# The effects of dark energy on the propagation of gravitational waves

Antonio Enea Romano  ICRANet, Piazza della Repubblica 10, I–65122 Pescara,
Italy

###### Abstract

An effective model independent approach for the study of gravitational waves
(GW) propagation is derived with two different methods, one based on the
effective stress-energy-momentum tensor (EST), and the other on the effective
action, generalizing the effective field theory (EFT) of dark energy to
systems with an arbitrary matter Lagrangian. The effective action is derived
in Einstein and Jordan frame, and it depends on an effective GW speed and an
effective Planck mass, which can both depend on time, frequency, and
polarization, due to the effects of higher order interaction terms. The GW
propagation friction term can be polarization and frequency dependent,
inducing a time, frequency and polarization dependent ratio between GW and
electromagnetic luminosity distance. For theories with matter coupled to the
Einstein frame metric the ratio depends on the effective speed, while for
Jordan frame matter coupling the ratio depends on the effective Planck mass.

###### pacs:

Valid PACS appear here

##  I Introduction

The detection of gravitational waves LIGOScientific:2016aoc  by the Laser
Interferometer Gravitational Wave Observatory (LIGO) and Virgo has opened a
new window on the Universe. While these observations are compatible with
general relativity predictions, they also allow to put constraints on modified
gravity theories (MGT) and dark energy models. The observation of
electromagnetic counterparts LIGOScientific:2017vwq  is particularly useful to
test MGTs Baker:2017hug ; Creminelli:2017sry ; Sakstein:2017xjx ;
Ezquiaga:2017ekz ; Wang:2017rpx , since they allow to set stringent
constraints on the difference between the speed of light $c$ and the GW speed
$c_{T}$.

In order to establish model independent constraints on dark energy models and
MGTs it can be useful to introduce a model independent equation based on using
an effective stress-energy-momentum tensor approach, Romano:2018frb ;
Romano:2023uwf . This approach consists in encoding in an appropriately
defined sound speed the effects due to the source terms in the perturbation
equations, associated to the interaction with other fields or self-
interaction. It has been successfully applied to multi-fields systems
Romano:2020oov  and MGTs Vallejo-Pena:2019hgv , giving the correct definition
of the momentum dependent sound speed and friction term able to account for
the effects of the entropy. In this paper we show that it can also be applied
to GWs, allowing to find the model independent relation between the friction
term and effective GW speed.

For any theory whose field equations can be put in an ”Einstein’s-like” form
we derive a general model independent equation for GW propagation in a
Friedman-Robertson-Walker (FRW) Universe, based on appropriately defining an
effective speed encoding the effects of the source term associated to the
anisotropic part of the EST. This effective description also involves the
modification of the friction term, which induces a frequency dependent
difference between electromagnetic waves and GW luminosity distance
$d(z)^{GW}_{L}$ and $d^{EM}_{L}(z)$.

Previous calculations of the GW propagation equation for MGTs Kobayashi:2011nu
; DeFelice:2011bh  based on the quadratic action were not including the
momentum and polarization dependency of the effective GW speed, because they
are due to source terms which arise only at higher order, such as in the cubic
action. The effective approach is in agreement and generalizes results
obtained in the effective fields theory of dark energy Gleyzes:2013ooa ;
Creminelli:2014wna . The momentum dependency of the GW effective speed is
consistent with including higher order terms in the EFT of dark energy
deRham:2018red ; Baker:2022eiz .

##  II Perturbations of the field equations

Let’s consider dark energy models or MGTs for which the variation of the
action with respect to the metric gives ’Einstein’s like’ field equations

$G_{\mu\nu}=T^{DE}_{\mu\nu}+T^{mat}_{\mu\nu}=T_{\mu\nu}^{eff}\,,$ (1)

where $G_{\mu\nu}$ is the Einstein’s tensor, $T^{DE}_{\mu\nu}$ and
$T^{mat}_{\mu\nu}$ are respectively the effective dark energy and matter ESTs,
and we are using units in which $8\pi G=c=1$. In this notation the effective
Planck mass which can arise in MGTs is included in the $T_{\mu\nu}^{eff}$.
Eq.(1) is true for a large class of theories, for example for Horndeski’s
theories Horndeski:1974wa  and nonlocal theories Jaccard:2013gla . The
calculation of $T_{\mu\nu}^{eff}$ can be cumbersome, and for this reason an
action approach is often preferred, but if we treat its components as
effective quantities we can derive model independent equations.

The advantage of perturbing the field equations is that the perturbations of
$T_{\mu\nu}^{eff}$ can be treated as effective model independent quantities,
under the assumption that they are compatible with the structure of the
Einstein’s tensor, and satisfy the energy conditions and the conservation
equations implied by the Bianchi identities. Since the field equations have
the same structure that in general relativity, the field equations of these
theories can be written in a ”Einstein-like” form, and this must apply also to
perturbations.

Contrary to EFT of dark energy, Gubitosi:2012hu ; Gleyzes:2013ooa  which is
assuming a single scalar degree of freedom, the scalar-vector-tensor (SVT)
decomposition for metric and EST perturbations allows to derive model
independent equations valid for an arbitrary number of fields, since the SVT
decomposition is purely geometrical, and applies to any metric and EST.

For the perturbed field equations $\delta G_{\mu\nu}=\delta T_{\mu\nu}^{eff}$
the SVT decomposition gives Kodama:1984ziu

$h_{A}^{\prime\prime}+2\mathcal{H}h_{A}^{\prime}+\nabla^{2}h_{A}=a^{2}\Pi^{eff}_{A}\,,$
(2)

where $A$ stands for the polarizations, the prime denotes the derivative with
respect to conformal time $\eta$, $a(\eta)$ is the scale factor,
$\mathcal{H}=a^{\prime}/a$, and $\Pi_{A}$ is related to the traceless
transverse anisotropic part of $T_{\mu\nu}^{eff}$.

The l.h.s. of the above equation is obtained by expanding the Einstein’s
tensor to first order in perturbations, and for this reason we get a linear
differential operator, while the r.h.s. is not obtained by a perturbative
expansion, but by treating the components of $\delta T_{\mu\nu}^{eff}$ as
effective quantities, respecting the structure of the Einstein’s equations.
This implies that the source term in eq.(2) is effectively including higher
order terms in perturbations, which would require to go beyond the quadratic
action. This point will be very important when comparing to previous results.

##  III Effective speed of gravitational waves

For comoving curvature perturbations it has been shown Romano:2018frb ;
Romano:2023uwf  that is possible to introduce an effective sound speed
encoding the effects of entropy perturbations in multi-fields systems
Romano:2020oov , and in MGTs Vallejo-Pena:2019hgv . We will show that a
similar effective speed can be defined for GWs. Let’s assume that
$\hat{h}_{A}$ is a solution of eq.(2). We can rewrite the equation as

$\frac{(\hat{h}^{\prime}_{A}a^{2})^{\prime}}{a^{2}}-\frac{(\hat{g}_{A}a^{2})^{\prime}}{a^{2}}-\nabla^{2}\hat{h}_{A}=0\,,$
(3)

where we have defined

$\hat{g}_{A}=\frac{1}{a^{2}}\int a^{4}\hat{\Pi}^{eff}_{A}\,d\eta\,,$ (4)

and the hat denotes quantities obtained by substituting the solutions of the
equations of motion. Note that $\Pi_{A}$ is related to the coupling of GWs
with other fields, so in general a system of coupled differential equations
has to be solved to obtain $\hat{\Pi}_{A}$.

After introducing the space dependent effective GW speed $c_{T,A}$ (SEGS) and
the effective scale factor $\alpha_{A}$

$\displaystyle
c_{T,A}^{2}(\eta,x^{i})=\Big{(}1-\frac{\hat{g}_{A}}{\hat{h}^{\prime}_{A}}\Big{)}^{-1}$
, $\displaystyle\alpha_{A}=\frac{a}{c_{T,A}}\,,$ (5)

we can rewrite eq.(3) as

$\frac{1}{a^{2}}(\alpha_{A}^{2}\hat{h}^{\prime}_{A})^{\prime}-\nabla^{2}\hat{h}_{A}=0\,.$
(6)

Note that the effective speed $c_{T,A}$ for a given physical system is
obtained by substituting the solutions of the equations of motion, and as such
it is a function of space and time, not a functional of $h_{A}$. Taking this
into account from eq.(6) we obtain

$\displaystyle\hat{h}_{A}^{\prime\prime}+2\frac{\alpha_{A}^{\prime}}{\alpha_{A}}\hat{h}^{\prime}_{A}-c_{T,A}^{2}\nabla^{2}\hat{h}_{A}$
$\displaystyle=$ $\displaystyle 0\,,$ (7)

showing that the SEGS is the correct definition of effective speed, since it
is the coefficient of the Laplacian.

We have shown that any solution of eq.(2) can be obtained by solving eq.(7)
with an appropriately chosen $c_{T,A}$, but the opposite is not guaranteed,
i.e. not all solutions of the effective equation (7) are also solutions of
eq.(2). In fact the source term $\Pi_{A}$ is not arbitrary, since it is
related to the EST, which has to satisfy the energy and conservation
conditions, and consequently not all forms of $c_{T,A}$ are physically
acceptable. These constraints on $c_{T,A}$ will be studied in a separate work.

Eq.(7) can be used as a general model independent equation, which can be used
for phenomenological analysis, treating $c_{T,A}$ as a function to be
determined by analyzing observational data.

Using eq.(5) and dropping the hat, we obtain

$\displaystyle
h_{A}^{\prime\prime}+2\mathcal{H}\Big{(}1-\frac{c_{T,A}^{\prime}}{\mathcal{H}c_{T,A}}\Big{)}h^{\prime}_{A}-c_{T,A}^{2}\nabla^{2}h_{A}$
$\displaystyle=$ $\displaystyle 0\,,$ (8)

showing that the friction term cannot be modified unless the the effective
speed is time dependent. We have dropped the hat because for phenomenological
applications $h_{A}$ in eq.(8) is an unknown function which can be determined
by solving the effective equation for different forms of the effective speed
$c_{T,A}$. For a given model the effective speed is given by eq.(5), obtained
by substituting the solutions of the equations of motion, but for model
independent phenomenological applications $c_{T,A}$ is a free function to be
constrained by observational data.

The advantage of the EST approach is that it allows to derive a general
equation for GWs, valid for any dark energy model or MGT, including the
effects of higher order terms of the anisotropic part of the EST. This is
particularly useful for phenomenological studies, since it gives a general
equation which can be used for model independent analyses.

##  IV Momentum effective speed

In momentum space eq.(2) takes the form

$\tilde{h}_{A}^{\prime\prime}+2\mathcal{H}\tilde{h}_{A}^{\prime}+k^{2}\tilde{h}_{A}=a^{2}\tilde{\Pi}^{eff}_{A}\,,$
(9)

where we are denoting with $\tilde{h}_{A}(\eta,k)$ and
$\tilde{\Pi}_{A}(\eta,k)$ the Fourier transform of $h_{A}$ and $\Pi_{A}$.
Following a similar procedure to the one used to derive eq.(8) we obtain

$\displaystyle\tilde{h}_{A}^{\prime\prime}+2\frac{\tilde{\alpha_{A}}^{\prime}}{\tilde{\alpha_{A}}}\tilde{h}_{A}^{\prime}+\tilde{c}^{2}_{T,A}k^{2}\tilde{h}_{A}$
$\displaystyle=$ $\displaystyle 0\,,$ (10)
$\displaystyle\tilde{h}^{\prime\prime}_{A}+2\mathcal{H}\Big{(}1-\frac{\tilde{c}^{\prime}_{T,A}}{\mathcal{H}\tilde{c}_{T,A}}\Big{)}\tilde{h}^{\prime}_{A}+\tilde{c}^{2}_{T,A}k^{2}\tilde{h}_{A}$
$\displaystyle=$ $\displaystyle 0\,,$ (11)

where we have defined the momentum effective GW speed (MEGS) according to

$\displaystyle\tilde{c}_{T,A}^{2}(\eta,k)$ $\displaystyle=$
$\displaystyle\Big{(}1-\frac{\tilde{g}_{A}}{\tilde{h}^{\prime}_{A}}\Big{)}^{-1}\,,$
(12) $\displaystyle\tilde{g}_{A}$ $\displaystyle=$
$\displaystyle\frac{1}{a^{2}}\int a^{4}\tilde{\Pi}_{A}^{eff}\,d\eta\,,$ (13)

and $\tilde{\alpha}_{A}^{2}=a^{2}/\tilde{c}_{T,A}^{2}$. Note that the MEGS
$\tilde{c}_{T}(\eta,k)$ defined in eq.(13) is not the Fourier transform of the
SEGS $c_{T}(\eta,x^{i})$ defined in eq.(5), since the Fourier transform of the
terms $c_{T}\nabla^{2}h_{A}$ and $h^{\prime}_{A}c_{T}^{\prime}/c_{T}$ in
eq.(8) are a convolution, not a product, of their transform. The definition of
MEGS is mathematically convenient since it involves a minimal modification of
eq.(2).

##  V Frequency dependency of the GW luminosity distance

In general relativity the amplitude of GWs is inversely proportional to the
electromagnetic luminosity distance $d^{EM}_{L}(z)$, but a modification of the
friction term of the GW propagation equation induces a difference
Belgacem:2017ihm  with respect to the GW luminosity distance $d_{L}^{GW}$. In
order to find the effects on the luminosity distance it is convenient to
rewrite eq.(11) as

$\tilde{\chi}_{A}^{\prime\prime}+\Big{(}\tilde{c}_{T,A}k^{2}-\frac{\tilde{\alpha}_{A}^{\prime\prime}}{\tilde{\alpha}_{A}}\Big{)}\tilde{\chi}_{A}=0\,,$
(14)

where we have defined $\tilde{h}_{A}=\tilde{\chi}_{A}/\tilde{\alpha}_{A}$.  On
sub-horizon scale $\alpha^{\prime\prime}/\alpha$ can be neglected, and since
the amplitude $\tilde{h}_{A}$ is proportional to $1/\tilde{\alpha}$ instead of
$1/a$, we get that

$\frac{d_{L}^{GW}}{d_{L}^{EM}}(z)=\frac{a(z)}{\tilde{\alpha}_{A}(z)}\frac{\tilde{\alpha}_{A}(0)}{a(0)}=\frac{\tilde{c}_{T,A}(z,k)}{\tilde{c}_{T,A}(0,k)}\,,$
(15)

where we have defined
$\tilde{\alpha}_{A}(z)=a(z)/\tilde{c}_{T,A}(\eta(z),k)=a(z)/\tilde{c}_{T,A}(z)$,
and used $d_{L}^{GW}=r\,\tilde{\alpha}(0)/\tilde{\alpha}(z)$,
$d_{L}^{EM}=r\,a(0)/a(z)$, assuming $(1+z)=a(0)/a(z)$, i.e. that matter is
minimally coupled to the Einstein frame metric. In general relativity
$\tilde{c}_{T,A}=1$, $\tilde{\alpha}(z)=a(z)$, and we recover the well known
result that $d_{L}^{GW}(z)=d_{L}^{EM}(z)$. Neglecting higher order interaction
effects the effective speed $\tilde{c}_{T,A}$ reduces to $c_{T}$, and we
recover the results obtained in Romano:2023ozy  using the EFT of dark energy.

##  VI Effective Lagrangian and metric

The Lagrangian density for which the Lagrange’s equation gives eq.(6) is

$\mathcal{L}^{eff}_{h}=\frac{a^{2}}{c_{T,A}^{2}}\Big{[}h^{\prime
2}_{A}-c_{T,A}^{2}(\nabla h_{A})^{2}\Big{]},$ (16)

in agreement, and generalizing, the EFT calculations in the Einstein frame
Creminelli:2014wna .

The effective Lagrangian can be obtained from the general relativity
Lagrangian density

$\mathcal{L}_{h}^{GR}=a^{2}\Big{[}h^{\prime 2}_{A}-c^{2}(\nabla
h_{A})^{2}\Big{]}=\sqrt{-g}(\partial_{\mu}h_{A}\partial^{\mu}h_{A})\,,$ (17)

via the transformation

$a\rightarrow\alpha_{A}=\frac{a}{c_{T,A}}\quad,\quad c\rightarrow c_{T,A}\,,$
(18)

where we have denoted with $c$ the speed of light, to avoid ambiguity. This in
agreement with eq.(7), which shows that $\alpha_{A}$ can be regarded as an
effective scale factor. The effective action can be written as

$\mathcal{L}^{eff}_{h}=\sqrt{-g_{A}}(\partial_{\mu}h_{A}\partial^{\mu}h_{A})\,,$
(19)

in terms of the effective metric

$ds^{2}_{A}=a^{2}\Big{[}c_{T,A}d\eta^{2}-\frac{\delta_{ij}}{{c_{T,A}}}dx^{i}dx^{j}\Big{]}\,,$
(20)

for which the wave eq.(8) can be written in terms of the covariant d’Alembert
operator as

$\square
h_{A}=\frac{1}{\sqrt{-g_{A}}}\partial_{\mu}(\sqrt{-g_{A}}\partial^{\mu}h_{A})=0\,.$
(21)

This equation shows that the effects of the interaction of the graviton with
other fields can be effectively described as the propagation in vacuum through
a space with the effective metric given in eq.(20). Note that the above
effective metric is only valid to describe GWs propagation, it is not a
modification of the background metric describing the full cosmological model.
For scalar perturbations for example a different effective metric and action
can be defined in terms of the momentum effective sound speed (MESS)
Romano:2018frb ; Romano:2023uwf .

In the action approach the EST on the r.h.s. of the Einstein’s equations
originates from the interaction of tensor perturbations with themselves or
other fields. Based on this we can obtain the effective action in eq.(16) by
introducing higher order interaction terms in the Einstein’s frame Lagrangian
as $\mathcal{L}(h_{A},\phi^{i})$

$\mathcal{L}_{h}=a^{2}\Big{[}h^{\prime 2}_{A}-(\nabla
h_{A})^{2}+\mathcal{L}(h_{A},\phi^{i})\Big{]}\,\\\ $ (22)

which can be rewritten in the form of eq.(16) by introducing the effective GW
speed as

$c^{2}_{T,A}=\Big{(}1+\frac{\mathcal{L}}{h^{\prime 2}_{A}}\Big{)}^{-1}\,,$
(23)

and $\phi^{i}$ denotes abstractly all the other fields the graviton is coupled
to, including itself, or another polarization. Note that effective speed is
obtained by substituting in eq.(23) the solutions of the equations of motion.
In the effective field theory of inflation Noumi:2014zqa ; Creminelli:2014wna
, denoting with $\delta K_{\mu\nu}$ the perturbation of the extrinsic
curvature of the spatial slices, the leading order term for $\mathcal{L}$ is

$\mathcal{L}^{(2)}\propto\bar{M}_{2}(\eta)\delta K_{\mu\nu}\delta
K^{\mu\nu}\propto\bar{M}_{2}(\eta)h^{\prime 2}_{A},$ (24)

which induces a time dependent $c_{T}(\eta)$, and an action in agreement with
eq.(16). Higher order terms $\mathcal{L}^{(i>2)}$ can induce a space and
polarization dependent $c_{T,A}(\eta,x)$, which encodes the effects of the
higher order interaction terms Note3 . By comparing eq.(5) and eq.(23) we
obtain the relation between the interaction Lagrangian $\mathcal{L}$ and the
anisotropic part of the EST $\mathcal{L}=g^{2}_{A}-2h^{\prime}_{A}g_{A}$,
which allows to relate the effective action approach with the EST.

##  VII Jordan frame formulation

Eq.(16) should be valid for any system satisfying eq.(1), and in particular
it can be shown that it is consistent with the Horndeski’s theory Lagrangian
density in Jordan frame Kobayashi:2011nu

$\mathcal{L}^{Hor}_{h,J}=\tilde{a}^{2}M_{*}^{2}\Big{[}h_{A}^{\prime
2}-c_{T}^{2}(\nabla h_{A})^{2}\Big{]}\,,$ (25)

where we are denoting with $\tilde{a}$ the Jordan frame scale factor. The
Lagrangian density in eq.(16) is in the Einstein’s frame, since it gives the
equations which were derived from the ”Einstein’s like” eq.(1), while eq.(25)
is in the Jordan frame. In order to find the transformation between the two
frames we can note that the non perturbed field equations for the Horndeski’s
theory can be written in the Jordan frame as Kobayashi:2011nu

$\displaystyle\Omega^{2}G_{\mu\nu}^{J}=T^{eff,J}_{\mu\nu}$ ,
$\displaystyle\Omega^{2}=G_{4}+\frac{1}{2}\nabla_{\lambda}G_{5}\nabla^{\lambda}\phi\,,$
(26)

from which we can find the conformal transformation from the Jordan to the
Einstein frame, $g_{\mu\nu}=\Omega^{2}\tilde{g}_{\mu\nu}$, with
$\Omega=M_{*}\,c_{T}$, in agreement with applications of the EFT of dark
energy Kennedy:2017sof . The Einstein’s frame effective Lagrangian can be
converted to the Jordan’s frame by the conformal transformation
$a=\Omega\,\tilde{a}$

$\displaystyle\mathcal{L}^{eff}_{h,J}$ $\displaystyle=$
$\displaystyle\frac{\tilde{a}^{2}\Omega^{2}}{c^{2}_{T,A}}\Big{[}h^{\prime
2}_{A}-c_{T,A}^{2}(\nabla h_{A})^{2}\Big{]}=$ (27) $\displaystyle=$
$\displaystyle\tilde{a}^{2}M_{A}^{2}\Big{[}h_{A}^{\prime 2}-c^{2}_{T,A}(\nabla
h_{A})^{2}\Big{]}\,,$

where $M_{A}(\eta,x_{i})$ is the space and polarization dependent effective
Planck mass in the Jordan frame, defined as

$\displaystyle M_{A}=\frac{\Omega}{c_{T,A}}=\frac{M_{*}}{1+\delta_{A}}$ ,
$\displaystyle c_{T,A}=c_{T}(1+\delta_{A})\,.$ (28)

The above action is in agreement with, and generalizes, the results obtained
in the EFT of dark energy in the Jordan frame Gleyzes:2013ooa . The
corresponding GW propagation equation is

$h_{A}^{\prime\prime}+2\tilde{\mathcal{H}}\Big{(}1+\frac{M_{A}^{\prime}}{\tilde{\mathcal{H}}M_{A}}\Big{)}h^{\prime}_{A}-c_{T,A}^{2}\nabla^{2}h_{A}=0\,.$
(29)

The above equation is consistent with eq.(7) with $\alpha_{A}=\tilde{a}M_{A}$.
At leading order, i.e. ignoring higher order interaction terms, we recover the
commonly used Lagos:2019kds  Jordan frame action and equation.

Analogously to the Einstein’s frame eq.(22), eq.(27) could also be obtained
from the Lagrangian

$\mathcal{L}_{J}=\tilde{a}^{2}M^{2}_{*}\Big{[}h_{A}^{\prime
2}-c^{2}_{T}(\nabla h_{A})^{2}+\mathcal{L}_{J}^{i>2}\Big{]}\,,$ (30)

by defining

$\displaystyle 1+\delta_{A}$ $\displaystyle=$
$\displaystyle\Big{(}1+\frac{\mathcal{L}_{J}^{i>2}}{h_{A}^{\prime
2}}\Big{)}^{-1/2},$ (31)

where $\mathcal{L}_{J}^{i>2}$ denotes the Jordan’s frame higher order
interaction terms. The effective metric in the Jordan’s frame is the conformal
transformation of that in the Einstein’s frame given in eq.(20)

$ds_{J,A}^{2}=\Omega^{2}ds_{E,A}^{2}\,.$ (32)

##  VIII Momentum space effective Lagrangian

In momentum space the effective Lagrangian density is

$\mathcal{L}^{eff}_{h,k}=\tilde{\alpha}_{A}^{2}\Big{[}\tilde{h}^{\prime
2}_{A}+\tilde{c}^{2}_{T,A}k^{2}\tilde{h}_{A}\Big{]}\,,$ (33)

which can be obtained from the general relativity action

$\mathcal{L}_{h,k}^{GR}=a^{2}\Big{[}\tilde{h}^{\prime
2}_{A}+c^{2}k^{2}\tilde{h}_{A}\Big{]}\,,$ (34)

by the transformation

$\displaystyle
a\rightarrow\tilde{\alpha}_{A}=\frac{a}{\tilde{c}_{T,A}}=\tilde{a}\tilde{M}_{A}$
, $\displaystyle c\rightarrow\tilde{c}_{T,A}\,,$ (35)
$\displaystyle\tilde{c}_{T,A}=c_{T}(1+\tilde{\delta}_{A})$ ,
$\displaystyle\tilde{M}_{A}=\frac{M_{*}}{1+\tilde{\delta}_{A}}\,,$ (36)

where $\tilde{M}_{A}(\eta,k)=\Omega/\tilde{c}_{T,A}$ is the momentum and
polarization dependent effective Planck mass and

$\displaystyle(1+\tilde{\delta}_{A})$ $\displaystyle=$
$\displaystyle\Big{(}1+\frac{\mathcal{\tilde{L}}_{J}^{i>2}}{\tilde{h}^{\prime
2}_{A}}\Big{)}^{-1}\,.$ (37)

This is in agreement with eq.(11), which shows that $\tilde{\alpha}_{A}$ can
be regarded as an effective scale factor, but in this case it is momentum
dependent. As previously mentioned, the quantities $\tilde{\alpha}_{A}$ and
$\tilde{\delta}_{A}$ are not the Fourier transform of $\alpha_{A}$ and
$\delta_{A}$. More details about the momentum space effective action
derivation can be found in Romano:2023uwf .

##  IX Frame dependence and matter coupling

Physical observables of a given theory are invariant under conformal
transformations, but changing the metric to which matter fields are minimally
coupled to in the Lagrangian corresponds to introducing different theories,
which have different observable predictions.

Under a general conformal transformation, defined by $a=\Omega\,\tilde{a}$,
the Einstein frame effective Lagrangian in eq.(16) can be written as

$\frac{\Omega^{2}\tilde{a}^{2}}{c_{T,A}^{2}}\Big{[}h^{\prime
2}_{A}-c_{T,A}^{2}(\nabla h_{A})^{2}\Big{]}$ (38)

giving the propagation equation

$h_{A}^{\prime\prime}+2\tilde{\mathcal{H}}\Big{(}1-\frac{c_{T,A}^{\prime}}{\tilde{\mathcal{H}}c_{T,A}}+\frac{\Omega^{\prime}}{\tilde{\mathcal{H}}\Omega}\Big{)}h^{\prime}_{A}-c_{T,A}^{2}\nabla^{2}h_{A}=0\,,$
(39)

where $\tilde{\mathcal{H}}=\tilde{a}^{\prime}/\tilde{a}$, from which we can
see that while the GW speed does not depend on the conformal transformation,
the friction terms does.

As expected, eq.(39) gives eq.(29) for $\Omega=c_{T}M_{*}$. Note that eq.(7)
is valid also in the new frame, since
$\alpha_{A}=a^{2}/c_{T,A}^{2}=\Omega^{2}\tilde{a}^{2}/c_{T,A}^{2}=\tilde{a}M_{A}$,
but the change of frame implies a change of the Hubble parameter.  If matter
is minimally coupled to the metric $\tilde{g}_{\mu\nu}$, the redshift is given
$1+z=\tilde{a}(0)/\tilde{a}(z)$, and

$\frac{d_{L,A}^{GW}}{d_{L,A}^{EM}}(z)=\frac{\tilde{a}(z)}{\tilde{a}(0)}\frac{\tilde{\alpha}_{A}(0)}{\tilde{\alpha}_{A}(z)}=\frac{\tilde{c}_{T,A}(z)}{{\tilde{c}_{T,A}(0)}}\frac{\Omega(0)}{\Omega(z)}\,,$
(40)

which in the case of the Jordan frame coupling, corresponding to
$\Omega=M_{*}\,c_{T}$, gives

$d_{L,A}^{GW}(z)=\frac{\tilde{M}_{A}(0,k)}{\tilde{M}_{A}(z,k)}d_{L}^{EM}(z)\,,$
(41)

which generalizes the results obtained using the EFT of dark energy
Romano:2023ozy , or those obtained in some specific modified gravity theories
Belgacem:2017ihm ; Dalang:2019rke . The momentum and polarization dependence
is due to including the effects of higher order interaction terms neglected in
leading order calculations.

In a frame different from the Einstein’s frame the friction term can be
modified even if $c_{T,A}$ is constant, but this is just due to the change in
the Hubble parameter due to the scale factor redefinition, not to a real
effect on the GW propagation, since tensor perturbations are invariant under
conformal transformations Tsujikawa:2014uza . In fact eq.(8) is valid in any
frame, since the friction term in different frames is the same function of
space and time $\alpha^{\prime}_{A}/\alpha_{A}$, written in different forms
according to
$\alpha_{A}=a^{2}/c_{T,A}^{2}=\Omega^{2}\tilde{a}^{2}/c_{T,A}^{2}$.

##  X Observational constraints

Eq.(15) and eq.(41) allow to constrain the effective GW speed and the
effective Planck mass for different types of theories, using the luminosity
distance ratio between $d_{L}^{GW}(z)$ and $d_{L}^{EM}(z)$. For a GW of given
frequency the observation of a single electromagnetic counterpart at redshift
$z_{ct}$ allows to set a single constraint on
$\tilde{c}_{T,A}(z_{ct},k_{ct})$, but only observations at different
frequencies and redshifts allow to fully constrain the MEGS. Combining data
from different GW detectors, sensitive to different frequency bands, such as
LISA, Einstein Telescope and Cosmic Explorer, would be particularly useful.

For dark sirens a statistical method similar to the one employed for the
estimation of the Hubble constant Gray:2021sew  can be adopted to estimate the
redshift, allowing to constrain $\tilde{c}_{T,A}$ with much more data than for
bright sirens, which are in fact expected to be a limited portion of the GWs
detections. The electromagnetic counterpart observation of GW170817
LIGOScientific:2017vwq  is imposing a tight constraint on the low redshift
value of $\tilde{c}_{T,A}(z)$ at the frequencies probed by LIGO-Virgo, so a
possible phenomenological ansatz for theories with Einstein frame matter
coupling could be

$\frac{d^{GW}_{L,A}(z)}{d^{EM}_{L}(z)}=1+\frac{\gamma_{A}(k)}{(1+z)^{n_{A}(k)}}\tanh\Big{[}\frac{k-k^{*}_{A}}{\sigma_{A}(k)}\Big{]}\,,$
(42)

where $k^{*}_{A}$ is the scale at which a transition takes place, $\gamma_{A}$
controls the size of the change, and $\sigma_{A}$ the sharpness of the
transition. Other constraints on the $\tilde{c}_{T,A}$ can be obtained from
the waveform Harry:2022zey , since stretching or squeezing is expected, and
they could be combined with those on the luminosity distance given in eq.(42).

For theories with Jordan frame coupling the distance ratio can be used to
constrain the effective Planck mass, testing its possible polarization and
frequency dependency. A possible model independent ansatz could be a a
generalization of the parametrizations studied in
LISACosmologyWorkingGroup:2019mwx , for example

$\displaystyle\frac{d^{GW}_{L,A}(z)}{d^{EM}_{L}(z)}$ $\displaystyle=$
$\displaystyle\Xi_{A}(k)+\frac{1-\Xi_{A}(k)}{(1+z)^{n_{A}(k)}}\,.$ (43)

##  XI Conclusions

We have derived effective equations and Lagrangians for the GWs propagation in
the Einstein and Jordan frame, which can be used for model independent
analysis of GWs observational data. We have provided two derivations, one
based on the EST, and the other on the effective action, generalizing the
results obtained in the EFT of dark energy Gleyzes:2013ooa . The effective
description is based on two quantities: the effective GW speed and the
effective Planck mass, which can both be polarization and frequency dependent,
due to higher order interaction terms. In the Einstein frame the effective
Lagrangian depends only on the effective GW speed. This effective approach
allows to derive model independent relations between the GW and
electromagnetic luminosity distance, which can be tested observationally with
dark or bright sirens.

While this model independent approach is useful for phenomenological analyses
of GWs observations, it should be noted that the EST includes all the effects
of matter and dark energy in a single object, and without other observations
it would not be possible to disentangle them. For example, the observation of
the difference between the luminosity distance of the two polarizations of a
GW event could be caused either by the propagation through an anisotropic
matter medium, or by a MGT. In order to test specific dark energy models it
will be important to perform higher order perturbations calculations, in order
to compute the anisotropy effects which are not included in the quadratic
action, and in the EST approach are treated as effective model independent
quantities.

##  XII Acknowledgments

I thank Misao Sasaki, Tessa Baker, Sergio Vallejo, Riccardo Sturani, Rogerio
Rosenfeld, Claudia de Rham, Nicola Tamanini, Suvodip Mukherjee and Mairi
Sakellariadou for interesting discussions. I thank the ICTP-SAIFR for the kind
hospitality.

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