\hskip
25.0pt\times\text{Re}(\Phi(\Gamma^{\alpha\beta})^{*}e^{i(v-w)}).$ (166)

Since $\bar{G}_{3,\varphi}=0$ in GR, we would conclude that the GW EMT
divergence vanishes in this case. But here this part doesn’t vanish, even
under the TT gauge and after the imposition of the equations of motion.

We proceed on computing the divergence of the second term in Eq. (157).
Replacing Eq. (155) on it:

$\displaystyle[2\bar{G}_{3,\varphi}\langle\epsilon^{2}\delta\varphi_{,\mu}(\delta\varphi_{,\nu})^{*}\rangle]^{;\nu}=2(\bar{G}_{3,\varphi}q_{\mu}q_{\nu}\Phi^{2})^{;\nu}$
(167)

where

$\displaystyle\Phi^{2}\mathrel{\hbox
to0.0pt{\raisebox{1.07639pt}{$\cdot$}\hss}\raisebox{-1.07639pt}{$\cdot$}}=\Phi\Phi^{*}.$
(168)

As in the case for tensorial waves, here we have, once more, that the dominant
term of the field equation for the perturbations give [24]:

$\displaystyle q_{\alpha}q^{\alpha}=0,$ (169)

which implies in

$\displaystyle q^{\alpha}{q_{\mu}}_{;\alpha}=0.$ (170)

Furthermore, the amplitude evolution for the scalar waves become [24]:

$\displaystyle
2\bar{G}_{3,\varphi}(2q^{\alpha}\Phi_{,\alpha}+{q^{\alpha}}_{;\alpha}\Phi)+2\bar{G}_{3,\varphi\varphi}\bar{\varphi}^{,\alpha}q_{\alpha}\Phi=0$
(171)

Multiplying the equation by $\Phi^{*}$ and summing with its complex conjugate
multiplied by $\Phi$, one obtains:

$\displaystyle(\bar{G}_{3,\varphi}q^{\alpha}\Phi^{2})_{;\alpha}=0.$ (172)

Using Eqs. (170) and (172), one concludes that:

$\displaystyle[2\bar{G}_{3,\varphi}\langle\delta\varphi_{,\mu}(\delta\varphi_{,\nu})^{*}\rangle]^{;\nu}=0.$
(173)

Summing Eqs.(166) and (173), one concludes that:

$\displaystyle{T^{GW}_{\mu\nu}}^{;\nu}\neq 0.$ (174)

This means that even in theories where scalar and tensorial waves are luminal
and where the TT gauge is achievable, the GW EMT can still have non-vanishing
divergence.

In [12], we have shown how, in the cosmological context, the vanishing of the
GW EMT divergence can be used to obtain the GW version of the duality relation
888This relation was originally obtained by other means in [30].:

$\displaystyle D_{L}^{(GW)}=(1+z)^{2}D_{A}^{(GW)},$ (175)

where $z$ is the redshift, $D_{L}^{(GW)}$ is the GW luminosity distance and
$D_{A}^{(GW)}$ the correspondent angular-diameter distance. This relation, in
turn, influence certain observables. The signal of a stochastic gravitational
wave background is related to the spectral GW energy density

$\displaystyle\langle
S\rangle=\mathcal{F}\bigg{(}\frac{d^{3}\rho_{gw}}{dfd^{2}\Omega}\bigg{)},$
(176)

where $f$ is the frequency and $\Omega$ is the solid angle of the observation.
It is possible to obtain the total spectral energy density as a summation of
the different galaxy sources contributions [32]:

$\displaystyle\frac{d^{3}\rho_{GW}}{dfd^{2}\Omega}$
$\displaystyle=\int\int\frac{1+z(\vartheta)}{4\pi}\left(\frac{D_{A}^{(GW)}}{D_{L}^{(GW)}}\right)^{2}\sqrt{p^{\mu}(\vartheta)p_{\mu}(\vartheta)}\times$
$\displaystyle\hskip 30.0pt\times
n_{G}(x^{\mu}(\vartheta),\theta_{G})\mathcal{L}_{G}(f_{G},\theta_{G})d\vartheta
d\theta_{G},$ (177)

where $p_{\mu}$ is the spatial projection of the tangent vector of the GW null
geodesics $x^{\mu}(\vartheta)$, $n_{G}$ is the number density of galaxies,
$\mathcal{L}_{G}$ are the GW luminosity of the galaxies and $\theta_{G}$ are a
set of variables related to the galaxy emitting the waves.

We have seen that in more general reduced Horndeski theories, the GW EMT has
non-vanishing divergence, which would imply in a modification to the duality
relation and a consequent change on the stochastic GW signal, by Eq. (177).
The stochastic GW background, in principle, can then serve as a test reduced
Horndeski theory. If any deviation from the expected signal predicted in the
GR theory ever arise, this can indicate that more general theories must be
considered to explain the phenomenon.

##  VII Conclusion

The main results of this work are Eqs. (93), (94), (100), (96), with the
subsequent coefficient definitions. It represents the off-shell GW EMT for the
family of reduced Horndeski theories, with scalar waves included. With it, we
were able to study some subcases of interest: those where scalar waves are
luminal, Eq.(115), and those where the TT gauge is achievable, Eq. (122). In
the first of these, the trace of the tensor was shown to vanish, as expected
for radiation fluids and in the second it was shown not to vanish trivially,
Eq. (123), without the imposition of the full field equations. As suggested by
studies of the GW EMT in $f(R)$ theories [21, 22], in the cosmological
context, this non-vanishing of the trace could indicate a candidate to dark
energy: GW curving background space-time could be part of the reason behind
the current accelerated expansion era.

The Brans-Dicke GW EMT was reobtained as a particular case, confirming
previous results from literature [11]. Different from previous literature, we
obtain the GW EMT in Brans-Dicke for both the Jordan and Einstein frames, Eqs.
(128) and (145), respectively, and compare them to conclude that the tensor,
and consequently the energy density of GWs, is not invariant under conformal
transformations (i.e. Eqs. (150) and (153)).

Furthermore, with the help of previous studies regarding GW propagation in
reduced Horndeski theories [23, 24], we can expect that, within particular
theories where the divergence of the GW EMT does not vanish in the geometrical
optics limit, the GW duality relation will be modified and, therefore, we get
a first glimpse of how our results can entail changes in the SGWB
observations, even after the imposition of the field equations: either (i) by
directly modifying the GW spectral energy density per solid angle calculated
in [32] as Eq. (177), in any background space-time, or (ii) by the change in
the relation between the SGWB detected signal and spectral energy density, as
exemplified in particular theories by [11]. We expect that different theories
will predict different SGWB signals for a fixed set of GW sources and that
this can be a way of probing deviations from the usual GR theory.

Our work complements the study made by [20] in the sense that it investigates
the GW EMT in a different set of scalar-tensor theories. In another sense, it
generalizes this same work by allowing any background space-time, not only
asymptotically-flat ones, which would make important cases such as FLRW
universes to be left aside.

Although full generality has been achieved in terms of the background metric,
we must alert for the limits of our approach. The weak-limit averaging process
used to compute the GW EMT can result in divergent terms in theories where
$\bar{G}_{3,X}\neq 0$. Those potential divergences are all present in the part
of the GW EMT given by Eq. (100). We have pointed out that such divergences
can still vanish in the most troubling cases after imposing the field
equations. When, even after this impositions,such terms remain, we point out
that the problem can be in our assumption that the GW amplitude and wavelength
are of the same order of smallness, as in Eq. (12). A more detailed treatment
where different limiting procedures are done for both perturbative parameters
might eliminate such divergences.

Future steps could be made in the direction of generalizing the GW EMT for
Horndeski theories in which tensorial perturbations are not luminal and for
theories beyond Horndeski. Further studies on the relevance of our results to
observations of the SGWB signal in Horndeski theories are worth being pursued
as well.

##  VIII Acknowledgments

J.C.L. thanks Brazilian funding agency FAPERJ for PhD scholarship 2016007634.

##  Appendix A Details regarding variation of action

Here we provide details on how to obtain each contribution of Eq. (92). As
discussed in the main text, the terms $T^{(i)}_{\mu\nu}$ are obtained by
varying the contributions inside the average of Eq. (90) that are directly
related with the functions $\bar{G}_{i}$. Using Eq. (71), (73) and (74), one
obtains:

$\displaystyle T^{(second,2)}_{\mu\nu}=\frac{1}{16\pi G}\bigg{\langle}$
$\displaystyle\epsilon^{2}\bigg{\\{}\bar{G}_{2,X}\delta\varphi_{,\mu}\delta\varphi_{,\nu}+\bar{G}_{2,XXX}(\bar{\varphi}_{,\alpha}\delta\varphi^{,\alpha})^{2}\frac{\bar{\varphi}_{,\mu}\bar{\varphi}_{,\nu}}{2}-2\bar{G}_{2,XX}\bigg{[}\bar{\varphi}_{(,\mu}\delta\varphi_{,\nu)}\bar{\varphi}_{,\lambda}+\delta\varphi_{,\lambda}\frac{\bar{\varphi}_{,\mu}\bar{\varphi}_{,\nu}}{4}\bigg{]}\delta\varphi^{,\lambda}\bigg{\\}}\bigg{\rangle}.$
(178)

The calculation of the remaining terms is a little more extensive. For the
ones involving $\bar{G}_{4}$, for example, using the results of Subsec. IV.2,
one concludes that:

$\displaystyle T^{(second,4)}_{\mu\nu}=\frac{-1}{16\pi G}\bigg{\langle}$
$\displaystyle\epsilon^{2}\frac{\bar{G}_{4}}{2}\bigg{\\{}2\bigg{(}\frac{\delta
h}{\delta\bar{g}^{\mu\nu}}\bigg{)}_{;\beta}h^{,\beta}+h_{,\mu}h_{,\nu}+2\bigg{(}2\bigg{(}\frac{\delta
h^{\alpha\gamma}}{\delta\bar{g}^{\mu\nu}}\bigg{)}^{;\beta}-\bigg{(}\frac{\delta
h^{\alpha\beta}}{\delta\bar{g}^{\mu\nu}}\bigg{)}^{;\gamma}\bigg{)}h_{\alpha\beta;\gamma}-2{h_{\mu\gamma}}^{;\beta}{h_{\nu\beta}}^{;\gamma}-h_{\alpha\beta;\mu}{h^{\alpha\beta}}_{;\nu}$
$\displaystyle\hskip
5.0pt+2h_{\mu\beta;\gamma}{h_{\nu}}^{\beta;\gamma}-2h_{,\beta}\bigg{(}\frac{\delta
h^{\beta\gamma}}{\delta\bar{g}^{\mu\nu}}\bigg{)}_{;\gamma}-2\bigg{(}\frac{\delta
h}{\delta\bar{g}^{\mu\nu}}\bigg{)}^{;\beta}{h_{\beta\gamma}}^{;\gamma}\bigg{\\}}+2\epsilon^{2}\bar{G}_{4,\varphi}\delta\varphi\bigg{[}\bigg{(}\frac{\delta
h^{\alpha\beta}}{\delta\bar{g}^{\mu\nu}}\bigg{)}_{;\alpha\beta}-\bar{\Box}\bigg{(}\frac{\delta
h}{\delta\bar{g}^{\mu\nu}}\bigg{)}$ $\displaystyle\hskip 370.0pt-
h_{;\mu\nu}\bigg{]}\bigg{\rangle}$ (179)

Using integration by parts (in the sense of Eq.(35)) in terms where
derivatives of perturbations are contracted, one will end up with
contributions where $\bar{\Box}\hat{\gamma}_{\alpha\beta}$ appears. For
example, using Eqs.(75):

$\displaystyle\bar{G}_{4}\bigg{\langle}\epsilon^{2}\bigg{(}\frac{\delta
h}{\delta\bar{g}^{\mu\nu}}\bigg{)}_{;\beta}h^{,\beta}\bigg{\rangle}$
$\displaystyle=-\bar{G}_{4}\bigg{\langle}2\epsilon^{2}h\bar{\Box}\bigg{(}\frac{\delta
h}{\delta\bar{g}^{\mu\nu}}\bigg{)}\bigg{\rangle}$
$\displaystyle=\bar{G}_{4}\bigg{\langle}2\epsilon^{2}h\bigg{[}\bar{\Box}\hat{\gamma}_{\mu\nu}-\bar{F}_{\mu\nu}\bar{\Box}\delta\varphi\bigg{]}\bigg{\rangle}.$
(180)

Since we are assuming luminal tensorial modes from the start, the equation for
the tensorial part of GWs is of the form [24]:

$\displaystyle\epsilon\bar{\Box}\hat{\gamma}_{\mu\nu}+\mathcal{O}(1)=0,$ (181)

where
$\epsilon\bar{\Box}\hat{\gamma}^{\alpha\beta}=\mathcal{O}(\epsilon^{-1})$. The
terms involving $\bar{\Box}\hat{\gamma}^{\alpha\beta}$ appearing in the GW EMT
are of the form
$\langle\epsilon^{2}W\bar{\Box}\hat{\gamma}^{\alpha\beta}\rangle$, with
$W=\mathcal{O}(1)$. Multiplying Eq.(181) by $\epsilon W$ and taking the weak
limit average:

$\displaystyle\langle\epsilon^{2}W\bar{\Box}\hat{\gamma}^{\alpha\beta}\rangle=0,$
(182)

so that these D’lembertians terms can be neglected. Of course, the same will
be true for terms with $\bar{\Box}\hat{\gamma}$. Notice, however, that we are
still interested in the case of non-luminal scalar waves, so that analogous
terms involving $\bar{\Box}\delta\varphi$ do not vanish in principle.

Another simplification comes from using integration by parts in terms where
divergences of $\hat{\gamma}_{\mu\nu}$ can appear. Because of the harmonic
gauge given by Eq. (24), these divergences vanish. For example, using Eqs.(20)
and (37):

$\displaystyle\langle-2\epsilon^{2}{h_{\mu\gamma}}^{;\beta}{h_{\nu\beta}}^{;\gamma}\rangle$
$\displaystyle=\langle-2\epsilon^{2}{h_{\mu\gamma}}^{;\gamma}{h_{\nu\beta}}^{;\beta}\rangle$
$\displaystyle=\bigg{\langle}\epsilon^{2}\bigg{\\{}-\frac{\hat{\gamma}_{,\mu}\hat{\gamma}_{,\nu}}{2}-2C_{\gamma(\mu}\hat{\gamma}_{,\nu)}\delta\varphi^{,\gamma}$
$\displaystyle\hskip
50.0pt-2C_{\mu\gamma}C_{\nu\beta}\delta\varphi^{,\gamma}\delta\varphi^{,\beta}\bigg{\\}}\bigg{\rangle},$
(183)

where the harmonic gauge was already imposed.

With these simplifications, expressing everything in Eq. (179) in terms of
$\hat{\gamma}_{\mu\nu}$ and $\delta\varphi$ by using Eqs.(20), (75) and (77)
and adding the terms present in $T^{(first)}_{\mu\nu}$ (given by Eq. (90))
that have the $\bar{G}_{4}$ function and its derivatives as factors, one is
able to obtain Eq.(96).

The last part of the GW EMT is $T^{(second,3)}_{\mu\nu}$. Different from the
previous parts, one needs to take care when varying covariant derivatives
here, as discussed in Subsec. IV.2. Variation will still commute with
covariant derivatives when the field being differentiated is a perturbation.
But in terms where derivatives act upon the background scalar (i.e. Eq. (84)),
this cannot be done. There are two terms where this occurs, namely those with
coefficients given by Eq. (86) and (87). Initially, notice that

$\displaystyle\delta(\bar{\varphi}_{;\mu\nu})=-\bar{\varphi}_{,\tau}\delta\bar{\Gamma}^{\tau}_{\mu\nu}.$
(184)

It is straightforward to show that

$\displaystyle\delta\bar{\Gamma}^{\tau}_{\mu\nu}=-\frac{1}{2}[\bar{g}_{\lambda\mu}(\delta\bar{g}^{\lambda\tau})_{;\nu}+\bar{g}_{\lambda\nu}(\delta\bar{g}^{\lambda\tau})_{;\mu}-\bar{g}_{\mu\gamma}\bar{g}_{\nu\sigma}(\delta\bar{g}^{\gamma\sigma})^{;\tau}]$
(185)

Then, Eq. (84) becomes, after integration by parts and using that
$L_{\alpha\beta}$ is a symmetric tensor in both cases of interest:

$\displaystyle
T_{2}=-\frac{\epsilon^{2}}{2}\int\sqrt{-\bar{g}}[2({L_{\lambda}}^{\beta}\bar{\varphi}_{,\sigma})_{;\beta}-(L_{\lambda\sigma}\bar{\varphi}_{,\gamma})^{;\gamma}]\delta\bar{g}^{\lambda\sigma}d^{4}x,$
(186)

from which we conclude that the contribution for the GW EMT coming from the
variation of connections are:

$\displaystyle
T_{\mu\nu}\sim\langle\epsilon^{2}[2({L^{\beta}}_{(\mu}\bar{\varphi}_{,\nu)})_{;\beta}-(L_{\mu\nu}\bar{\varphi}_{,\gamma})^{;\gamma}]\rangle.$
(187)

For the case in which Eq. (86) is valid:

$\displaystyle T_{\mu\nu}\sim-\langle\epsilon^{2}[$
$\displaystyle\bar{G}_{3,X}\delta\varphi_{,\gamma}\delta\varphi^{,\gamma}\bar{\varphi}_{(,\mu}]_{;\nu)}\rangle$
$\displaystyle\hskip
10.0pt+\frac{\bar{g}_{\mu\nu}}{2}\langle\epsilon^{2}[\bar{G}_{3,X}\delta\varphi_{,\gamma}\delta\varphi^{,\gamma}\bar{\varphi}_{,\phi}]^{;\phi}\rangle.$
(188)

For the case of Eq. (87):

$\displaystyle T_{\mu\nu}\sim\langle\epsilon^{2}[$
$\displaystyle\bar{G}_{3,XX}(\delta\varphi_{,\gamma}\bar{\varphi}^{,\gamma})^{2}\bar{\varphi}_{(,\mu}]_{;\nu)}\rangle$
$\displaystyle\hskip
10.0pt-\frac{\bar{g}_{\mu\nu}}{2}\langle\epsilon^{2}[\bar{G}_{3,XX}(\delta\varphi_{,\gamma}\bar{\varphi}^{,\gamma})^{2}\bar{\varphi}_{,\phi}]^{;\phi}\rangle.$
(189)

We confirm that these terms will not vanish in the weak limit average,
different from the variations of connections coming from covariant derivatives
of perturbations.

With these terms in mind and varying the rest of the $\bar{G}_{3}$ related
terms of Eq. (90), one obtains:

$\displaystyle T^{(second,3)}_{\mu\nu}=\frac{1}{16\pi G}\bigg{\langle}$
$\displaystyle\epsilon^{2}\bigg{\\{}\bar{G}_{3,\varphi
X}\bar{\varphi}_{,\mu}\bar{\varphi}_{,\nu}\delta\varphi\bar{\Box}\delta\varphi-2\bar{G}_{3,\varphi}\delta\varphi\delta\varphi_{;\mu\nu}+\bar{G}_{3,XXX}\frac{\bar{\varphi}_{,\mu}\bar{\varphi}_{,\nu}}{2}\bar{\Box}\bar{\varphi}(\bar{\varphi}_{,\gamma}\delta\varphi^{,\gamma})^{2}+\frac{\bar{g}_{\mu\nu}}{2}[\bar{G}_{3,X}\delta\varphi_{,\gamma}\delta\varphi^{,\gamma}\bar{\varphi}_{,\delta}]^{;\delta}$
$\displaystyle-[\bar{G}_{3,X}\delta\varphi_{,\gamma}\delta\varphi^{,\gamma}\bar{\varphi}_{,(\mu}]_{;\nu)}+[\bar{G}_{3,XX}(\bar{\varphi}_{,\gamma}\delta\varphi^{,\gamma})^{2}\bar{\varphi}_{(,\mu}]_{;\nu)}-\frac{\bar{g}_{\mu\nu}}{2}[\bar{G}_{3,XX}(\bar{\varphi}_{,\gamma}\delta\varphi^{,\gamma})^{2}\bar{\varphi}_{\beta}]^{;\beta}$
$\displaystyle-\bar{G}_{3,X}\bigg{\\{}\bar{\varphi}_{,\alpha}\bar{\varphi}_{,\beta}\bigg{[}\frac{\delta
h^{\alpha\beta}}{\delta\bar{g}^{\mu\nu}}\bar{\Box}\delta\varphi+h^{\alpha\beta}\delta\varphi_{;\mu\nu}-\delta\varphi^{,\alpha}\bigg{[}\bigg{(}\frac{\delta
h}{\delta\bar{g}^{\mu\nu}}\bigg{)}^{;\beta}-2\bigg{(}\frac{\delta
h^{\beta\gamma}}{\delta\bar{g}^{\mu\nu}}\bigg{)}_{;\gamma}\bigg{]}\bigg{]}$
$\displaystyle\hskip
40.0pt-\bar{\varphi}_{,\beta}\bar{\varphi}_{(,\mu}\delta\varphi_{,\nu)}(h^{,\beta}-2{h^{\beta\gamma}}_{;\gamma})-\delta\varphi^{,\alpha}h_{(,\mu}\bar{\varphi}_{,\nu)}\bar{\varphi}_{,\alpha}-\delta\varphi_{,\mu}\delta\varphi_{,\nu}\bar{\Box}\bar{\varphi}-\delta\varphi_{,\alpha}\delta\varphi^{,\alpha}\bar{\varphi}_{;\mu\nu}$
$\displaystyle\hskip
245.0pt-\bar{\varphi}_{,(\mu}\delta\varphi_{,\nu)}\bar{\Box}\delta\varphi-\bar{\varphi}_{,\alpha}\delta\varphi^{,\alpha}\delta\varphi_{;\mu\nu}\bigg{\\}}$
$\displaystyle-2\bar{G}_{3,XX}\bigg{\\{}\frac{(\bar{\varphi}_{,\gamma}\delta\varphi^{,\gamma})^{2}}{2}\bar{\varphi}_{;\mu\nu}-\frac{\bar{\varphi}_{,\mu}\bar{\varphi}_{,\nu}}{4}\\{\bar{\varphi}_{,\alpha}\bar{\varphi}_{,\beta}[h^{\alpha\beta}\bar{\Box}\delta\varphi-\delta\varphi^{,\alpha}(h^{,\beta}-2{h^{\beta\gamma}}_{;\gamma})]-\delta\varphi_{,\alpha}\delta\varphi^{,\alpha}\bar{\Box}\bar{\varphi}$
$\displaystyle\hskip
210.0pt-\delta\varphi_{,\alpha}\bar{\varphi}^{,\alpha}\bar{\Box}\delta\varphi\\}+\bar{\varphi}_{,\gamma}\delta\varphi^{,\gamma}\bar{\varphi}_{(,\mu}\delta\varphi_{,\nu)}\bar{\Box}\bar{\varphi}\bigg{\\}}\bigg{\\}}\bigg{\rangle},$
(190)

By expressing the above equation in terms of $\hat{\gamma}_{\mu\nu}$ and
$\delta\varphi$ and adding the terms present in $T^{(first)}_{\mu\nu}$ (given
by Eq. (90)) having the $\bar{G}_{3}$ function and its derivatives as factors,
one is able to obtain Eq.(100).

## References

  * [1] G. Agazie et al., ApJL 951 L8 (2023).
  * [2] G. Agazie et al., ApJL 952 L37 (2023).
  * [3] A. Afzal et al., ApJL 951 L11 (2023).
  * [4] P.Amaro-Seoane et al., arXiv:1702.00786 (2017).
  * [5] J.Baker et al., arXiv:1907.06482 (2020).
  * [6] E. Barausse, Gen. Relativ. and Grav. 52:81 (2020).
  * [7] P.Amaro-Seoane et al., Liv. Rev. Relativ. 26, 2 (2023).
  * [8] P. Auclair et al., Liv. Rev. Relativ. 26, 5 (2023).
  * [9] J. Romano and N. Cornish, Liv. Rev. Relativ. 20, 2 (2017).
  * [10] B. P. Abbot et al. (Virgo and LIGO Scientific Collaborations), Phys. Rev. Lett. 120, 201102 (2018).
  * [11] M. Isi and L. C. Stein, Phys. Rev. D 98, 104025 (2018).
  * [12] J. C. Lobato, I. S. Matos, M. O. Calvão and I. Waga, Phys. Rev. D 106, 104048 (2022).
  * [13] B. P. Abbott et al.(Virgo and LIGO Scientific Collaborations),Phys. Rev. Lett. 119, 161101 (2017).
  * [14] A. Goldstein et al., Astrophys. J. 848, L14 (2017).
  * [15] R. A. Isaacson, Phys. Rev. 166, 1263 (1968).
  * [16] R. A. Isaacson, Phys. Rev. 166, 1272 (1968).
  * [17] S. R. Green and R. M. Wald, Phys. Rev. D 83, 084020 (2011).
  * [18] S. R. Green and R. M. Wald, Phys. Rev. D 87, 124037 (2013).
  * [19] G. A. Burnett, J. Math. Phys. 30, 90 (1989).
  * [20] L. C. Stein and N. Yunes, Phys. Rev. D 83, 064038 (2011).
  * [21] A. W. H. Preston, T. R. Morris, JCAP 09 (2014) 017.
  * [22] A. W. H. Preston, JCAP 08 (2016) 038.
  * [23] C. Dalang, P. Fleury and L.Lombriser, Phys. Rev. D 102, 044036 (2020).
  * [24] C. Dalang, P. Fleury and L. Lombriser, Phys. Rev. D 103, 064075 (2021).
  * [25] T. Kobayashi, Rep. Prog. Phys (2019).
  * [26] A. Garoffolo, G. Tasinato, C. Carbone, D. Bertacca, and S. Matarrese, arXiv:1912.08093.
  * [27] C. W. Misner, K. S. Thorne, and J. A. Wheeler, “Gravitation” (W. H. Freeman and Company, San Francisco, 1973), pp. 964.
  * [28] M. Maggiore, Gravitational Waves, Vol. 1: Theory and Experiments (Oxford University Press, New York, 2007).
  * [29] R. D’inverno, Introducing Einstein’s relativity (Oxford University Press, Oxford, England, 1992).
  * [30] G. Tasinato et al., JCAP 06 (2021) 050.
  * [31] S. Capozziello and V. Faraoni, Beyond Einstein gravity: A survey of gravitational theories for cosmology and astrophysics (Springer, New York, United States, 2011).
  * [32] G. Cusin, C. Pitrou and J.-P. Uzan, Phys. Rev. D 96, 103019 (2017).