# Quadrupole formulae with cosmological constant: comparison

Denis Dobkowski-Ryłko<EMAIL_ADDRESS>Institute of
Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and
Informatics, University of Gdansk, Wita Stwosza 57, 80-308, Gdańsk, Poland
Jerzy Lewandowski<EMAIL_ADDRESS>Faculty of Physics, University
of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland

###### Abstract

We consider three different approaches (by Ashtekar, Bonga and Kesavan; Hoque
and Virmani; and Dobkowski-Ryłko and Lewandowski) to investigate gravitational
radiation produced by time changing matter source in de Sitter spacetime. All
of them lead to generalizations of the quadrupole formula, however, due to
different gauge conditions and choices of the hypersurfaces, across which the
energy flux is computed, it is nontrivial to see that they all coincide, as
one would expect from the symplectic theory. Each of the expressions for the
radiated energy in the form of gravitational waves is expressed in terms of
the mass and pressure quadrupole moments and written explicitly up to the
linear order in $\sqrt{\Lambda}$, or equvalently in Hubble parameter $H$. It
is shown that up to the first order all three of the generalizations of the
quadrupole formula agree.

##  I Introduction

Due to the current observational evidence for very small, however, positive
value of the cosmological constant $\Lambda$, there have been many recent
developments in the radiation theory for spacetimes with $\Lambda>0$. In
particular, there is a set of papers by Ashtekar et al. ABK1 ; ABK2 ; ABK3 ,
which presents a thorough description of the asymptotics of spacetimes with
positive cosmological constant. One of its main results is the generalization
of the quadrupole formula for the energy carried by gravitational waves, where
the spacelike conformal boundary of de Sitter spacetime was used ABK3 . There
have also been other approaches to obtaining the expression for the radiated
energy of the compact source, and one of the proposals is to use the
cosmological horizon, which is a null surface, as a generalization of the
conformal boundary in the radiation theory in Minkowski spacetime. This
framework has been investigated, among others, by Houqe and Virmani (HV) HV
and Dobkowski-Ryłko and Lewandowski (DRL) DDR&JL . There are other works, were
the asymptotic symmetries and charges in de Sitter spacetime KL2 ; ANS ; CFR1
; PST  and charges and fluxes on cosmological DGGP , black hole DG  and more
general, nonexpanding horizons AKKL1 ; AKKL2  were studied in the presence of
the cosmological constant. The time changing compact source in de Sitter
background was also investigated in Hoque1 , where Isaacson prescription was
used in order to define an effective gravitational stress tensor and compute
the energy flux. Not only it was shown that the two results of HV  and Hoque1
agree, but also that the flux computed on a cosmological horizon agrees with
the one evaluated at conformal boundary $\mathcal{I}^{+}$ of de Sitter
spacetime.

The linearized solutions obtained by Ashtekar, Bonga and Kesavan (ABK) ABK3
in the generalized Lorentz gauge were also derived in Bondi gauge in CHK .
Moreover, the odd parity quadrupole moments were included in the computations
and displacement memory effect have been investigated. The linearized waves
obtained in CHK  were compared to other works on gravitational radiation in de
Sitter spacetime Teukolsky , in particular by Bonga et al. BBP  where the
Bondi-Sachs framework was used to study the linear as well as non-linear
perturbation theory. Despite of different boundary conditions, not only the
linearized solutions from CHK  and BBP  are shown to be equivalent, but also
to the results by Loganayagam and Sheyte LS  on gravitational perturbations in
de Sitter spacetime, where a master variable formalism was applied.
Furthermore, a detailed analysis of the asymptotic structure of (anti-)de
Sitter spacetime was performed in partial Bondi gauge in GZ .

Below we briefly present the common part of the derivation of each of the
three generalizations of the quadrupole formula considered here. It consists
of a set of assumptions used in computations, the general solution to the
linearized Einstein’s equations with cosmological constant $\Lambda$ in the
presence of a first order linearized source and its form written in terms of
the mass and pressure quadrupole moments.

First, consider a compact, time changing, matter source radiating energy in
the form of gravitational waves in de Sitter spacetime. Fig. 1 presents a
conformal diagram of the spacetime of interest. Let $\bar{g}_{\alpha\beta}$ be
the background de Sitter metric, defined on the upperleft triangle – future
Poincaré patch – containing the causal future of the source, of the form

$\displaystyle\bar{g}_{\alpha\beta}dx^{\alpha}dx^{\beta}=\tfrac{1}{H^{2}\eta^{2}}\mathring{g}_{\alpha\beta}dx^{\alpha}dx^{\beta}=\tfrac{1}{H^{2}\eta^{2}}(-d\eta^{2}+dx^{2}+dy^{2}+dz^{2}),$
(1)

where $H:=\sqrt{\Lambda/3}$ is the Hubble parameter and $\Lambda$ is the
cosmological constant. Notice, that in this coordinates the cosmological
horizon is defined as $\eta+r=0$, whereas on spacelike future infinity
$\mathcal{I}^{+}$ we have $\eta=0$. Consider the first order perturbations of
de Sitter spacetime

$\displaystyle
g_{\alpha\beta}=\bar{g}_{\alpha\beta}+\epsilon\gamma_{\alpha\beta},$ (2)

where $\epsilon$ is the smallness parameter. Similarily as in the radiation
theory in Minkowski spacetime, it is convenient to consider the trace-reversed
metric perturbation, namely

$\displaystyle\bar{\gamma}_{\alpha\beta}:=\gamma_{\alpha\beta}-\tfrac{1}{2}\bar{g}_{\alpha\beta}\gamma.$
(3)

Consequently, the linearized Einstein’s equations in the presence of a first
order linearized source $T_{\alpha\beta}$ take the following form

$\displaystyle\bar{\Box}\bar{\gamma}_{\alpha\beta}-2\bar{\nabla}_{(\alpha}\bar{\nabla}^{\mu}\bar{\gamma}_{\beta)\mu}+\bar{g}_{\alpha\beta}\bar{\nabla}^{\mu}\bar{\nabla}^{\nu}\bar{\gamma}_{\mu\nu}-\tfrac{2}{3}\Lambda(\bar{\gamma}_{\alpha\beta}-\bar{g}_{\alpha\beta}\bar{\gamma})=-16\pi
T_{\alpha\beta},$ (4)

where $\bar{\nabla}$ and $\bar{\Box}$ are the derivative and the d’Alembertian
operators with respect to the de Sitter metric $\bar{g}_{\alpha\beta}$.

Figure 1: Compact source radiating energy in the form of gravitational waves.
Its causal future covers the future Poincaré patch (triangle with vertices at
$i^{-}$, $i^{0}$ and $i^{+}$). We assume no incoming radiation across the past
boundary $E^{+}(i^{-})$ (future event horizon of $i^{-}$) of the future
Poincaré patch. $\mathcal{H}$ denotes the cosmological horizon.

Equation (4) have been first derived by Vega et al. Vega  and also presented
more recently in ABK3 , therefore, here we just provide the final results.
Following ABK3  we introduce a vector field $\eta^{\alpha}$, which is normal
to $\eta=\text{const}$ hypersurface satisfying

$\displaystyle\eta^{\alpha}\nabla_{\alpha}\eta=1,$ (5)

and choose the gauge condition

$\displaystyle\bar{\nabla}^{\alpha}\bar{\gamma}_{\alpha\beta}=2Hn^{\alpha}\bar{\gamma}_{\alpha\beta},$
(6)

where $n^{\alpha}:=-H\eta\eta^{\alpha}$. Then, we decompose the metric
perturbation and stress-energy tensor as follows

$\displaystyle\tilde{\chi}:=(\eta^{\alpha}\eta^{\beta}+\mathring{q}^{\alpha\beta})\bar{\chi}_{\alpha\beta},$
$\displaystyle\chi_{\alpha}:=\eta^{\sigma}\mathring{q_{\alpha}}^{\beta}\bar{\chi}_{\beta\sigma},$
$\displaystyle\chi_{\alpha\beta}:=\mathring{q_{\alpha}}^{\mu}\mathring{q_{\beta}}^{\nu}\bar{\chi}_{\mu\nu},$
(7)
$\displaystyle\mathcal{\tilde{T}}:=(\eta^{\alpha}\eta^{\beta}+\mathring{q}^{\alpha\beta})T_{\alpha\beta},$
$\displaystyle\mathcal{T}_{\alpha}:=\eta^{\sigma}\mathring{q_{\alpha}}^{\beta}T_{\beta\sigma},$
$\displaystyle\mathcal{T}_{\alpha\beta}:=\mathring{q_{\alpha}}^{\mu}\mathring{q_{\beta}}^{\nu}T_{\mu\nu}.$
(8)

where $\mathring{q}^{\alpha\beta}$ is the flat, spacial metric induced by
$\mathring{g}_{\alpha\beta}$ on a cosmological slice $\eta=\text{const}$, and
$\bar{\chi}_{\alpha\beta}$ is the rescaled trace-reversed metric perturbation
defined as
$\bar{\chi}_{\alpha\beta}:=H^{2}\eta^{2}\bar{\gamma}_{\alpha\beta}$.
Consequently, the solutions to the Einstein’s equations may be written in the
following form

$\displaystyle\tilde{\chi}(\eta,\vec{x})$
$\displaystyle=4\eta\int\frac{d^{3}\vec{x}^{\prime}}{|\vec{x}-\vec{x}^{\prime}|}\frac{1}{\eta_{{}_{\text{Ret}}}}\mathcal{\tilde{T}}(\eta_{\text{Ret}},\vec{x}^{\prime}),$
(9) $\displaystyle\chi_{a}(\eta,\vec{x})$
$\displaystyle=4\eta\int\frac{d^{3}\vec{x}^{\prime}}{|\vec{x}-\vec{x}^{\prime}|}\frac{1}{\eta_{{}_{\text{Ret}}}}\mathcal{T}_{a}(\eta_{\text{Ret}},\vec{x}^{\prime}),$
(10) $\displaystyle\chi_{ab}(\eta,\vec{x})$
$\displaystyle=4\int\frac{d^{3}\vec{x}^{\prime}}{|\vec{x}-\vec{x}^{\prime}|}\mathcal{T}_{ab}(\eta_{{}_{\text{Ret}}},\vec{x}^{\prime})+4\int
d^{3}\vec{x}^{\prime}\int_{-\infty}^{\eta_{{}_{\text{Ret}}}}d\eta^{\prime}\frac{1}{\eta^{\prime}}\partial_{\eta^{\prime}}\mathcal{T}_{ab}(\eta^{\prime},\vec{x}^{\prime}),$
(11)

where $\eta_{{}_{\text{Ret}}}:=\eta-{|\vec{x}-\vec{x}^{\prime}|}$. Tensors
defined on the cosmological slices $\eta=\text{const}$ carry indices denoted
by lower Latin letters: $a,b,c,...=1,2,3$, as in the expressions above.

Following (ABK3 ) we make several assumption on the radiating isolated system
of interest:

  * •

physical size of the system $D(\eta)$ is uniformly bounded by $D_{0}$ on all
$\eta=\text{const}$ hypersurfaces (which we refer to as cosmological slices);

  * •

physical size of the system is much smaller than the cosmological radius:
$D_{0}\ll 1/H$;

  * •

system is stationary at distant past and future, namely:
$\mathcal{L}_{T}T_{\alpha\beta}=0$, where $T^{\alpha}$ is the Killing vector
field:

$\displaystyle T^{\alpha}\partial_{\alpha}$
$\displaystyle=-H\Big{(}\eta\partial_{\eta}+x\partial_{x}+y\partial_{y}+z\partial_{z}\Big{)};$
(12) $\displaystyle=\partial_{t}-Hr\partial_{r}$ (13)

  * •

no incoming radiation across the past boundary $E^{+}(i^{-})$ (future event
horizon of $i^{-}$) of the future Poincaré patch;

  * •

the velocity of the source is small: $v\ll 1$ (we use the units where
$c=1=G$).

Then, the trace reversed rescaled metric perturbation can be put in the form

$\displaystyle\chi_{ab}(\eta,\vec{x})=\frac{4}{r}\int{d^{3}\vec{x}^{\prime}}\mathcal{T}_{ab}(\eta_{{}_{\text{ret}}},\vec{x}^{\prime})+4\int
d^{3}\vec{x}^{\prime}\int_{-\infty}^{\eta_{{}_{\text{ret}}}}d\eta^{\prime}\frac{1}{\eta^{\prime}}\partial_{\eta^{\prime}}\mathcal{T}_{ab}(\eta^{\prime},\vec{x}^{\prime}),$
(14)

where $\eta_{\text{ret}}=\eta-r$.

Mass and pressure quadrupole moments are defined as

$\displaystyle Q_{ab}^{(\rho)}(\eta)$
$\displaystyle:=\int_{\Sigma}d^{3}V\rho(\eta)\bar{x}_{a}\bar{x}_{b},$ (15)
$\displaystyle Q_{ab}^{(p)}(\eta)$
$\displaystyle:=\int_{\Sigma}d^{3}V\big{(}p_{1}(\eta)+p_{2}(\eta)+p_{3}(\eta)\big{)}\bar{x}_{a}\bar{x}_{b},$
(16)

where $\Sigma$ is any cosmological slice, $d^{3}V$ denotes proper volume
element and $\bar{x}_{a}:=-\tfrac{1}{H\eta}x_{a}$. Matter density $\rho$ and
pressure $p_{i}$ are defined as usual, that is

$\displaystyle\rho=T_{\alpha\beta}n^{\alpha}n^{\beta},$ $\displaystyle
p_{i}=T^{ab}\partial_{a}x_{i}\partial_{b}x_{i}.$ (17)

Retarded solution (14) may be expressed in terms of the quadrupole moments in
a similar way as in Minkowski case (for details see ABK3 )

$\displaystyle\chi_{ab}$
$\displaystyle=-\tfrac{2H}{r}\eta_{\text{ret}}[\ddot{Q}_{ab}^{(\rho)}+2H\dot{Q}_{ab}^{(\rho)}+H\dot{Q}_{ab}^{(p)}+2H^{2}Q_{ab}^{(p)}]({\eta_{\text{ret}}})$
$\displaystyle-2H\int^{\eta_{\text{ret}}}_{-\infty}\frac{d\eta^{\prime}}{\eta^{\prime}}\partial_{\eta}^{\prime}\bigg{(}\eta^{\prime}[\ddot{Q}_{ab}^{(\rho)}+2H\dot{Q}_{ab}^{(\rho)}+H\dot{Q}_{ab}^{(p)}+2H^{2}Q_{ab}^{(p)}](\eta^{\prime})\bigg{)}$
(18)

where the dot corresponds to the lie derivative with respect to the time
translation Killing vector field $T$.

##  II Comparison of the generalizations of quadrupole formulae

Over the last decade, the generalization of the radiation framework for
spacetimes with positive cosmological constant attracted attention of many
relativists. In ABK3  found a generalized quadrupole formula on future
infinity $\mathcal{I}^{+}$ in the TT-gauge was found (ABK). In (HV) and (DRL)
approach, on the other hand, the cosmological Killing horizon, which is a null
surface, was used as the conformal boundary from Minkowski spacetime. Then,
the energy flux across such generalization of the conformal boundary was
calculated. However, despite the fact, that the same null surface has been
chosen, the two derivations differ and the resulting expressions are not
manifestly equivalent. In particular, (HV) uses the symplectic current density
of the covarian phase space approach, that was introduced in WH , together
with the tt-projection in their derivation of the quadrupole formula, whereas
(DDR) concentrates on Wald-Zoupas WZ  and Chandrasekaran-Flanagan-Prabhu CFP
theory of radiation through the null surfaces has been applied and a
generalization of the quadrupole formula has been derived up to linear terms
in the Hubble parameter $H$. This allowed to explicitly show that in the limit
when $\Lambda\rightarrow 0$ the result recovers Einstein’s quadrupole formula.
The goal of the current paper is to check whether the three expressions for
generalized quadrupole formulae agree, at least, up to linear order terms in
$H$ (or equivalently $\sqrt{\Lambda}$).

###  II.1 Generalization of the quadrupole formula by Dobkowski-Ryłko and
Lewandowski

We begin with (DRL) as in this approach the result is already written in the
desired form, namely, such that allows to take a limit as $\Lambda$ approaches
zero just by setting $\sqrt{\Lambda}=0$ (or $H=0$). Since $(\eta,\vec{x})$
coordinates are not suitable to take such limit, one needs to move to a
different coordinate system. Therefore, we introduce coordinates
$(t,r,\theta,\varphi)$, which are defined as

$\displaystyle\eta=-\tfrac{1}{H}e^{-Ht},$ $\displaystyle x=r\cos\theta,$
$\displaystyle y=r\sin\theta\cos\varphi,$ $\displaystyle
z=r\sin\theta\sin\varphi,$ (19)

in which the metric tensor $\bar{g}_{\alpha\beta}$ takes the following form

$\displaystyle\bar{g}_{\alpha\beta}dx^{\alpha}dx^{\beta}=-dt^{2}+e^{2Ht}(dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2}).$ (20)

Notice, that now, by setting $H=0$, the metric coefficients simplify to those
of the Minkowski metric. In (DRL) approach, authors used the general formula
for the energy flux through a null surface derived by Chandrasekaran et al.
CFP . It reads

$\displaystyle E_{T}$
$\displaystyle=\frac{1}{8\pi}\int_{\mathcal{H}}d^{3}V\bigg{(}\sigma_{AB}\sigma^{AB}-\tfrac{1}{2}\theta^{2}\bigg{)}$
(21)

where $\sigma_{AB}$ and $\theta$ are shear and expansion, respectively, of the
Killing vector field $T^{\alpha}$, whereas $\mathcal{H}$ is the cosmological
horizon with the proper volume form $d^{3}V$. A suitable gauge condition needs
to be implemented, for the formula (21) to be applicable. Namely, the
perturbed cosmological horizon defined by $r=\exp(-Ht)/H$ must remain
null111For details see DDR&JL .. Consequently, a generalization of the
quadrupole formula has been derived up to terms linear in $H$

$\displaystyle E_{T}=\frac{1}{45}\int
dt\sum_{i,j=1}^{3}\Bigg{[}\bigg{(}\frac{d^{3}q^{(\rho)}_{ij}}{dt^{3}}\bigg{)}^{2}+2H\Bigg{(}\frac{d^{3}q^{(\rho)}_{ij}}{dt^{3}}\bigg{(}\frac{d^{2}q^{(p)}_{ij}}{dt^{2}}{-7}\frac{d^{2}q^{(\rho)}_{ij}}{dt^{2}}\bigg{)}\Bigg{)}\Bigg{]}(t_{\text{ret}})+\mathcal{O}(H^{2}),$
(22)

where

$\displaystyle q^{(i)}_{ab}:=$
$\displaystyle\mathring{q}^{ab}Q^{(i)}_{ab}-\tfrac{1}{3}\mathring{q}_{ab}Q^{(i)},$
(23)

are the traceless quadrupole moments and $i$ stands for $\rho$ (mass
quadrupole moment) or $p$ (pressure quadrupole moment). Here the retared time
$t_{\text{ret}}$ is defined as:

$\displaystyle t_{\text{ret}}=$
$\displaystyle-\frac{1}{H}\ln(e^{-Ht}+Hr)=-\frac{1}{H}\ln(2e^{-Ht})$ (25)

and the last equality holds on the horizon $\mathcal{H}$. Expression (22) is
invariant under transformations $t\rightarrow t+\frac{1}{H}c$, where
$c=\text{const}$. By setting $c=\ln 2$, we obtain the same expression as (22),
but now with the square bracket evalutated at $t$ instead of $t_{\text{ret}}$.
Furthermore, notice, that the last term in the square bracket in (22) maybe in
turned into the total derivative, namely

$\displaystyle\frac{d^{3}q^{(\rho)}_{ij}}{dt^{3}}\frac{d^{2}q^{(\rho)}_{ij}}{dt^{2}}=\frac{1}{2}\frac{d}{dt}\bigg{(}\frac{d^{2}q^{(\rho)}_{ij}}{dt^{2}}\bigg{)}^{2},$
(26)

which integrated vanishes for the source of our interest – stationary at
distant past and future. Therefore, this version of the generalized quadrupole
formula simplifies to

$\displaystyle E_{T}=\frac{1}{45}\int
dt\sum_{i,j=1}^{3}\Bigg{[}\bigg{(}\frac{d^{3}q^{(\rho)}_{ij}}{dt^{3}}\bigg{)}^{2}+2H\bigg{(}\frac{d^{3}q^{(\rho)}_{ij}}{dt^{3}}\frac{d^{2}q^{(p)}_{ij}}{dt^{2}}\bigg{)}\Bigg{]}+\mathcal{O}(H^{2}).$
(27)

Taking a limit for the vanishing cosmological constant is straightforward and
requires only setting $H$ to zero, the result reads

$\displaystyle E_{T}=\frac{1}{45}\int
dt\sum_{i,j=1}^{3}\bigg{(}\frac{d^{3}q^{(\rho)}_{ij}}{dt^{3}}\bigg{)}^{2}$
(28)

and coincides with the famous Einstein quadrupole formula Wald .

###  II.2 Generalization of the quadrupole formula by Hoque and Virmani

In this subsection we investigate the (HV) version of the generalized
quadrupole formula, which is also an expression for energy flux across the
cosmological horizon. It was shown that, by fully exhausting gauge freedom,
one may obtain the following constraint

$\displaystyle\partial^{i}\chi_{ij}={\chi^{i}}_{i}=0=\chi_{00}.$ (29)

To extract the transverse traceless part of the metric perturbation, the tt-
projection is applied:

$\displaystyle\chi^{\text{tt}}_{ij}={\Lambda_{ij}}^{kl}\chi_{kl}$ (30)

where:

$\displaystyle{\Lambda^{kl}}_{ij}$
$\displaystyle=\tfrac{1}{2}({P_{i}}^{k}{P_{j}}^{l}+{P_{i}}^{l}{P_{j}}^{k}-P_{ij}P^{kl}),$
(31) $\displaystyle{P_{i}}^{j}$
$\displaystyle={\delta_{i}}^{j}-\tilde{x}_{i}\tilde{x}^{j},$ (32)
$\displaystyle\tilde{x}^{i}$ $\displaystyle=x^{i}/r.$ (33)

In HV  it is stressed that the radial tt-projection is not an appropriate
operation to extract transverse-traceless part of the metric perturbation
$\chi_{ij}$, however, it is expected that for the radial distances far from
the source, it seems reasonable that the tt-projection will provide good
results. The correct notion to analyse asymptotically de Sitter spacetimes is
the TT-gauge which is provided by the decomposition of $\chi_{ij}$, namely

$\displaystyle\chi_{ij}=\frac{1}{3}\delta_{ij}\delta^{kl}\chi_{kl}+\Big{(}\partial_{i}\partial_{j}-\tfrac{1}{3}\delta_{ij}\nabla^{2}\Big{)}B+\partial_{i}B_{j}^{T}+\partial_{j}B^{T}_{i}+\chi_{ij}^{TT}.$
(34)

In the above decomposition $B_{i}^{T}$ is transverse, that is
$\partial^{i}B_{i}^{T}=0$, and $\chi_{ij}^{TT}$ is the gauge invariant term,
which is transverse-traceless

$\displaystyle\partial^{i}\chi_{ij}^{TT}=\delta^{ij}\chi_{ij}^{TT}=0.$ (35)

For more details on the conceptual difference of the tt-projection and TT-
gauge see AB . The final expression for the energy flux across the
cosmological horizons $\mathcal{H}$ obtained by (HV) approach is of the form

$\displaystyle E_{T}=\frac{1}{8\pi}\int d\tau\int_{S_{2}}d\Omega\
R^{\text{tt}}_{ij}\ R^{\text{tt}}_{kl}\ \delta^{ik}\delta^{jl},$ (36)

where:

$\displaystyle R_{ij}$
$\displaystyle=\dot{A}_{ij}+HA_{ij}=T^{\mu}\partial_{\mu}A_{ij}-HA_{ij},$ (37)
$\displaystyle A_{ij}$
$\displaystyle=\ddot{Q}^{(\rho)}_{ij}+2H\dot{Q}^{(\rho)}_{ij}+H\dot{Q}^{(p)}_{ij}+2H^{2}Q^{(p)}_{ij},$
(38)

and the dots indicate the Lie derivatives with respect to the Killing vector
field $T^{\mu}$, namely $\dot{X}_{ab}:=\mathcal{L}_{T}X_{ab}$, whereas $\tau$
is the Killing parameter along the integral curves of $T^{\mu}$. We choose
$(\tau,\theta,\varphi)$ to be the coordinates on the the cosmological horizon
$\mathcal{H}$, such that

$\displaystyle\frac{\partial\eta}{\partial\tau}=-H\eta,$
$\displaystyle\frac{\partial x^{i}}{\partial\tau}=-Hx^{i}.$ (39)

Consequently, it is convenient to express $T^{\mu}\partial_{\mu}$ as a
derivative with respect to $\tau$

$\displaystyle A_{ij}$
$\displaystyle=\frac{d^{2}}{d\tau^{2}}Q^{(\rho)}_{ij}-2H\frac{d}{d\tau}Q^{(\rho)}_{ij}+H\frac{d}{d\tau}Q^{(p)}_{ij},$
(40) $\displaystyle R_{ij}$
$\displaystyle=\frac{d^{3}}{d\tau^{3}}Q^{(\rho)}_{ij}-2H\frac{d^{2}}{d\tau^{2}}Q^{(\rho)}_{ij}+H\frac{d^{2}}{d\tau^{2}}Q^{(p)}_{ij}-H\frac{d^{2}}{d\tau^{2}}Q^{(\rho)}_{ij}+2H^{2}\frac{d}{d\tau}Q^{(\rho)}_{ij}-H^{2}\frac{d}{d\tau}Q^{(p)}_{ij}$
$\displaystyle=\frac{d^{3}}{d\tau^{3}}Q^{(\rho)}_{ij}-3H\frac{d^{2}}{d\tau^{2}}Q^{(\rho)}_{ij}+H\frac{d^{2}}{d\tau^{2}}Q^{(p)}_{ij}+\mathcal{O}(H^{2}).$
(41)

We start by computing the expression for $R^{\text{tt}}_{ij}$ in terms of
$R_{ij}$ and $R=\delta^{ij}R_{ij}$ that is

$\displaystyle R^{\text{tt}}_{ij}$ $\displaystyle={\Lambda_{ij}}^{kl}R_{kl}$
(42)
$\displaystyle=R_{ij}-R_{il}\tilde{x}_{j}\tilde{x}^{l}-R_{kj}\tilde{x}_{i}\tilde{x}^{k}+\tfrac{1}{2}R_{kl}\tilde{x}_{i}\tilde{x}^{k}\tilde{x}_{j}\tilde{x}^{l}-\tfrac{1}{2}\delta_{ij}(R-R_{kl}\tilde{x}^{k}\tilde{x}^{l})+\tfrac{1}{2}R\tilde{x}_{i}\hat{x}_{j}.$
(43)

We may then expand the expression under the integral in (36) to

$\displaystyle R^{\text{tt}}_{ij}\ R^{\text{tt}}_{kl}\delta^{ik}\delta^{jl}$
$\displaystyle=R^{\text{tt}}_{xx}\ R^{\text{tt}}_{xx}+2R^{\text{tt}}_{xy}\
R^{\text{tt}}_{xy}+2R^{\text{tt}}_{xz}\ R^{\text{tt}}_{xz}+R^{\text{tt}}_{yy}\
R^{\text{tt}}_{yy}+2R^{\text{tt}}_{yz}\ R^{\text{tt}}_{yz}+R^{\text{tt}}_{zz}\
R^{\text{tt}}_{zz}.$ (44)

Each term on the right hand side has to be integrated over the cosmological
horizon $\mathcal{H}$. Here we show how to perform the integration of
$R^{\text{tt}}_{xx}R^{\text{tt}}_{xx}$, namely

$\displaystyle\frac{1}{8\pi}\int d\tau\int_{S_{2}}d\Omega R^{\text{tt}}_{xx}\
R^{\text{tt}}_{xx}$ $\displaystyle=\frac{1}{24}\int
d\tau(\tfrac{128}{105}R^{2}_{xx}+\tfrac{4}{7}R^{2}_{yy}+\tfrac{4}{7}R^{2}_{zz}+\tfrac{32}{105}R^{2}_{xy}+\tfrac{32}{105}R^{2}_{xz}+\tfrac{16}{15}R^{2}_{yz}$
$\displaystyle-\tfrac{128}{105}R_{xx}R_{yy}-\tfrac{128}{105}R_{xx}R_{zz}+\tfrac{8}{105}R_{yy}R_{zz}).$
(45)

In a similar fashion we find the other terms and sum them up. The result reads

$\displaystyle\frac{1}{8\pi}\int d\tau\int_{S_{2}}d\Omega\ R^{\text{tt}}_{ij}\
R^{\text{tt}}_{kl}\delta^{ik}\delta^{jl}$ $\displaystyle=\tfrac{2}{15}\int
d\tau\Big{(}R^{2}_{xx}+R^{2}_{yy}+R^{2}_{zz}+3(R^{2}_{xy}+R^{2}_{xz}+R^{2}_{yz})$
$\displaystyle-R_{xx}R_{yy}-R_{xx}R_{zz}-R_{yy}R_{zz}\Big{)}.$ (46)

Using the expression (II.2) to express $R_{ij}$ in terms of the mass and
pressure quadrupole moments yields:

$\displaystyle E_{T}$ $\displaystyle=\tfrac{2}{15}\int
d\tau\Bigg{[}(\partial_{\tau}^{3}Q_{xx}^{(\rho)})^{2}+(\partial_{\tau}^{3}Q_{yy}^{(\rho)})^{2}+(\partial_{\tau}^{3}Q_{zz}^{(\rho)})^{2}+3(\partial_{t}^{3}Q_{xy}^{(\rho)})^{2}+3(\partial_{t}^{3}Q_{xz}^{(\rho)})^{2}+3(\partial_{t}^{3}Q_{yz}^{(\rho)})^{2}$
$\displaystyle-(\partial_{\tau}^{3}Q_{xx}^{(\rho)})(\partial_{\tau}^{3}Q_{yy}^{(\rho)})-(\partial_{\tau}^{3}Q_{xx}^{(\rho)})(\partial_{t}^{3}Q_{zz}^{(\rho)})-(\partial_{\tau}^{3}Q_{yy}^{(\rho)})(\partial_{\tau}^{3}Q_{zz}^{(\rho)})$
$\displaystyle+2H\partial_{\tau}^{3}Q_{xx}^{(\rho)}\Big{(}\partial_{\tau}^{2}Q_{xx}^{(p)}-3\partial_{\tau}^{2}Q_{xx}^{(\rho)}\Big{)}+2H\partial_{\tau}^{3}Q_{yy}^{(\rho)}\Big{(}\partial_{\tau}^{2}Q_{yy}^{(p)}-3\partial_{\tau}^{2}Q_{yy}^{(\rho)}\Big{)}$
$\displaystyle+2H\partial_{\tau}^{3}Q_{zz}^{(\rho)}\Big{(}\partial_{\tau}^{2}Q_{zz}^{(p)}-3\partial_{\tau}^{2}Q_{zz}^{(\rho)}\Big{)}+6H\partial_{\tau}^{3}Q_{xy}^{(\rho)}\Big{(}\partial_{\tau}^{2}Q_{xy}^{(p)}-3\partial_{\tau}^{2}Q_{xy}^{(\rho)}\Big{)}$
$\displaystyle+6H\partial_{\tau}^{3}Q_{xz}^{(\rho)}\Big{(}\partial_{\tau}^{2}Q_{xz}^{(p)}-3\partial_{\tau}^{2}Q_{xz}^{(\rho)}\Big{)}+6H\partial_{\tau}^{3}Q_{yz}^{(\rho)}\Big{(}\partial_{\tau}^{2}Q_{yz}^{(p)}-3\partial_{\tau}^{2}Q_{yz}^{(\rho)}\Big{)}$
$\displaystyle-H\partial_{\tau}^{3}Q_{xx}^{(\rho)}\Big{(}\partial_{\tau}^{2}Q_{yy}^{(p)}-3\partial_{\tau}^{2}Q_{yy}^{(\rho)}\Big{)}-H\partial_{\tau}^{3}Q_{yy}^{(\rho)}\Big{(}\partial_{\tau}^{2}Q_{xx}^{(p)}-3\partial_{\tau}^{2}Q_{xx}^{(\rho)}\Big{)}$
$\displaystyle-H\partial_{\tau}^{3}Q_{xx}^{(\rho)}\Big{(}\partial_{\tau}^{2}Q_{zz}^{(p)}-3\partial_{\tau}^{2}Q_{zz}^{(\rho)}\Big{)}-H\partial_{\tau}^{3}Q_{zz}^{(\rho)}\Big{(}\partial_{\tau}^{2}Q_{xx}^{(p)}-3\partial_{\tau}^{2}Q_{xx}^{(\rho)}\Big{)}$
$\displaystyle-H\partial_{\tau}^{3}Q_{yy}^{(\rho)}\Big{(}\partial_{\tau}^{2}Q_{zz}^{(p)}-3\partial_{\tau}^{2}Q_{zz}^{(\rho)}\Big{)}-H\partial_{\tau}^{3}Q_{zz}^{(\rho)}\Big{(}\partial_{\tau}^{2}Q_{yy}^{(p)}-3\partial_{\tau}^{2}Q_{yy}^{(\rho)}\Big{)}\Bigg{]}+\mathcal{O}(H^{2}).$
(47)

Furthermore, similarly as in DDR&JL , we introduce the traceless quadrupole
moments $q^{(i)}_{ab}$ via (23) to simplify the above expression.

Finally, the generalized quadrupole formula by Hoque et al. (36) expanded in
the Hubble parameter and written up to linear order in $H$ is of the form

$\displaystyle E_{T}=\tfrac{1}{45}\int
d\tau\sum_{i,j=1}^{3}\Bigg{[}\bigg{(}\tfrac{d^{3}q^{(\rho)}_{ij}}{d\tau^{3}}\bigg{)}^{2}+2H\bigg{(}\tfrac{d^{3}q^{(\rho)}_{ij}}{d\tau^{3}}\Big{(}\tfrac{d^{2}q^{(p)}_{ij}}{d\tau^{2}}{-3}\tfrac{d^{2}q^{(\rho)}_{ij}}{d\tau^{2}}\Big{)}\bigg{)}\Bigg{]}+\mathcal{O}(H^{2}).$
(48)

Notice, that here we may also express the last term in the square bracket
above as a total derivative (as in (26) but with a different constant factor),
which after integration yields the boundary terms that vanish due to the
assumption of the stationarity of the source in far past and future.
Consequently, the result coincides with the formula (27) found in DDR&JL  up
to the linear order terms in Hubble parameter $H=\sqrt{\Lambda/3}$.

###  II.3 Generalization of the quadrupole formula by Ashtekar, Bonga and
Kesavan

The last energy flux formula for gravitational radiation in de Sitter
background that we consider in this paper was calculated in (ABK) approach at
future infinity $\mathcal{I}^{+}$ and is of the form

$\displaystyle E_{T}\hat{=}\frac{1}{16\pi
H}\int_{\mathcal{I}^{+}}d^{3}x\mathcal{E}_{cd}\big{(}\mathcal{L}_{T}\chi_{ab}+2H\chi_{ab}\big{)}\mathring{q}^{ac}\mathring{q}^{bd},$
(49)

where ”$\hat{=}$” denotes the value at $\mathcal{I}^{+}$ and

$\displaystyle\mathcal{E}_{ab}=\frac{1}{2H\eta}\bigg{(}\mathring{q}_{a}^{c}\mathring{q}_{b}^{d}-\tfrac{1}{3}\mathring{q}_{ab}\mathring{q}^{cd}\bigg{)}\bigg{[}\tfrac{1}{2}\mathring{D}_{c}\mathring{D}_{d}\tilde{\chi}-\mathring{D}_{(c}\mathring{D}^{m}\chi_{d)m}-\mathring{D}_{(c}\partial_{\eta}\chi_{d)}+\Big{(}\partial_{\eta}^{2}-\tfrac{1}{\eta}\partial_{\eta}\Big{)}\chi_{cd}\bigg{]}$
(50)

is the perturbed electric part of the Weyl curvature. It can be shown, that
due to the gauge condition (6) the above expression for $\mathcal{E}_{ab}$
simplifies to

$\displaystyle\mathcal{E}_{ab}=\frac{1}{2H\eta}\bigg{(}\partial_{\eta}^{2}-\frac{1}{\eta}\bigg{)}\chi_{ab}^{TT}.$
(51)

Furthermore, the energy flux across future infinity $\mathcal{I}^{+}$
simplifies to:

$\displaystyle
E_{T}\hat{=}\frac{1}{8\pi}\int_{\mathcal{I^{+}}}dTd^{2}S\Big{[}R_{ab}R_{cd}^{TT}\mathring{q}^{ac}\mathring{q}^{bd}\Big{]},$
(52)

where $T$ is the affine parameter along integral curves of the Killing vector
field $T^{a}$ satisfying $dT=-dr/(rH)$ at future infinity and the radiation
field $R_{ab}$ is again of the form (37). Due to the fact that extracting TT
part of a tensor is a highly nontrivial procedure, instead of expanding eq.
(52) in Hubble parameter, we will use the intermediate form of the quadrupole
formula (II.3).

Therefore, we proceed by calculating each term present in the expression (50)
for the electric part of Weyl tensor $\mathcal{E}_{ab}$. Gauge condition (6)
can be split into two separate equations of the following form

$\displaystyle\mathring{D}^{a}\chi_{ab}=\partial_{\eta}\chi_{b}-\tfrac{2}{\eta}\chi_{b},$
$\displaystyle\mathring{D}^{a}\chi_{a}=\partial_{\eta}\big{(}\tilde{\chi}-\chi\big{)}-\tfrac{1}{\eta}\tilde{\chi},$

where $\chi=\mathring{q}^{ab}\chi_{ab}$. These equations may be solved and
provide the expressions for $\chi_{a}$ and $\tilde{\chi}$ written in terms of
the space-space component of the metric perturbation $\chi_{ab}$, namely

$\displaystyle\tilde{\chi}$
$\displaystyle=\eta\int^{\eta}_{-\infty}\frac{\mathring{D}^{a}\chi_{a}+\partial_{\eta^{\prime}}\chi}{\eta^{\prime}}d\eta^{\prime},$
(53) $\displaystyle\chi_{a}$
$\displaystyle=\eta^{2}\int^{\eta}_{-\infty}\frac{\mathring{D}^{a}\chi_{ab}}{\eta^{\prime
2}}d\eta^{\prime}.$ (54)

For the convenience we introduce the following notation

$\displaystyle\mathbb{T}_{ab}(\eta):=\int{d^{3}\vec{x}^{\prime}}\mathcal{T}_{ab}(\eta,\vec{x}^{\prime}),$
(55)

which allows to write the metric perturbation $\chi_{ab}$ in a more compact
form

$\displaystyle\chi_{ab}(\eta,\vec{x})=\frac{4}{r}\mathbb{T}_{ab}(\eta_{\text{ret}})+4\int_{-\infty}^{\eta_{{}_{\text{ret}}}}d\eta^{\prime}\frac{1}{\eta^{\prime}}\partial_{\eta^{\prime}}\mathbb{T}_{ab}(\eta^{\prime}).$
(56)

We begin with calculating the divergence of the metric perturbation
$\chi_{ab}$ with respect to the covariant derivative $\mathring{D}_{a}$ (of
the spatial metric $\mathring{q}_{ab}$)

$\displaystyle\mathring{D}^{a}\chi_{ab}$
$\displaystyle=-\frac{4x^{a}}{r^{3}}\mathbb{T}_{ab}(\eta_{\text{ret}})+\frac{4}{r}D^{a}\mathbb{T}_{ab}(\eta_{\text{ret}})+4D^{a}\int_{-\infty}^{\eta_{{}_{\text{ret}}}}d\eta^{\prime}\frac{1}{\eta^{\prime}}\partial_{\eta^{\prime}}\mathbb{T}_{ab}(\eta^{\prime})$
$\displaystyle=-\frac{4x^{a}}{r^{3}}\mathbb{T}_{ab}(\eta_{\text{ret}})-\frac{4x^{a}\eta}{r^{2}\eta_{\text{ret}}}\partial_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}}).$
(57)

Then, we input the above to the eq. (54) and find

$\displaystyle\chi_{b}$
$\displaystyle=-\eta^{2}\int^{\eta}_{-\infty}\bigg{(}\frac{4x^{a}}{\eta^{\prime
2}r^{3}}\mathbb{T}_{ab}(\eta^{\prime}_{\text{ret}})+\frac{4x^{a}}{r^{2}\eta\eta_{\text{ret}}}\partial_{\eta}\mathbb{T}_{ab}(\eta^{\prime}_{\text{ret}})\bigg{)}d\eta^{\prime}$
$\displaystyle=-\frac{4x^{a}\eta^{2}}{r^{2}}\int^{\eta}_{-\infty}\bigg{(}\frac{1}{\eta^{\prime
2}r}\mathbb{T}_{ab}(\eta^{\prime}_{\text{ret}})-\frac{r-2\eta^{\prime}}{\eta^{\prime
2}\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{ab}(\eta^{\prime}_{\text{ret}})\bigg{)}d\eta^{\prime}-\frac{4x^{a}\eta}{r^{2}{\eta_{\text{ret}}}}\mathbb{T}_{ab}(\eta_{\text{ret}})$
$\displaystyle=-\frac{4x^{a}\eta^{2}}{r^{3}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{ab}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}-\frac{4x^{a}\eta}{r^{2}{\eta_{\text{ret}}}}\mathbb{T}_{ab}(\eta_{\text{ret}}),$
(58)

where in the second line we integrated by parts. Next, we calculate the
partial derivative with respect to $\eta$ of the above expression, which
yields:

$\displaystyle\partial_{\eta}\chi_{b}$
$\displaystyle=-\frac{8x^{a}\eta}{r^{3}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{ab}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}-\frac{4(\eta+r)x^{a}}{r^{3}\eta_{\text{ret}}}\mathbb{T}_{ab}(\eta_{\text{ret}})-\frac{4x^{b}\eta}{\eta_{\text{ret}}r^{2}}\partial_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}}).$
(59)

Then, by adding equations (II.3) and (59) we obtain

$\displaystyle\mathring{D}^{a}\chi_{ab}+\partial_{\eta}\chi_{b}$
$\displaystyle=-\frac{8x^{a}\eta}{r^{3}\eta_{\text{ret}}}\mathbb{T}_{ab}(\eta_{\text{ret}})-\frac{8x^{a}\eta}{r^{2}\eta_{\text{ret}}}\partial_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}})-\frac{8x^{a}\eta}{r^{3}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{ab}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}.$
(60)

Finally, we calculate the covariant derivative of the above expression to find
the middle two terms in the square bracket of eq. (50), which yields

$\displaystyle\mathring{D}_{c}(D^{a}\chi_{ab}+\partial_{\eta}\chi_{b})$
$\displaystyle=-\frac{8\eta}{r^{3}\eta_{\text{ret}}}\mathbb{T}_{bc}(\eta_{\text{ret}})+\frac{24x^{a}x_{c}\eta}{r^{5}\eta_{\text{ret}}}\mathbb{T}_{ab}(\eta_{\text{ret}})+\frac{8(3\eta-4r)\eta
x^{a}x_{c}}{r^{4}\eta^{2}_{\text{ret}}}\partial_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}})$
$\displaystyle-\frac{8\eta}{r^{2}\eta_{\text{ret}}}\partial_{\eta}\mathbb{T}_{bc}(\eta_{\text{ret}})+\frac{8x_{c}x^{a}\eta}{r^{3}\eta_{\text{ret}}}\partial^{2}_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}})-\frac{8\eta}{r^{3}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{bc}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}$
$\displaystyle+\frac{24x_{c}x^{a}\eta}{r^{5}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{ab}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}.$
(61)

Notice the overall factor of $1/\eta$ in (50), therefore, before evaluating
the value of (II.3) at $\mathcal{I}^{+}$, we have to first divide it by $\eta$
and then set $\eta=0$, namely

$\displaystyle\frac{1}{\eta}\mathring{D}_{c}(D^{a}\chi_{ab}+\partial_{\eta}\chi_{b})$
$\displaystyle\hat{=}\frac{8}{r^{4}}\mathbb{T}_{bc}(\eta_{\text{ret}})-\frac{24x^{a}x_{c}}{r^{6}}\mathbb{T}_{ab}(\eta_{\text{ret}})-\frac{32x^{a}x_{c}}{r^{5}}\partial_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}})$
$\displaystyle+\frac{8}{r^{3}}\partial_{\eta}\mathbb{T}_{bc}(\eta_{\text{ret}})-\frac{8x_{c}x^{a}}{r^{4}}\partial^{2}_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}})-\frac{8}{r^{3}}\int^{0}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{bc}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}$
$\displaystyle+\frac{24x_{c}x^{a}}{r^{5}}\int^{0}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{ab}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}.$
(62)

Moreover, since we are interested in expanding (49) to zeroth and first order
terms in Hubble parameter $H$, it is sufficient only to consider the leading
order terms in $1/r$ of (II.3), which will become clear when we use the affine
parameter of the integral curves of $T^{a}$ as a coordinate on
$\mathcal{I}^{+}$. With that in mind, we compute

$\displaystyle\frac{1}{\eta}\mathring{D}_{c}(D^{a}\chi_{ab}+\partial_{\eta}\chi_{b})$
$\displaystyle\hat{=}-\frac{8x_{c}x^{a}}{r^{4}}\partial^{2}_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}})+\mathcal{O}(H^{3}).$
(63)

Here, we will not calculate the remaining two terms of (50) explicitly, but
only present the leading order terms in Hubble parameter $H$ of their final
expressions (for a detailed calculation see the Appendix), that is

$\displaystyle\tfrac{1}{\eta}D_{c}D_{d}\tilde{\chi}$
$\displaystyle\hat{=}-\frac{4x_{c}x_{d}}{r^{4}\eta_{\text{ret}}}\partial^{2}_{\eta}\mathbb{T}(\eta_{\text{ret}})-\frac{4x^{a}x^{b}x_{c}x_{d}}{r^{6}}\partial^{2}_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}})+\mathcal{O}(H^{3}),$
(64)

and

$\displaystyle\frac{1}{\eta}\Big{(}\partial_{\eta}^{2}-\tfrac{1}{\eta}\partial_{\eta}\Big{)}\chi_{cd}\hat{=}-\frac{4}{r}\partial_{r}\bigg{(}\frac{1}{r}\partial_{r}\mathbb{T}_{cd}(\eta_{\text{ret}})\bigg{)}.$
(65)

Consequently, by combining expressions (63), (64) and (65), and using the
identity
$\partial_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}})=-\partial_{r}\mathbb{T}_{ab}(\eta_{\text{ret}})$
we calcualate the perturbed electric part of the Weyl tensor and write it
explicitly up linear terms in $H$

$\displaystyle\mathcal{E}_{ab}$
$\displaystyle=\bigg{(}\mathring{q}_{a}^{c}\mathring{q}_{b}^{d}-\tfrac{1}{3}\mathring{q}_{ab}\mathring{q}^{cd}\bigg{)}\bigg{[}-\frac{x^{m}x^{n}x_{c}x_{d}}{Hr^{6}}\partial_{r}^{2}\mathbb{T}_{mn}(\eta_{\text{ret}})-\frac{x_{c}x_{d}}{Hr^{4}}\partial_{r}^{2}\mathbb{T}(\eta_{\text{ret}})d\eta^{\prime}$
$\displaystyle+\frac{2x^{m}x_{c}}{Hr^{4}}\partial^{2}_{r}\mathbb{T}_{dm}(\eta_{\text{ret}})+\frac{2x^{m}x_{d}}{Hr^{4}}\partial^{2}_{r}\mathbb{T}_{cm}(\eta_{\text{ret}})-\frac{2}{Hr^{2}}\partial^{2}_{r}\mathbb{T}_{cd}(\eta_{\text{ret}})\bigg{]}+\mathcal{O}(H^{2}).$
(66)

It is also convenient to compute the following expression

$\displaystyle\mathcal{L}_{T}\chi_{ab}+2H\chi_{ab}$
$\displaystyle=-H(\eta\partial_{\eta}+r\partial_{r})\chi_{ab}$ $\displaystyle\
\hat{=}\frac{4H}{r}\mathbb{T}_{ab}(\eta_{\text{ret}}).$ (67)

The formula of (ABK) for energy emitted by gravitational waves now simplifies
to

$\displaystyle E_{T}$
$\displaystyle\hat{=}\frac{1}{4\pi}\int_{\mathcal{I}^{+}}drd\Omega\
\bigg{(}\mathring{q}^{ec}\mathring{q}^{df}-\tfrac{1}{3}\mathring{q}^{ef}\mathring{q}^{cd}\bigg{)}\bigg{[}-\frac{\tilde{x}^{m}\tilde{x}^{n}\tilde{x}_{c}\tilde{x}_{d}}{Hr}\partial_{r}^{2}\mathbb{T}_{mn}(\eta_{\text{ret}})-\frac{\tilde{x}_{c}\tilde{x}_{d}}{Hr}\partial_{r}^{2}\mathbb{T}(\eta_{\text{ret}})$
$\displaystyle+\frac{4\tilde{x}^{m}\tilde{x}_{c}}{Hr}\partial^{2}_{r}\mathbb{T}_{dm}(\eta_{\text{ret}})-\frac{2}{Hr}\partial^{2}_{r}\mathbb{T}_{cd}(\eta_{\text{ret}})\bigg{]}\mathbb{T}_{ef}(\eta_{\text{ret}})+\mathcal{O}(H^{2})$
$\displaystyle\hat{=}\frac{1}{4\pi}\int_{\mathcal{I}^{+}}\frac{drd\Omega}{Hr}\
\bigg{(}\mathring{q}^{ec}\mathring{q}^{df}-\tfrac{1}{3}\mathring{q}^{ef}\mathring{q}^{cd}\bigg{)}\bigg{[}\tilde{x}^{m}\hat{x}^{n}\tilde{x}_{c}\tilde{x}_{d}\partial_{r}\mathbb{T}_{mn}(\eta_{\text{ret}})+\tilde{x}_{c}\hat{x}_{d}\partial_{r}\mathbb{T}(\eta_{\text{ret}})$
$\displaystyle-4\tilde{x}^{m}\tilde{x}_{c}\partial_{r}\mathbb{T}_{dm}(\eta_{\text{ret}})+2\partial_{r}\mathbb{T}_{cd}(\eta_{\text{ret}})\bigg{]}\partial_{r}\mathbb{T}_{ef}(\eta_{\text{ret}})+\mathcal{O}(H^{2}),$
(68)

where we again used the notion (33), integrated by parts and kept only terms
at most linear in $H$. Recall, that $T$ is the affine parameter along the
integral curves of $T^{a}$ and satisfies $dT=-dr/(rH)$ at $\mathcal{I}^{+}$,
in particular we may use $T$ as a new coordinate defined by the transformation

$\displaystyle r\hat{=}\frac{1}{H}e^{-HT}.$ (69)

Finally, integrating over the angular coordinates of (II.3) yields

$\displaystyle E_{T}$ $\displaystyle=\frac{2}{15}\int dT\
\bigg{(}4(\partial_{r}\mathbb{T}_{xx})^{2}+4(\partial_{r}\mathbb{T}_{yy})^{2}+4(\partial_{r}\mathbb{T}_{zz})^{2}+12(\partial_{r}\mathbb{T}_{xy})^{2}+12(\partial_{r}\mathbb{T}_{xz})^{2}+12(\partial_{r}\mathbb{T}_{yz})^{2}$
(70)
$\displaystyle-4\partial_{r}\mathbb{T}_{xx}\partial_{r}\mathbb{T}_{yy}(\eta_{\text{ret}})-4\partial_{r}\mathbb{T}_{xx}\partial_{r}\mathbb{T}_{zz}(\eta_{\text{ret}})-4\partial_{r}\mathbb{T}_{yy}\partial_{r}\mathbb{T}_{zz}\bigg{)}+\mathcal{O}(H^{2}).$
(71)

where

$\displaystyle\partial_{r}\mathbb{T}_{ab}$
$\displaystyle\hat{=}-\frac{1}{2}\bigg{[}\partial_{T}^{3}Q_{ab}^{(\rho)}-3H\partial_{T}^{2}Q_{ab}^{(\rho)}+H\partial_{T}^{2}Q_{ab}^{(p)}\bigg{]}+\mathcal{O}(H^{2}).$
(72)

Thus, we obtain the generalized quadrupole formula at $\mathcal{I}^{+}$
written up to the terms linear in the Hubble parameter $H$ (or equivalently
$\sqrt{\Lambda/3}$)

$\displaystyle E_{T}$ $\displaystyle=\tfrac{2}{15}\int
dT\Bigg{[}(\partial_{T}^{3}Q_{xx}^{(\rho)})^{2}+(\partial_{T}^{3}Q_{yy}^{(\rho)})^{2}+(\partial_{T}^{3}Q_{zz}^{(\rho)})^{2}+3(\partial_{T}^{3}Q_{xy}^{(\rho)})^{2}+3(\partial_{T}^{3}Q_{xz}^{(\rho)})^{2}+3(\partial_{T}^{3}Q_{yz}^{(\rho)})^{2}$
$\displaystyle-(\partial_{T}^{3}Q_{xx}^{(\rho)})(\partial_{T}^{3}Q_{yy}^{(\rho)})-(\partial_{T}^{3}Q_{xx}^{(\rho)})(\partial_{T}^{3}Q_{zz}^{(\rho)})-(\partial_{T}^{3}Q_{yy}^{(\rho)})(\partial_{T}^{3}Q_{zz}^{(\rho)})$
$\displaystyle+2H\partial_{T}^{3}Q_{xx}^{(\rho)}\Big{(}\partial_{T}^{2}Q_{xx}^{(p)}-3\partial_{T}^{2}Q_{xx}^{(\rho)}\Big{)}+2H\partial_{T}^{3}Q_{yy}^{(\rho)}\Big{(}\partial_{T}^{2}Q_{yy}^{(p)}-3\partial_{T}^{2}Q_{yy}^{(\rho)}\Big{)}$
$\displaystyle+2H\partial_{T}^{3}Q_{zz}^{(\rho)}\Big{(}\partial_{T}^{2}Q_{zz}^{(p)}-3\partial_{T}^{2}Q_{zz}^{(\rho)}\Big{)}+6H\partial_{T}^{3}Q_{xy}^{(\rho)}\Big{(}\partial_{T}^{2}Q_{xy}^{(p)}-3\partial_{T}^{2}Q_{xy}^{(\rho)}\Big{)}$
$\displaystyle+6H\partial_{T}^{3}Q_{xz}^{(\rho)}\Big{(}\partial_{T}^{2}Q_{xz}^{(p)}-3\partial_{T}^{2}Q_{xz}^{(\rho)}\Big{)}+6H\partial_{T}^{3}Q_{yz}^{(\rho)}\Big{(}\partial_{T}^{2}Q_{yz}^{(p)}-3\partial_{T}^{2}Q_{yz}^{(\rho)}\Big{)}$
$\displaystyle-H\partial_{T}^{3}Q_{xx}^{(\rho)}\Big{(}\partial_{T}^{2}Q_{yy}^{(p)}-3\partial_{T}^{2}Q_{yy}^{(\rho)}\Big{)}-H\partial_{T}^{3}Q_{yy}^{(\rho)}\Big{(}\partial_{T}^{2}Q_{xx}^{(p)}-3\partial_{T}^{2}Q_{xx}^{(\rho)}\Big{)}$
$\displaystyle-H\partial_{T}^{3}Q_{xx}^{(\rho)}\Big{(}\partial_{T}^{2}Q_{zz}^{(p)}-3\partial_{T}^{2}Q_{zz}^{(\rho)}\Big{)}-H\partial_{T}^{3}Q_{zz}^{(\rho)}\Big{(}\partial_{T}^{2}Q_{xx}^{(p)}-3\partial_{T}^{2}Q_{xx}^{(\rho)}\Big{)}$
$\displaystyle-H\partial_{T}^{3}Q_{yy}^{(\rho)}\Big{(}\partial_{T}^{2}Q_{zz}^{(p)}-3\partial_{T}^{2}Q_{zz}^{(\rho)}\Big{)}-H\partial_{T}^{3}Q_{zz}^{(\rho)}\Big{(}\partial_{T}^{2}Q_{yy}^{(p)}-3\partial_{T}^{2}Q_{yy}^{(\rho)}\Big{)}\Bigg{]}+\mathcal{O}(H^{2}).$
(73)

which again may be written in terms of traceless quadrupole moments via (23)
and the final expression is of the form:

$\displaystyle E_{T}=\tfrac{1}{45}\int
dT\sum_{i,j=1}^{3}\Bigg{[}\bigg{(}\frac{d^{3}q^{(\rho)}_{ij}}{dT^{3}}\bigg{)}^{2}+2H\bigg{(}\frac{d^{3}q^{(\rho)}_{ij}}{dT^{3}}\frac{d^{2}q^{(p)}_{ij}}{dT^{2}}\bigg{)}\Bigg{]}+\mathcal{O}(H^{2}).$
(74)

Therefore, the above energy flux of gravitational radiation across future
infinity $\mathcal{I}^{+}$ manifestly agrees with the other two formulas, (27)
and (36), computed on the cosmological horizon $\mathcal{H}$ via (DRL) and
(HV) approaches.

##  III Summary

We have investigated three different derivations of the generalizations of the
quadrupole formula. In (DRL) approach DDR&JL , it was calculated on
cosmological horizon $\mathcal{H}$ instead of the conformal boundary used in
radiation theory on the background of Minkowski spacetime and written
explicitly up to terms linear in $\sqrt{\Lambda}$. Its form allows to take a
limit as $\Lambda$ goes to zero simply by setting the Hubble parameter $H=0$
and the result recovers the famous Einstein’s quadrupole formula. Our goal was
two express the other to generalizations of the quadrupole formula by (HV) HV
and (ABK) ABK3  via the expansion in $\sqrt{\Lambda}$ and calculate the zeroth
and first order terms. Even though those two quadrupole formulas, (36) and
(52), look very similar, they come from conceptually different derivations and
it requires some tedious calculations to show that they do coincide, at least
up to terms linear in $\sqrt{\Lambda}$. Expression (36) is written in a tt-
projection and evaluated over the cosmological horizon, whereas (52) was
calculated in a TT-gauge at future infinity $\mathcal{I^{+}}$. The two
notions, tt-projection and TT-gauge, are generically distinct, therefore,
should not be used interchangeably. Even though, they do coincide at null
infinity $\mathcal{I}^{+}$ of asymptotically flat spacetimes and as recently
showed in HA  they also yield the same result for the energy flux computations
for binary system in de Sitter background HA , it is the TT-gauge that is the
correct approach to extract transverse-traceless part of rank-2 symmetric
tensor. Nevertheless, we expand the two expression for the radiated energy of
the compact source and put them in an analogous form as (27) found in DDR&JL .
Therefore, each of them written up to linear terms in $\sqrt{\Lambda}$, (48)
and (74), not only admits the correct limit for the vanishing cosmological
constant $\Lambda\rightarrow 0$, but also agree with the (27), as expected due
to the symplectic theory on the space of solutions to Einstein’s equations.
This shows, that all three approaches, (ABK), (HV) and (DRL), provide the same
generalization of the quadrupole formula up to terms linear in
$\sqrt{\Lambda}$ (or equivalently in Hubble parameter $H$).

Notice, however, that the terms linear in $\sqrt{\Lambda}$ are proportional to
the second derivatives of the traceless pressure quadrupole moment
$q^{(p)}_{ab}$, and in studying physical systems of interest (such as a binary
system) it is often set to zero. Therefore, the first correction to the
quadrupole formula is negligible and higher order terms are necessary to study
the impact of the expansion of the Universe on the radiated energy of the time
changing compact system. Higher order terms were taken into consideration in
BH , where the radiated power of the inspiring binary on circular orbit has
been calculated in the high-frequency limit (where the expansion of the
Universe is neglected during the orbital cycles of the system). Even though
the result showed that the standard Einstein quadrupole formula is sufficient
for studying binary systems in high-frequency approximation, one could go
beyond this approximation and probe the cosmological constant $\Lambda$ by
measuring radiated energy by in the form of gravitational waves due to the
binaries of supermassive black holes. The high-frequency approximation does
not hold for such systems, and the corrections due to the background curvature
could then become non-negligible. Recent results on the search of low-
frequency gravitational waves through radio pulsar timing nanograv  give hope
for new insight for the study of supermassive black hole binaries and
significance of cosmological constant in gravitational radiation theory for
expanding Universe.

Acknowledgements: JL was supported by 2021/43/B/ST2/02950 of the Polish
National Science Centre.

##  IV APPENDIX

Here, we present a more detailed calculation of the expansion of the
generalization of the quadrupole formula derived in ABK3 . Finding the
explicit expression for the electric part of the Weyl tensor
$\mathcal{E}_{ab}$ in terms of the metric perturbation $\chi_{ab}$ requires
calculating $\tilde{\chi}$ from

$\displaystyle\tilde{\chi}$
$\displaystyle=\eta\int^{\eta}_{-\infty}\frac{\mathring{D}^{a}\chi_{a}+\partial_{\eta^{\prime}}\chi}{\eta^{\prime}}d\eta^{\prime}.$
(75)

We start with

$\displaystyle\chi_{b}$
$\displaystyle=-\frac{4x^{a}\eta^{2}}{r^{3}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{ab}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}-\frac{4x^{a}\eta}{r^{2}{\eta_{\text{ret}}}}\mathbb{T}_{ab}(\eta_{\text{ret}})$
(76)

and calcualte its derivative

$\displaystyle D^{b}\chi_{b}$
$\displaystyle=-\frac{4\delta^{ab}\eta^{2}}{r^{3}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{ab}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}+\frac{12x^{a}x^{b}\eta^{2}}{r^{5}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{ab}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}-\frac{4\delta^{ab}\eta}{r^{2}{\eta_{\text{ret}}}}\mathbb{T}_{ab}(\eta_{\text{ret}})$
$\displaystyle+\frac{12x^{a}x^{b}\eta}{r^{4}{\eta_{\text{ret}}}}\mathbb{T}_{ab}(\eta_{\text{ret}})+\frac{4x^{a}x^{b}\eta}{r^{3}{\eta_{\text{ret}}}}\partial_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}}).$
(77)

Next, using the expression (56) we compute:

$\displaystyle\partial_{\eta}\chi$
$\displaystyle=\frac{4}{r}\partial_{\eta}\mathbb{T}(\eta_{\text{ret}})+4\frac{1}{\eta_{\text{ret}}}\partial_{\eta}\mathbb{T}(\eta_{\text{ret}})$
$\displaystyle=\frac{4\eta}{r\eta_{\text{ret}}}\partial_{\eta}\mathbb{T}(\eta_{\text{ret}}).$
(78)

Then, combining eqs. (IV) and (IV) yields:

$\displaystyle\frac{\mathring{D}^{a}\chi_{a}+\partial_{\eta^{\prime}}\chi}{\eta^{\prime}}$
$\displaystyle=-\frac{4\delta^{ab}\eta}{r^{3}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{ab}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}+\frac{12x^{a}x^{b}\eta}{r^{5}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{ab}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}-\frac{4\delta^{ab}}{r^{2}{\eta_{\text{ret}}}}\mathbb{T}_{ab}(\eta_{\text{ret}})$
$\displaystyle+\frac{12x^{a}x^{b}}{r^{4}{\eta_{\text{ret}}}}\mathbb{T}_{ab}(\eta_{\text{ret}})+\frac{4x^{a}x^{b}}{r^{3}{\eta_{\text{ret}}}}\partial_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}})+\frac{4}{r\eta_{\text{ret}}}\partial_{\eta}\mathbb{T}(\eta_{\text{ret}}).$
(79)

Taking the derivative $\mathring{D}_{d}$ of the above gives

$\displaystyle\mathring{D}_{d}\frac{\mathring{D}^{a}\chi_{a}+\partial_{\eta^{\prime}}\chi}{\eta^{\prime}}$
$\displaystyle=\frac{12x_{d}\eta}{r^{5}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}+\frac{12x_{d}}{r^{4}{\eta_{\text{ret}}}}\mathbb{T}(\eta_{\text{ret}})+\frac{4x_{d}}{r^{2}\eta^{2}_{\text{ret}}}\partial_{\eta}\mathbb{T}(\eta_{\text{ret}})-\frac{4x_{d}}{r^{2}\eta_{\text{ret}}}\partial^{2}_{\eta}\mathbb{T}(\eta_{\text{ret}})$
$\displaystyle+\frac{24\delta^{a}_{d}x^{b}\eta}{r^{5}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{ab}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}-\frac{60x^{a}x^{b}x_{d}\eta}{r^{7}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{ab}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}-\frac{60x^{a}x^{b}x_{d}}{r^{6}{\eta_{\text{ret}}}}\mathbb{T}_{ab}(\eta_{\text{ret}})$
$\displaystyle+\frac{24\delta^{a}_{d}x^{b}}{r^{4}{\eta_{\text{ret}}}}\mathbb{T}_{ab}(\eta_{\text{ret}})+\frac{8\delta_{d}^{a}x^{b}}{r^{3}{\eta_{\text{ret}}}}\partial_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}})+\frac{4(-6\eta+7r)x^{a}x^{b}x_{d}}{r^{5}{\eta^{2}_{\text{ret}}}}\partial_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}})$
$\displaystyle-\frac{4x^{a}x^{b}x_{d}}{r^{4}{\eta_{\text{ret}}}}\partial^{2}_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}}).$
(80)

Then, by computing yet another derivative of the above we find

$\displaystyle
D_{c}D_{d}\frac{\mathring{D}^{a}\chi_{a}+\partial_{\eta^{\prime}}\chi}{\eta^{\prime}}$
$\displaystyle=\frac{12\delta_{cd}\eta}{r^{5}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}-\frac{60x_{c}x_{d}\eta}{r^{7}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}-\frac{60x_{c}x_{d}}{r^{6}{\eta_{\text{ret}}}}\mathbb{T}(\eta_{\text{ret}})$
$\displaystyle+\frac{12\delta_{cd}}{r^{4}{\eta_{\text{ret}}}}\mathbb{T}(\eta_{\text{ret}})+\frac{4x_{c}x_{d}\eta(-3\eta+4r)}{r^{5}{\eta^{3}_{\text{ret}}}}\partial_{\eta}\mathbb{T}(\eta_{\text{ret}})+\frac{4\delta_{cd}}{r^{2}\eta^{2}_{\text{ret}}}\partial_{\eta}\mathbb{T}(\eta_{\text{ret}})$
$\displaystyle+\frac{4x_{c}x_{d}(2\eta-4r)}{r^{4}\eta^{2}_{\text{ret}}}\partial^{2}_{\eta}\mathbb{T}(\eta_{\text{ret}})-\frac{4\delta_{cd}}{r^{2}\eta_{\text{ret}}}\partial^{2}_{\eta}\mathbb{T}(\eta_{\text{ret}})+\frac{4x_{c}x_{d}}{r^{3}\eta_{\text{ret}}}\partial^{3}_{\eta}\mathbb{T}(\eta_{\text{ret}})$
$\displaystyle+\frac{24\eta}{r^{5}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{cd}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}-\frac{120x_{c}x^{b}\eta}{r^{7}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{bd}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}$
$\displaystyle-\frac{120x^{b}x_{d}\eta}{r^{7}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{cb}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}-\frac{60x^{a}x^{b}\delta_{cd}\eta}{r^{7}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{ab}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}$
$\displaystyle+\frac{420x^{a}x^{b}x_{c}x_{d}\eta}{r^{9}}\int^{\eta}_{-\infty}\frac{1}{\eta^{\prime
2}_{\text{ret}}}\mathbb{T}_{ab}(\eta^{\prime}_{\text{ret}})d\eta^{\prime}+\frac{420x^{a}x^{b}x_{c}x_{d}}{r^{8}{\eta_{\text{ret}}}}\mathbb{T}_{ab}(\eta_{\text{ret}})$
$\displaystyle-\frac{120x^{b}x_{d}}{r^{6}{\eta_{\text{ret}}}}\mathbb{T}_{cb}(\eta_{\text{ret}})-\frac{120x^{b}x_{c}}{r^{6}{\eta_{\text{ret}}}}\mathbb{T}_{bd}(\eta_{\text{ret}})-\frac{60x^{a}x^{b}\delta_{cd}}{r^{6}{\eta_{\text{ret}}}}\mathbb{T}_{ab}(\eta_{\text{ret}})$
$\displaystyle+\frac{4(45\eta^{2}-100\eta
r+57r^{2})x^{a}x^{b}x_{c}x_{d}}{r^{7}{\eta^{3}_{\text{ret}}}}\partial_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}})+\frac{24}{r^{4}{\eta_{\text{ret}}}}\mathbb{T}_{cd}(\eta_{\text{ret}})$
$\displaystyle+\frac{8}{r^{3}{\eta_{\text{ret}}}}\partial_{\eta}\mathbb{T}_{cd}(\eta_{\text{ret}})+\frac{8(-6\eta+7r)x^{b}x_{c}}{r^{5}{\eta^{2}_{\text{ret}}}}\partial_{\eta}\mathbb{T}_{bd}(\eta_{\text{ret}})$
$\displaystyle+\frac{8(-6\eta+7r)x^{b}x_{d}}{r^{5}{\eta^{2}_{\text{ret}}}}\partial_{\eta}\mathbb{T}_{cb}(\eta_{\text{ret}})+\frac{4(-6\eta+7r)x^{a}x^{b}\delta_{cd}}{r^{5}{\eta^{2}_{\text{ret}}}}\partial_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}})$
$\displaystyle+\frac{8(3\eta-4r)x^{a}x^{b}x_{c}x_{d}}{r^{6}{\eta^{2}_{\text{ret}}}}\partial^{2}_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}})-\frac{8x^{b}x_{d}}{r^{4}{\eta_{\text{ret}}}}\partial^{2}_{\eta}\mathbb{T}_{cb}(\eta_{\text{ret}})-\frac{8x^{b}x_{c}}{r^{4}{\eta_{\text{ret}}}}\partial^{2}_{\eta}\mathbb{T}_{bd}(\eta_{\text{ret}})$
$\displaystyle-\frac{4x^{a}x^{b}\delta_{cd}}{r^{4}{\eta_{\text{ret}}}}\partial^{2}_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}})+\frac{4x^{a}x^{b}x_{c}x_{d}}{r^{5}{\eta_{\text{ret}}}}\partial^{3}_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}}).$
(81)

Finally, we integrate the above expression, divide it by $\eta$ and find its
value at conformal infinity of de Sitter spacetime $\mathcal{I}^{+}$ by
setting $\eta=0$:

$\displaystyle D_{c}D_{d}\tilde{\chi}$
$\displaystyle=\eta\int^{\eta}_{-\infty}D_{c}D_{d}\frac{\mathring{D}^{a}\chi_{a}+\partial_{\eta^{\prime}}\chi}{\eta^{\prime}}d\eta^{\prime}$
$\displaystyle\hat{=}-\frac{4x_{c}x_{d}}{r^{4}\eta_{\text{ret}}}\partial^{2}_{\eta}\mathbb{T}(\eta_{\text{ret}})-\frac{4x^{a}x^{b}x_{c}x_{d}}{r^{6}}\partial^{2}_{\eta}\mathbb{T}_{ab}(\eta_{\text{ret}})+\mathcal{O}(H^{3}).$
(82)

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