# Complete gravitational-waveform amplitude modes for quasi-circular compact
binaries to the 3.5PN order

Quentin Henry<EMAIL_ADDRESS>Max Planck Institute for Gravitational
Physics
(Albert Einstein Institute), D-14476 Potsdam, Germany

###### Abstract

We complete the computation of the gravitational waveform amplitude for non-
spinning compact binaries to the third and a half post-Newtonian (PN) order in
the quasi-circular case. This is done by performing a spin-weighted spherical
harmonics decomposition of the amplitude. This computation is achieved using
the post-Newtonian-Multipolar-post-Minkowskian formalism. The amplitude modes
are written in a PN expanded way as well as a factorized form suitable for EOB
template building. Combining this work with previous ones, we provide in the
supplemental materials the expressions of the modes including all non-spinning
and spinning effects to the 3.5PN order for non-precessing quasi-circular
binaries.

###### pacs:

04.25.Nx, 04.25.dg, 04.30.-w, 97.80.-d, 97.60.Jd, 95.30.Sf

##  I Introduction

The gravitational wave (GW) detections by current detectors LIGO-Virgo-KAGRA
[1, 2, 3], as well as next generation detectors, such as LISA [4] and Einstein
Telescope [5] require precise modeling of the waveform in order to extract the
signal from the noise. Although only numerical relativity can provide accurate
waveforms close to the merger, several analytical methods produce theoretical
predictions of the inspiral phase. To model this phase, three main types of
expansion exist. First of all, the post-Newtonian (PN) one consists in
performing a Taylor expansion of the equations using the small parameter
$v^{2}/c^{2}$, where $v$ is the relative velocity of the two companions of a
binary system and $c$ the speed of light. In the standard nomenclature, a
$n$PN quantity is of order $\mathcal{O}(1/c^{2n})$. The second one is the
post-Minkowskian (PM) expansion, in which we consider the small parameter
$GM/rc^{2}$, where $G$ is Newton’s constant, $M$ the total mass of the system
and $r$ the separation. The last one consists in performing an expansion using
the small parameter is the mass ratio $q=m_{1}/m_{2}$. Different frameworks
using these types of approximations provide analytical solutions for physical
observables such as the radiated energy flux or the GW amplitude and phase.
These partial results overlap and can be combined to numerical relativity
notably using the effective-one-body (EOB) formalism [6, 7] which produces
accurate waveforms over the entire parameter space.

In the radiative sector, the energy flux as well as the GW phase are known to
the 3.5PN order beyond the Einstein quadrupole formula [8, 9] and are now
being pushed to 4PN [10, 11, 12, 13, 14, 15]. However, only partial results
are currently available for the amplitude of the gravitational waveform to the
3.5PN order. The amplitude can be computed in the form of a spin-weighted
spherical harmonics mode decomposition, parametrised as usual by two numbers
$(\ell,m)$.

The test-mass limit approximation provides extremely useful results in the
amplitude modes at the leading order in the mass ratio. In the parameter space
where both the PN and the test-mass limit approximations are valid, these
results should agree, which constitutes a very powerful check. The amplitude
modes in the test-mass limit are displayed in, e.g. [16, 17, 18].

For non-spinning binaries, the amplitude modes in the PN approximation have
been derived consistently to the 3PN order [19]. Regarding the 3.5PN accuracy,
the $(\ell,m)=(2,2)$ mode was published in [20], the (3,3) and (3,1) modes in
[21] and the (2,1) mode in [11]. Some other modes were derived in [22],
especially some for high $\ell$ at the relative 1PN order, matching the 3.5PN
accuracy in the waveform. These modes have been used for EOB template
building. It has been found that factorizing their expressions in several
blocks increases the convergence of the EOB model when compared to numerical
relativity [23]. Thus, the PN expanded modes at the third PN order were
factorized in, e.g. [24, 25] to give more accurate EOB templates. Although the
factorized form of the (2,2), (3,3) and (3,1) modes are known for the 3.5PN
accuracy in the waveform, the one of the (2,1) mode was never published.
Furthermore, it was found out recently that an error in a published version of
the (2,1) mode at the 2.5PN order propagated in the EOB models (see [26] for
justification). The PN expanded form of the (2,1) mode eventually got
corrected in the erratum of [19] but the mistake in the EOB models remained at
the time of publication of this work.

In order to prevent new propagation of mistakes and to avoid multiplying
references for different terms in the PN expanded modes as well as their
factorized version, this paper aims at completing the full gravitational
waveform to the 3.5PN order, including spin effects for a quasi-circular and
non-precessing motion. The spin effects in the waveform modes to this order
were derived in the recent work [26] and we now complete the missing non-
spinning coefficients. We also write the modes in a convenient way for EOB
template building. Due to the length of the expressions, the spin
contributions to the amplitude modes are not be displayed here but can be
found in [26]. However, the Supplemental Materials attached to this paper
contain both non-spinning and spinning coefficients to the 3.5PN order [27].

In the present paper, we use the post-Newtonian-Multipolar-post-Minkowskian
(PN-MPM) formalism; see [28] for a detailed review of this approach. This
method combines the PN and the PM expansions, so that we consider the small
parameters $v^{2}/c^{2}\sim GM/c^{2}r\ll 1$. We will qualitatively describe
this method in Sec. II.2.

The paper is organized as follows. In Sec. II, we define the spherical modes
and recall their relation with the radiative multipole moments. We also
explain qualitatively how to derive the radiative moments within the PN-MPM
scheme. We refer to previous works for technical details of this derivation.
In Sec. III, we give the results for the non-spinning terms in amplitude
modes, written in a conventional Taylor expansion form in Sec. III.1, and
factorized conveniently for EOB usage in Sec. III.2. Appendix A contains
generalized formulas to compute some particular hereditary effects. Appendix B
contains the different quantities appearing in the factorized modes. We also
provide our results for the full waveform modes including non-spinning and
spinning effects to the 3.5PN order as _Mathematica_ files in the Supplemental
Materials [27].

##  II Brief overview of the method

###  II.1 Spherical harmonics decomposition of the gravitational field

The transverse-traceless (TT) projection $h_{ij}^{\text{TT}}$ of the
gravitational field can be uniquely decomposed in terms of a set of symmetric
trace-free (STF) multipole moments $U_{L}$ and $V_{L}$, called radiative
multipole moments, as [29]111We use the signature $(-,+,+,+)$. Latin indices
stand for spatial coordinates, i.e. $i=1,2,3$ and the multi-index notation
$L=i_{1}\dots i_{\ell}$. The weighted symmetrization operator is noted by
parenthesis around indices.

$\displaystyle h_{ij}^{\text{TT}}$
$\displaystyle=\frac{4G}{c^{2}R}\,\mathcal{P}_{ijkl}(\bm{N})\sum^{+\infty}_{\ell=2}\frac{1}{c^{\ell}\ell!}\left\\{N_{L-2}\,U_{klL-2}(T_{R})-\frac{2\ell}{c(\ell+1)}\,N_{aL-2}\,\varepsilon_{ab(k}\,V_{l)bL-2}(T_{R})\right\\}+\mathcal{O}\left(\frac{1}{R^{2}}\right)\,,$
(1)

where $R$ is the distance between the source and the observer, $\bm{N}$ is the
direction of propagation of the GW and $T_{R}=T-R/c$ is the retarded time in
some radiative gauge in which $T_{R}$ is asymptotically null. The quantity
$\mathcal{P}_{ijkl}=\mathcal{P}_{i(k}\mathcal{P}_{l)j}-\frac{1}{2}\mathcal{P}_{ij}\mathcal{P}_{kl}$
is the TT projection operator, where
$\mathcal{P}_{ij}=\delta_{ij}-N_{i}N_{j}$. The polarization waveforms are
defined by

$\displaystyle h_{+}$
$\displaystyle=\frac{1}{2}(P_{i}P_{j}-Q_{i}Q_{j})h_{ij}^{\text{TT}},$ (2a)
$\displaystyle h_{\times}$
$\displaystyle=\frac{1}{2}(P_{i}Q_{j}+Q_{i}P_{j})h_{ij}^{\text{TT}},$ (2b)

where the vectors $(\bm{P},\bm{Q},\bm{N})$ form an orthonormal triad properly
defined in, e.g. Sec. II. A. of [26]. As usual, we decompose
$h_{+}-\mathrm{i}h_{\times}$ in a spin-weighted spherical harmonics basis of
weight -2 [30]

$h\equiv
h_{+}-\mathrm{i}h_{\times}=\sum_{l=0}^{\infty}\sum_{m=-\ell}^{\ell}h_{\ell
m}Y^{\ell m}_{-2}(\Theta,\Phi),$ (3)

where the two angles $(\Theta,\Phi)$ characterize the direction of propagation
$\bm{N}$. The amplitude modes $h_{\ell m}$ are then linked to the radiative
multipole moments through

$h_{\ell m}=-\frac{G}{\sqrt{2}Rc^{\ell+2}}\left(U^{\ell
m}-\frac{\mathrm{i}}{c}V^{\ell m}\right).$ (4)

In our convention, $U^{\ell m}$ and $V^{\ell m}$ are given by222Note that the
choice of definitions in Ref. [30, 31] on $h_{\ell m}$ differs from ours by a
global minus sign for each $\ell$ and $m$ due to a different definition of the
vector basis $(\bm{P},\bm{Q},\bm{N})$.

$\displaystyle U^{\ell m}$
$\displaystyle=\frac{4}{\ell!}\,\sqrt{\frac{(\ell+1)(\ell+2)}{2\ell(\ell-1)}}\,\alpha_{L}^{\ell
m}\,U_{L}\,,$ (5a) $\displaystyle V^{\ell m}$
$\displaystyle=-\frac{8}{\ell!}\,\sqrt{\frac{\ell(\ell+2)}{2(\ell+1)(\ell-1)}}\,\alpha_{L}^{\ell
m}\,V_{L}\,,$ (5b)

with $\alpha_{L}^{\ell
m}\equiv\int\mathrm{d}\Omega\,\hat{N}_{L}\,\overline{Y}^{\,\ell m}$ being
defined from the complex conjuguate of the ordinary spherical harmonics
$Y^{\ell m}$. Its explicit expression is displayed in Eq. (4.7) of Ref. [11].

As we can see from Eqs. (4)-(5), one needs to derive the radiative multipole
moments $U_{L}$ and $V_{L}$ to consistent order to obtain the amplitude modes
$h_{\ell m}$. The derivation of these moments are described in the following
section.

###  II.2 Computational method

In this section, we describe qualitatively how to derive the radiative
multipole moments required for the knowledge of the full waveform at the 3.5PN
order. We will not give the technical details of the intermediate computations
as they were broadly described in other papers cited below. A more detailed
overview of each step can be found in, e.g. Sec. III. of [26].

The PN-MPM formalism provides an efficient way to compute the radiative
multipole moments defined in (1), see Ref. [28] for a review. The first step
of this computation consists in deriving the equations of motion (EOM) of the
individual compact bodies. They are displayed to the 3.5PN order in [20] in
harmonic coordinates and have been derived more recently to the 4PN order in
[32] in the same gauge. In the case of quasi-circular orbits, one can extract
the orbital frequency of the system $\omega$ from the EOM which allows
defining the orbital phase through

$\phi=\int\mathrm{d}t\,\omega(t).$ (6)

To derive the phase to a given PN order, one needs to know both the conserved
energy and the radiated energy flux at the same order. As mentionned in the
introduction, this phase is known to the 3.5PN order. However, the PN
precision required on the radiative multipole moments to derive the radiated
flux, hence the phase, is lower than the one required to derive the amplitude
modes. In other words, one needs more PN information to derive the modes than
the energy flux at a given PN order.

In this project, we computed additional PN terms in the $U_{L}$ for
$\ell\in\\{4,\dots,9\\}$ and $V_{L}$ for $\ell\in\\{3,\dots,8\\}$. The mass
quadrupole, mass octupole and current quadrupole were already obtained in
previous works to consistent order [20, 21, 11].

For this project, we first computed the so-called source multipole moments.
They are defined by constructing the most general solution of Einstein’s
equations in vacuum in the form of a multipolar post-Minkowskian expansion
outside the system. They are then matched to the PN metric, valid in a spatial
zone near the system. By doing so, we find that the source multipole moments
are integrals over quantities that contain the stress-energy tensor and the PN
metric. The explicit expressions of the source moments are given in, e.g. Eqs.
(2.31) and (2.33) of [11]. The metric as well as the stress-energy tensor were
already known from previous works, see e.g. [10], in which these quantities
were derived for the 4PN accuracy of the source mass quadrupole.

In order to perform this matching between different representations of the
gravitational field, we must resort to a regularisation scheme. We used in
this paper the Hadamard partie finie regularisation. However, it was found out
that this regularisation starts to fail at high PN orders. In fact, only
dimensional regularisation has been proved to be satisfactory. It was shown in
previous works that for non-spinning binaries, these two regularisations start
to differ to the 3PN order [33, 9, 34, 35, 12, 13]. In this project, the
missing multipoles to complete the full non-spinning waveform at the 3.5PN
order were at most required to the 2.5PN order so that we could use the
Hadamard partie finie regularisation safely. The integration methods for this
regularisation scheme are provided in, e.g. [36].

This integration leads to the expression of the source multipole moments in an
arbitrary frame for general orbits. The next step consists in reducing the
expressions to the center of mass (CM) frame and impose a quasi-circular
motion. To do so, we use the CM position displayed in, e.g. [37] and neglect
eccentricities.

As a last step, we deduce the radiative moments from the source moments. Due
to the non-linearity of the gravitational field, they differ by some non-
linear interactions of different moments. We find two main types of
contributions: the so-called instantaneous interactions, which correspond to
at least quadratic products of source multipoles evaluated at retarded time;
and hereditary interactions which contain for example tail and memory
integrals over the past history of the source preceding the retarded time. The
general expressions of the interactions were derived entirely to the 3.5PN
order in [21]. By using these general formulas and the integration methods
presented in Sec. VII. of [19], we obtained all the radiative multipole
moments that were needed to compute the full waveform to the 3.5PN order. In
Appendix A, we give the generalization to higher PN order of some integrals
required for computing the memory interactions.

Finally, we inserted the values of these moments in Eqs. (4)-(5) to obtain the
GW modes displayed in the following section.

##  III Results

In this section, we display the results for the waveform modes written in two
different ways. In Sec. III.1, we give their expressions in the conventional
PN expanded way in the form of, e.g. [19]. In Sec. III.2, we write them in a
factorized form convenient for the EOB method.

###  III.1 Spin-weighted spherical modes

The amplitude modes defined in Eq. (4) can be written in terms of the phase
variable $\phi$ defined in (6). In order to simplify the explicit expressions
of the modes, we introduce the new phase variable

$\psi\equiv\phi-\frac{2G\mathcal{M}\omega}{c^{3}}\ln\left(\frac{\omega}{\omega_{0}}\right),$
(7)

where $\mathcal{M}$ is the ADM mass and where the scale constant $\omega_{0}$
is introduced in the definition of the tail integrals [19]. This redefinition
absorbs most of the logarithms of the orbital frequency in the oscillatory
part of the modes. The GW modes appearing in (3) read

$\displaystyle h_{\ell
m}=\frac{2GM\,\nu\,x}{R\,c^{2}}\,\sqrt{\frac{16\pi}{5}}\,\hat{H}_{\ell
m}\,e^{-\mathrm{i}m\psi}\,,$ (8)

where $M$ is the total mass of the system, $\nu=m_{1}m_{2}/M^{2}$ is the
symmetric mass ratio and where we introduced the convenient PN parameter

$x=\left(\frac{GM\omega}{c^{3}}\right)^{2/3}.$ (9)

The values of the $\hat{H}_{\ell m}$’s for the full waveform produced by
binary a binary system of non-spinning compact objects to the 3.5PN order are
given by

$\displaystyle\hat{H}_{22}$
$\displaystyle=1+x\left(-\frac{107}{42}+\frac{55\nu}{42}\right)+2\pi
x^{3/2}+x^{2}\left(-\frac{2173}{1512}-\frac{1069\nu}{216}+\frac{2047\nu^{2}}{1512}\right)$
$\displaystyle+x^{5/2}\left(-\frac{107\pi}{21}-24\mathrm{i}\nu+\frac{34\pi\nu}{21}\right)+x^{3}\bigg{(}\frac{27027409}{646800}-\frac{856\gamma_{E}}{105}+\frac{428\mathrm{i}\pi}{105}+\frac{2\pi^{2}}{3}$
$\displaystyle+\left(-\frac{278185}{33264}+\frac{41\pi^{2}}{96}\right)\nu-\frac{20261\nu^{2}}{2772}+\frac{114635\nu^{3}}{99792}-\frac{428}{105}\ln(16x)\bigg{)}$
$\displaystyle+x^{7/2}\left(-\frac{2173}{756}\pi+\left(-\frac{2495}{378}\pi+\frac{14333}{162}\mathrm{i}\right)\nu+\left(\frac{40}{27}\pi-\frac{4066}{945}\mathrm{i}\right)\nu^{2}\right)\,,$
(10a) $\displaystyle\hat{H}_{21}$
$\displaystyle=\frac{1}{3}\mathrm{i}\,\delta\bigg{[}x^{1/2}+x^{3/2}\left(-\frac{17}{28}+\frac{5\nu}{7}\right)+x^{2}\left(\pi+\mathrm{i}\left(-\frac{1}{2}-2\ln
2\right)\right)$
$\displaystyle+x^{5/2}\left(-\frac{43}{126}-\frac{509\nu}{126}+\frac{79\nu^{2}}{168}\right)+x^{3}\bigg{(}-\frac{17\pi}{28}+\frac{3\pi\nu}{14}$
$\displaystyle+\mathrm{i}\left(\frac{17}{56}+\nu\left(-\frac{353}{28}-\frac{3\ln
2}{7}\right)+\frac{17\ln 2}{14}\right)\bigg{)}$
$\displaystyle+x^{7/2}\biggl{(}\frac{15223771}{1455300}+\frac{\pi^{2}}{6}-\frac{214}{105}\gamma_{\text{E}}-\frac{107}{105}\ln(4x)-\ln
2-2(\ln 2)^{2}$
$\displaystyle+\nu\biggl{(}-\frac{102119}{2376}+\frac{205}{128}\pi^{2}\biggr{)}-\frac{4211}{8316}\nu^{2}+\frac{2263}{8316}\nu^{3}+\mathrm{i}\pi\biggl{(}\frac{109}{210}-2\ln
2\biggr{)}\biggr{)}\biggr{]}\,,$ (10b) $\displaystyle\hat{H}_{20}$
$\displaystyle=-\frac{5}{14\sqrt{6}}\left[1+x\left(-\frac{4075}{4032}+\frac{67\nu}{48}\right)\right]+\mathcal{O}\left(\frac{1}{c^{4}}\right)\,,$
(10c) $\displaystyle\hat{H}_{33}$
$\displaystyle=-\frac{3}{4}\mathrm{i}\sqrt{\frac{15}{14}}\,\delta\bigg{[}x^{1/2}+x^{3/2}(-4+2\nu)+x^{2}\left(3\pi+\mathrm{i}\left(-\frac{21}{5}+6\ln\left(3/2\right)\right)\right)$
$\displaystyle+x^{5/2}\left(\frac{123}{110}-\frac{1838\nu}{165}+\frac{887\nu^{2}}{330}\right)+x^{3}\bigg{(}-12\pi+\frac{9\pi\nu}{2}$
$\displaystyle+\mathrm{i}\left(\frac{84}{5}-24\ln\left(3/2\right)+\nu\left(-\frac{48103}{1215}+9\ln\left(3/2\right)\right)\right)\bigg{)}\bigg{]}$
$\displaystyle+x^{7/2}\biggl{(}\frac{19388147}{280280}+\frac{492}{35}\ln\left(3/2\right)-18\ln^{2}(3/2)-\frac{78}{7}\gamma_{\text{E}}+\frac{3}{2}\pi^{2}+6\mathrm{i}\pi\left(-\frac{41}{35}+3\ln(3/2)\right)$
$\displaystyle+\frac{\nu}{8}\left(-\frac{7055}{429}+\frac{41}{8}\pi^{2}\right)-\frac{318841}{17160}\nu^{2}+\frac{8237}{2860}\nu^{3}-\frac{39}{7}\ln(16x)\biggr{)}\bigg{]}$
(10d) $\displaystyle\hat{H}_{32}$
$\displaystyle=\frac{1}{3}\sqrt{\frac{5}{7}}\bigg{[}x(1-3\nu)+x^{2}\left(-\frac{193}{90}+\frac{145\nu}{18}-\frac{73\nu^{2}}{18}\right)+x^{5/2}\left(2\pi-6\pi\nu+\mathrm{i}\left(-3+\frac{66\nu}{5}\right)\right)$
$\displaystyle+x^{3}\left(-\frac{1451}{3960}-\frac{17387\nu}{3960}+\frac{5557\nu^{2}}{220}-\frac{5341\nu^{3}}{1320}\right)+x^{7/2}\left(\frac{193}{30}\mathrm{i}-\frac{193}{45}\pi\right.$
$\displaystyle\left.+\nu\left(-\frac{258929}{5400}\mathrm{i}+\frac{136}{9}\pi\right)+\nu^{2}\left(\frac{33751}{450}\mathrm{i}-\frac{46}{9}\pi\right)\right)\bigg{]}\,,$
(10e) $\displaystyle\hat{H}_{31}$
$\displaystyle=\frac{\mathrm{i}\,\delta}{12\sqrt{14}}\bigg{[}x^{1/2}+x^{3/2}\left(-\frac{8}{3}-\frac{2\nu}{3}\right)+x^{2}\left(\pi+\mathrm{i}\left(-\frac{7}{5}-2\ln
2\right)\right)$
$\displaystyle+x^{5/2}\left(\frac{607}{198}-\frac{136\nu}{99}-\frac{247\nu^{2}}{198}\right)+x^{3}\bigg{(}-\frac{8\pi}{3}-\frac{7\pi\nu}{6}$
$\displaystyle+\mathrm{i}\left(\frac{56}{15}+\frac{16\ln
2}{3}+\nu\left(-\frac{1}{15}+\frac{7\ln 2}{3}\right)\right)\bigg{)}\bigg{]}$
$\displaystyle+x^{7/2}\biggl{(}\frac{10753397}{1513512}-2\ln
2\left(\frac{212}{105}+\ln
2\right)-\frac{26}{21}\gamma_{\text{E}}+\frac{\pi^{2}}{6}-2\mathrm{i}\pi\left(\frac{41}{105}+\ln
2\right)$
$\displaystyle\qquad+\frac{\nu}{8}\bigg{(}-\frac{1738843}{19305}+\frac{41}{8}\pi^{2}\bigg{)}+\frac{327059}{30888}\nu^{2}-\frac{17525}{15444}\nu^{3}-\frac{13}{21}\ln
x\biggr{)}\bigg{]}\,,$ (10f) $\displaystyle\hat{H}_{30}$
$\displaystyle=-\frac{2}{5}\mathrm{i}\sqrt{\frac{6}{7}}\nu\left[x^{5/2}+x^{7/2}\left(-\frac{5017}{1296}-\frac{25}{108}\nu\right)\right]\,,$
(10g) $\displaystyle\hat{H}_{44}$
$\displaystyle=-\frac{8}{9}\sqrt{\frac{5}{7}}\bigg{[}x(1-3\nu)+x^{2}\left(-\frac{593}{110}+\frac{1273\nu}{66}-\frac{175\nu^{2}}{22}\right)$
$\displaystyle+x^{5/2}\left(4\pi-12\pi\nu+\mathrm{i}\left(-\frac{42}{5}+\nu\left(\frac{1193}{40}-24\ln
2\right)+8\ln 2\right)\right)$
$\displaystyle+x^{3}\left(\frac{1068671}{200200}-\frac{1088119\nu}{28600}+\frac{146879\nu^{2}}{2340}-\frac{226097\nu^{3}}{17160}\right)$
$\displaystyle+x^{7/2}\left(\frac{12453}{275}\mathrm{i}-\frac{1186}{55}\pi-\frac{2372}{55}\mathrm{i}\ln
2+\nu\left(-\frac{31525499}{140800}\mathrm{i}+\frac{2480}{33}\pi+\frac{4960}{33}\mathrm{i}\ln
2\right)\right.$
$\displaystyle\left.+\nu^{2}\left(\frac{4096237}{21120}\mathrm{i}-\frac{284}{11}\pi-\frac{568}{11}\mathrm{i}\ln
2\right)\right)\bigg{]}\,,$ (10h) $\displaystyle\hat{H}_{43}$
$\displaystyle=-\frac{9\mathrm{i}\,\delta}{4\sqrt{70}}\bigg{[}x^{3/2}(1-2\nu)+x^{5/2}\left(-\frac{39}{11}+\frac{1267\nu}{132}-\frac{131\nu^{2}}{33}\right)$
$\displaystyle+x^{3}\left(3\pi-6\pi\nu+\mathrm{i}\left(-\frac{32}{5}+\nu\left(\frac{16301}{810}-12\ln\left(3/2\right)\right)+6\ln\left(3/2\right)\right)\right)$
$\displaystyle+x^{7/2}\left(\frac{7206}{5005}-\frac{82869}{5720}\nu+\frac{104839}{3432}\nu^{2}-\frac{2987}{572}\nu^{3}\right)\bigg{]}\,,$
(10i) $\displaystyle\hat{H}_{42}$
$\displaystyle=\frac{1}{63}\sqrt{5}\bigg{[}x(1-3\nu)+x^{2}\left(-\frac{437}{110}+\frac{805\nu}{66}-\frac{19\nu^{2}}{22}\right)+x^{5/2}\bigg{(}2\pi-6\pi\nu$
$\displaystyle+\mathrm{i}\left(-\frac{21}{5}+\frac{84\nu}{5}\right)\bigg{)}+x^{3}\left(\frac{1038039}{200200}-\frac{606751\nu}{28600}+\frac{400453\nu^{2}}{25740}+\frac{25783\nu^{3}}{17160}\right)$
$\displaystyle+x^{7/2}\left(\frac{9177}{550}\mathrm{i}-\frac{437}{55}\pi+\nu\left(-\frac{83029}{880}\mathrm{i}+\frac{772}{33}\pi\right)+\nu^{2}\left(\frac{93081}{1100}\mathrm{i}+\frac{14}{11}\pi\right)\right)\bigg{]}\,,$
(10j) $\displaystyle\hat{H}_{41}$
$\displaystyle=\frac{\mathrm{i}\,\delta}{84\sqrt{10}}\bigg{[}x^{3/2}(1-2\nu)+x^{5/2}\left(-\frac{101}{33}+\frac{337\nu}{44}-\frac{83\nu^{2}}{33}\right)$
$\displaystyle+x^{3}\left(\pi-2\pi\nu+\mathrm{i}\left(-\frac{32}{15}-2\ln
2+\nu\left(\frac{1661}{30}+4\ln 2\right)\right)\right)$
$\displaystyle+x^{7/2}\left(\frac{42982}{15015}-\frac{513989}{51480}\nu+\frac{196957}{10296}\nu^{2}-\frac{1195}{572}\nu^{3}\right)\bigg{]}\,,$
(10k) $\displaystyle\hat{H}_{40}$
$\displaystyle=-\frac{1}{504\sqrt{2}}\left[1+x\left(-\frac{180101}{29568}+\frac{27227}{1056}\nu\right)\right]+\mathcal{O}\left(\frac{1}{c^{4}}\right)\,,$
(10l) $\displaystyle\hat{H}_{55}$
$\displaystyle=\frac{625\mathrm{i}\,\delta}{96\sqrt{66}}\bigg{[}x^{3/2}(1-2\nu)+x^{5/2}\left(-\frac{263}{39}+\frac{688\nu}{39}-\frac{256\nu^{2}}{39}\right)$
$\displaystyle+x^{3}\left(5\pi-10\pi\nu+\mathrm{i}\left(-\frac{181}{14}+\nu\left(\frac{105834}{3125}-20\ln\left(5/2\right)\right)+10\ln\left(5/2\right)\right)\right)$
$\displaystyle+x^{7/2}\left(\frac{9185}{819}-\frac{188765}{3276}\nu+\frac{54428}{819}\nu^{2}-\frac{10567}{819}\nu^{3}\right)\bigg{]}\,,$
(10m) $\displaystyle\hat{H}_{54}$
$\displaystyle=-\frac{32}{9\sqrt{165}}\bigg{[}x^{2}\left(1-5\nu+5\nu^{2}\right)+x^{3}\left(-\frac{4451}{910}+\frac{3619\nu}{130}-\frac{521\nu^{2}}{13}+\frac{339\nu^{3}}{26}\right)$
$\displaystyle+x^{7/2}\left(-\frac{52}{5}\mathrm{i}+4\pi+8\mathrm{i}\ln
2+\nu\left(\frac{3351011}{53760}\mathrm{i}-20\pi-40\mathrm{i}\ln
2\right)\right.$
$\displaystyle\left.+\nu^{2}\left(-\frac{10923}{128}\mathrm{i}+20\pi+40\mathrm{i}\ln
2\right)\right)\bigg{]}\,,$ (10n) $\displaystyle\hat{H}_{53}$
$\displaystyle=-\frac{9}{32}\mathrm{i}\sqrt{\frac{3}{110}}\,\delta\bigg{[}x^{3/2}(1-2\nu)+x^{5/2}\left(-\frac{69}{13}+\frac{464\nu}{39}-\frac{88\nu^{2}}{39}\right)$
$\displaystyle+x^{3}\left(3\pi-6\pi\nu+\mathrm{i}\left(-\frac{543}{70}+\nu\left(\frac{83702}{3645}-12\ln\left(3/2\right)\right)+6\ln\left(3/2\right)\right)\right)$
$\displaystyle+x^{7/2}\left(\frac{12463}{1365}-\frac{56969}{1820}\nu+\frac{2172}{91}\nu^{2}-\frac{365}{273}\nu^{3}\right)\bigg{]}\,,$
(10o) $\displaystyle\hat{H}_{52}$
$\displaystyle=\frac{2}{27\sqrt{55}}\bigg{[}x^{2}\left(1-5\nu+5\nu^{2}\right)+x^{3}\left(-\frac{3911}{910}+\frac{3079\nu}{130}-\frac{413\nu^{2}}{13}+\frac{231\nu^{3}}{26}\right)$
$\displaystyle+x^{7/2}\left(-\frac{26}{5}\mathrm{i}+2\pi+\nu\left(\frac{16237}{336}\mathrm{i}-10\pi\right)+\nu^{2}\left(-\frac{1861}{20}\mathrm{i}+10\pi\right)\right)\bigg{]}\,,$
(10p) $\displaystyle\hat{H}_{51}$
$\displaystyle=\frac{\mathrm{i}\,\delta}{288\sqrt{385}}\bigg{[}x^{3/2}(1-2\nu)+x^{5/2}\left(-\frac{179}{39}+\frac{352\nu}{39}-\frac{4\nu^{2}}{39}\right)$
$\displaystyle+x^{3}\left(\pi-2\pi\nu+\mathrm{i}\left(-\frac{181}{70}-2\ln
2+\nu\left(\frac{626}{5}+4\ln 2\right)\right)\right)$
$\displaystyle+x^{7/2}\left(\frac{5023}{585}-\frac{49447}{2340}\nu+\frac{68}{9}\nu^{2}+\frac{287}{117}\nu^{3}\right)\bigg{]}\,,$
(10q) $\displaystyle\hat{H}_{50}$
$\displaystyle=\frac{4117\mathrm{i}\nu}{7560\sqrt{462}}x^{7/2}\left[1-\frac{21588}{4117}\nu\right]\,,$
(10r) $\displaystyle\hat{H}_{66}$
$\displaystyle=\frac{54}{5\sqrt{143}}\bigg{[}x^{2}\left(1-5\nu+5\nu^{2}\right)+x^{3}\left(-\frac{113}{14}+\frac{91\nu}{2}-64\nu^{2}+\frac{39\nu^{3}}{2}\right)$
$\displaystyle+x^{7/2}\left(-\frac{249}{14}\mathrm{i}+6\pi+12\mathrm{i}\ln
3+\nu\left(\frac{21787499}{217728}\mathrm{i}-30\pi-60\mathrm{i}\ln
3\right)\right.$
$\displaystyle\left.+\nu^{2}\left(-\frac{323903}{2592}\mathrm{i}+30\pi+60\mathrm{i}\ln
3\right)\right)\bigg{]}\,,$ (10s) $\displaystyle\hat{H}_{65}$
$\displaystyle=\frac{3125\mathrm{i}\,\delta}{504\sqrt{429}}\bigg{[}x^{5/2}\left(1-4\nu+3\nu^{2}\right)+x^{7/2}\left(-\frac{149}{24}+\frac{349}{12}\nu-\frac{409}{12}\nu^{2}+\frac{29}{3}\nu^{3}\right)\bigg{]}\,,$
(10t) $\displaystyle\hat{H}_{64}$
$\displaystyle=-\frac{128}{495}\sqrt{\frac{2}{39}}\bigg{[}x^{2}\left(1-5\nu+5\nu^{2}\right)+x^{3}\left(-\frac{93}{14}+\frac{71\nu}{2}-44\nu^{2}+\frac{19\nu^{3}}{2}\right)$
$\displaystyle+x^{7/2}\left(-\frac{83}{7}\mathrm{i}+4\pi+8\mathrm{i}\ln
2+\nu\left(\frac{3432215}{49152}\mathrm{i}-20\pi-40\mathrm{i}\ln
2\right)\right.$
$\displaystyle\left.+\nu^{2}\left(-\frac{382365}{4096}\mathrm{i}+20\pi+40\mathrm{i}\ln
2\right)\right)\bigg{]}\,,$ (10u) $\displaystyle\hat{H}_{63}$
$\displaystyle=-\frac{81\mathrm{i}\,\delta}{616\sqrt{65}}\bigg{[}x^{5/2}\left(1-4\nu+3\nu^{2}\right)+x^{7/2}\left(-\frac{133}{24}+\frac{301}{12}\nu-\frac{329}{12}\nu^{2}+7\nu^{3}\right)\bigg{]}\,,$
(10v) $\displaystyle\hat{H}_{62}$
$\displaystyle=\frac{2}{297\sqrt{65}}\bigg{[}x^{2}\left(1-5\nu+5\nu^{2}\right)+x^{3}\left(-\frac{81}{14}+\frac{59\nu}{2}-32\nu^{2}+\frac{7\nu^{3}}{2}\right)$
$\displaystyle+x^{7/2}\left(-\frac{83}{14}\mathrm{i}+2\pi+\nu\left(\frac{799789}{13440}\mathrm{i}-10\pi\right)+\nu^{2}\left(-\frac{19193}{160}\mathrm{i}+10\pi\right)\right)\bigg{]}\,,$
(10w) $\displaystyle\hat{H}_{61}$
$\displaystyle=\frac{\mathrm{i}\,\delta}{8316\sqrt{26}}\bigg{[}x^{5/2}\left(1-4\nu+3\nu^{2}\right)+x^{7/2}\left(-\frac{125}{24}+\frac{277}{12}\nu-\frac{289}{12}\nu^{2}+\frac{17}{3}\nu^{3}\right)\bigg{]}\,,$
(10x) $\displaystyle\hat{H}_{60}$
$\displaystyle=\frac{4195x}{1419264\sqrt{273}}\left[1-\frac{3612}{839}\nu\right]+\mathcal{O}\left(\frac{1}{c^{4}}\right)\,,$
(10y) $\displaystyle\hat{H}_{77}$
$\displaystyle=-\frac{16807\mathrm{i}\,\delta}{1440}\sqrt{\frac{7}{858}}\bigg{[}x^{5/2}\left(1-4\nu+3\nu^{2}\right)+x^{7/2}\left(-\frac{319}{34}+\frac{2225}{51}\nu-\frac{2558}{51}\nu^{2}+\frac{230}{17}\nu^{3}\right)\bigg{]}\,,$
(10z) $\displaystyle\hat{H}_{76}$
$\displaystyle=\frac{81}{35}\sqrt{\frac{3}{143}}x^{3}\bigg{[}1-7\nu+14\nu^{2}-7\nu^{3}\bigg{]}\,,$
(10aa) $\displaystyle\hat{H}_{75}$
$\displaystyle=\frac{15625\mathrm{i}\,\delta}{26208\sqrt{66}}\bigg{[}x^{5/2}\left(1-4\nu+3\nu^{2}\right)+x^{7/2}\left(-\frac{271}{34}+\frac{1793}{51}\nu-\frac{1838}{51}\nu^{2}+\frac{134}{17}\nu^{3}\right)\bigg{]}\,,$
(10ab) $\displaystyle\hat{H}_{74}$
$\displaystyle=-\frac{128x^{3}}{1365}\sqrt{\frac{2}{33}}\bigg{[}1-7\nu+14\nu^{2}-7\nu^{3}\bigg{]}\,,$
(10ac) $\displaystyle\hat{H}_{73}$
$\displaystyle=-\frac{243\mathrm{i}\,\delta}{160160}\sqrt{\frac{3}{2}}\bigg{[}x^{5/2}\left(1-4\nu+3\nu^{2}\right)+x^{7/2}\left(-\frac{239}{34}+\frac{1505}{51}\nu-\frac{1358}{51}\nu^{2}+\frac{70}{17}\nu^{3}\right)\bigg{]}\,,$
(10ad) $\displaystyle\hat{H}_{72}$
$\displaystyle=\frac{x^{3}}{3003\sqrt{3}}\bigg{[}1-7\nu+14\nu^{2}-7\nu^{3}\bigg{]}\,,$
(10ae) $\displaystyle\hat{H}_{71}$
$\displaystyle=\frac{\mathrm{i}\,\delta}{864864\sqrt{2}}\bigg{[}x^{5/2}\left(1-4\nu+3\nu^{2}\right)+x^{7/2}\left(-\frac{223}{34}+\frac{1361}{51}\nu-\frac{1118}{51}\nu^{2}+\frac{38}{17}\nu^{3}\right)\bigg{]}\,,$
(10af) $\displaystyle\hat{H}_{70}$
$\displaystyle=\mathcal{O}\left(\frac{1}{c^{8}}\right)\,,$ (10ag)
$\displaystyle\hat{H}_{88}$
$\displaystyle=-\frac{16384}{63}\sqrt{\frac{2}{85085}}x^{3}\bigg{[}1-7\nu+14\nu^{2}-7\nu^{3}\bigg{]}\,,$
(10ah) $\displaystyle\hat{H}_{87}$
$\displaystyle=-\frac{117649\,\mathrm{i}\,\delta}{5184}\sqrt{\frac{7}{24310}}x^{7/2}\bigg{[}1-6\nu+10\nu^{2}-4\nu^{3}\bigg{]}\,,$
(10ai) $\displaystyle\hat{H}_{86}$
$\displaystyle=\frac{243}{35}\sqrt{\frac{3}{17017}}x^{3}\bigg{[}1-7\nu+14\nu^{2}-7\nu^{3}\bigg{]}\,,$
(10aj) $\displaystyle\hat{H}_{85}$
$\displaystyle=\frac{78125\,\mathrm{i}\,\delta}{36288\sqrt{4862}}x^{7/2}\bigg{[}1-6\nu+10\nu^{2}-4\nu^{3}\bigg{]}\,,$
(10ak) $\displaystyle\hat{H}_{84}$
$\displaystyle=-\frac{128}{4095}\sqrt{\frac{2}{187}}x^{3}\bigg{[}1-7\nu+14\nu^{2}-7\nu^{3}\bigg{]}\,,$
(10al) $\displaystyle\hat{H}_{83}$
$\displaystyle=-\frac{81\,\mathrm{i}\,\delta}{5824}\sqrt{\frac{3}{1870}}x^{7/2}\bigg{[}1-6\nu+10\nu^{2}-4\nu^{3}\bigg{]}\,,$
(10am) $\displaystyle\hat{H}_{82}$
$\displaystyle=\frac{x^{3}}{9009\sqrt{85}}\bigg{[}1-7\nu+14\nu^{2}-7\nu^{3}\bigg{]}\,,$
(10an) $\displaystyle\hat{H}_{81}$
$\displaystyle=\frac{\mathrm{i}\,\delta}{741312\sqrt{238}}x^{7/2}\bigg{[}1-6\nu+10\nu^{2}-4\nu^{3}\bigg{]}\,,$
(10ao) $\displaystyle\hat{H}_{80}$
$\displaystyle=\mathcal{O}\left(\frac{1}{c^{4}}\right)\,,$ (10ap)
$\displaystyle\hat{H}_{99}$
$\displaystyle=\frac{1594323\,\mathrm{i}\,\delta}{7168\sqrt{20995}}x^{7/2}\bigg{[}1-6\nu+10\nu^{2}-4\nu^{3}\bigg{]}\,,$
(10aq) $\displaystyle\hat{H}_{97}$
$\displaystyle=-\frac{5764801\,\mathrm{i}\,\delta}{1410048\sqrt{1235}}x^{7/2}\bigg{[}1-6\nu+10\nu^{2}-4\nu^{3}\bigg{]}\,,$
(10ar) $\displaystyle\hat{H}_{95}$
$\displaystyle=\frac{390625\,\mathrm{i}\,\delta}{4935168\sqrt{247}}x^{7/2}\bigg{[}1-6\nu+10\nu^{2}-4\nu^{3}\bigg{]}\,,$
(10as) $\displaystyle\hat{H}_{93}$
$\displaystyle=-\frac{81\,\mathrm{i}\,\delta}{113152}\sqrt{\frac{3}{665}}x^{7/2}\bigg{[}1-6\nu+10\nu^{2}-4\nu^{3}\bigg{]}\,,$
(10at) $\displaystyle\hat{H}_{91}$
$\displaystyle=\frac{\mathrm{i}\,\delta}{9165312\sqrt{2090}}x^{7/2}\bigg{[}1-6\nu+10\nu^{2}-4\nu^{3}\bigg{]}\,.$
(10au)

The modes written without a remainder are all known up to order
$\mathcal{O}(1/c^{7})$. Note that some modes for $(\ell,0)$ are written with a
lower order remainder. As discussed in [26], the contributions to these modes
only arise from the memory integrals in the mass-type radiative moments.
Although the PN-MPM does not allow to compute them to high orders, they have
been derived to the 3PN order beyond their leading contribution in [22]. Other
modes not appearing in (10) are of order $\mathcal{O}(1/c^{8})$ and are thus
neglected here. Finally, the modes for negative $m$ can be recovered using
$\hat{H}_{\ell,-m}=(-1)^{\ell}\hat{H}_{\ell m}^{*}$ where the star notation
refers to the complex conjugate operation.

The modes $(2,2)$, $(2,1)$, $(3,3)$ and $(3,1)$ were already known for the
3.5PN waveform and derived in [20, 21, 11]. We recover these published
results. Furthermore, the computation has been pushed 0.5PN higher for the
other modes which completes the full waveform at the 3.5PN order. The overlaps
agree with the existing literature [30, 19], and notably with the modes
derived for a test particle around a Schwarzschild black hole [16, 17, 18, 31,
25].

###  III.2 Effective-One-Body factorized modes

In this section, we recall the conventions and definitions of the resummed
form of the modes convenient for EOB application. Following Refs. [38, 23, 31,
24], we rewrite the PN-expanded waveform modes of Eqs. (8)–(10) in a
factorized, resummed form as333In this section, we pose $c=G=1$.

$h_{\ell m}^{\text{F}}=h_{\ell
m}^{(N,\epsilon_{p})}\hat{S}_{\text{eff}}^{(\epsilon_{p})}T_{\ell
m}e^{\mathrm{i}\delta_{\ell m}}f_{\ell m}\,,$ (11)

where $\epsilon_{p}$ is the parity of $\ell+m$: $\epsilon_{p}=0$ if $\ell+m$
is even, and $\epsilon_{p}=1$ if $\ell+m$ is odd. We see that the modes are
factorized in five blocks. The first one, $h_{\ell m}^{(N,\epsilon_{p})}$ is
the leading PN order contribution. It is known analytically for each $\ell$
and $m$ [29, 30].

The second factor is the effective source term $\hat{S}_{\text{eff}}$ which,
depending on $\epsilon_{p}$, is either the effective energy $E_{\text{eff}}$
or the orbital angular momentum $p_{\phi}$,

$\hat{S}_{\text{eff}}=\left\\{\begin{array}[]{ll}\frac{E_{\text{eff}}(v)}{\mu},&\quad\ell+m\text{
even}\\\ v\,\frac{p_{\phi}(v)}{\mu M},&\quad\ell+m\text{
odd}\end{array}\right.,$ (12)

where $\mu\equiv m_{1}m_{2}/M$ is the reduced mass,
$v\equiv\left(M\omega\right)^{1/3}=\sqrt{x}$ and $E_{\text{eff}}$ is related
to the total energy $E$ via the EOB energy map
$E=M\sqrt{1+2\nu\left(E_{\text{eff}}/\mu-1\right)}$.

Similarly to the phase redefinition (7), the factor $T_{\ell m}$ allows to
absorb the “leading logarithms” induced by tail effects [39, 40, 41]. It is
given by

$T_{\ell
m}=\frac{\Gamma\left(\ell+1-2\mathrm{i}\hat{k}\right)}{\Gamma(\ell+1)}e^{\pi\hat{k}}e^{2\mathrm{i}\hat{k}\ln(2m\omega
r_{0})},$ (13)

where $\Gamma(...)$ is the Euler gamma function, $\hat{k}\equiv m\omega E$ and
the constant $r_{0}$, related to $\omega_{0}$, takes the value $2M/\sqrt{e}$.

The remaining part of the factorized modes is expressed as an amplitude
$f_{\ell m}$ and a phase $\delta_{\ell m}$, which are computed such that the
expansion of $h_{\ell m}^{\text{F}}$ agrees with the PN-expanded modes in Eq.
(8). We can further resum the amplitude term by introducing $\rho_{\ell
m}=(f_{\ell m})^{1/\ell}$ which improves the agreement with numerical-
relativity waveforms [24, 31]. For spinning binaries, the non-spinning and
spin contributions are separated for the odd $m$ modes, such that

$\displaystyle f_{\ell m}=\left\\{\begin{array}[]{ll}\rho_{\ell
m}^{\ell},&\quad m\text{ even}\\\ (\rho_{\ell m}^{\text{NS}})^{\ell}+f_{\ell
m}^{\text{S}},&\quad m\text{ odd}\end{array}\right.,$ (16)

where $\rho_{\ell m}^{\text{NS}}$ is the non-spinning part of $\rho_{\ell m}$,
while $f_{\ell m}^{\text{S}}$ is the spinning part of $f_{\ell m}$.

The explicit results for $\rho_{\ell m}$, $f_{\ell m}$ and $\delta_{\ell m}$
are displayed in Appendix B. The energy $E$ in $\delta_{\ell m}$ is replaced
by the Hamiltonian in EOB waveform models.

As discussed in [26], we remind that an error in the literature [24, 25] has
been found in the $\mathcal{O}(\nu v^{5})$ term of $\delta_{21}$. The
coefficient -493/42 in these papers, coming from the radiation reaction
contribution, should read -25/2, as we see in Eq. (29b). We also provide new
PN terms: although the (2,1) PN expanded mode was published in [11], its
factorized form was not. We finally obtain new coefficients in the (3,2) and
$\ell\geq 4$ modes. These expressions are available in a _Mathematica_ file in
the Supplemental Materials [27] including spinning effects.

##  IV Summary

In this paper, we computed the full gravitational waveform for non-spinning
binaries in quasi-circular motion to the 3.5PN order. To do so, we used the
PN-MPM formalism to derive the radiative multipole moments. All the
computational methods were already developed in previous works (except for the
generalization to next-to-leading order of some integration formulas,
displayed in Appendix A, in order to compute the memory terms).

The waveform modes have been written in the conventional PN expanded way as
well as a factorized form convenient for EOB template building. We already
derived in [26] all spin effects in the full waveform to the 3.5PN order in
the quasi-circular and non-precessing case. Combining these results, we obtain
the waveform modes to the 3.5PN order including non-spinning and spinning
effects.

We recall that in [26], we found some mistakes in the literature, notably some
spin coefficients $\mathcal{O}(v^{6}\chi^{2}\nu^{2})$ in the (2,1) mode. We
also corrected a non-spinning term in $\delta_{21}$. In the present work, we
agree fully on the overlap with the literature, except for this discrepancy in
$\delta_{21}$.

We provide in the Supplemental Material [27] the full expressions of the modes
for non-spinning and spinning effects. The notations and details regarding the
spinning part of the modes are given in [26].

## Acknowledgments

The author is extremely grateful to grateful to Mohammed Khalil and François
Larrouturou who double checked some results. The author is also grateful to
Luc Blanchet and Guillaume Faye for interesting discussions.

##  Appendix A Generalization of formulas to compute memory terms

The memory integrals are of the form

$\int_{-\infty}^{T_{R}}\mathrm{d}\tau A_{L}(\tau)B_{K}(\tau),$ (17)

where $A_{L}$ and $B_{L}$ are time derivatives of mass-type or current-type
STF multipole moments. To compute them, one needs to know the value of
integrals of the type

$\int_{-\infty}^{T_{R}}\mathrm{d}\tau\frac{e^{in\phi(\tau)}}{r^{p}(\tau)},$
(18)

where we recall that $\phi$ is the orbital phase, $r$ is the separation, $n$
is an integer and $p$ is an integer or half-integer. The memory integrals have
been derived in the 3PN full non-spinning waveform paper [19]. At this order,
only the leading PN order is required. However, for the 3.5PN waveform, we
need to generalize the formula to the next-to-leading order (NLO). To do so,
we introduce the adiabatic parameter evaluated at retarded time $T_{R}$

$\xi(T_{R})\equiv\frac{1}{(T_{c}-T_{R})\,\omega(T_{R})},$ (19)

where $T_{c}$ is the coalescence time. This quantity is a small parameter of
order $\mathcal{O}(c^{-5})$. At NLO, the adiabatic parameter reads

$\xi(T_{R})=\frac{256}{5}\nu
x^{5/2}(T_{R})\bigl{[}1+\kappa\,x(T_{R})+\mathcal{O}(x^{2})\bigr{]},$ (20)

where $\kappa=-\tfrac{743}{252}-\tfrac{11}{3}\nu$. Next, we perform the
following change of variable

$y\equiv\frac{T_{R}-\tau}{T_{c}-T_{R}},$ (21)

and we can express the phase and the separation in terms of $y$ at the NLO

$\displaystyle r(\tau)$
$\displaystyle=r(T_{R})(1+y)^{1/4}\left[1+\eta\,x(T_{R})\left((1+y)^{-1/4}-1\right)\right],$
(22a) $\displaystyle\phi(\tau)$
$\displaystyle=\phi(T_{R})-\frac{8}{5\xi(T_{R})}\bigg{\\{}\left[(1+y)^{5/8}-1\right]$
(22b)
$\displaystyle\qquad\qquad\qquad\qquad\qquad+x(T_{R})\Big{[}\zeta\left[(1+y)^{5/8}-1\right]+\lambda\left[(1+y)^{3/8}-1\right]\Big{]}\bigg{\\}},$

where $\eta=-\tfrac{1751}{1008}-\tfrac{7}{12}\nu$,
$\zeta=-\tfrac{743}{672}-\tfrac{11}{8}\nu$ and
$\lambda=\tfrac{3715}{2016}+\tfrac{55}{24}\nu$. After inserting these
expressions in (18), performing the new change of variable
$z=\tfrac{8}{5}[(1+y)^{5/8}-1]$ and truncating the integrand to the NLO, we
find

$\int_{-\infty}^{T_{R}}\mathrm{d}\tau\frac{e^{\mathrm{i}n\phi(\tau)}}{r^{p}(\tau)}=\frac{e^{\mathrm{i}n\phi(T_{R})}}{r^{p}(T_{R})\,\omega(T_{R})\,\xi(T_{R})}\int_{0}^{\infty}\mathrm{d}z\frac{e^{-\frac{\mathrm{i}n}{\xi}z}}{\left(1+\frac{5z}{8}\right)^{\frac{2p-3}{5}}}\left[1+x(T_{R})f(z)+\mathcal{O}(x^{2})\right],$
(23)

where $f$ is a function of $z$, straightforward to compute, but lengthy to
display. Finally, by performing two integrations by part and using the fact
that $\xi=\mathcal{O}(c^{-5})$, we find for $n\neq 0$

$\int_{-\infty}^{T_{R}}\mathrm{d}\tau\frac{e^{\mathrm{i}n\phi(\tau)}}{r^{p}(\tau)}=\frac{e^{\mathrm{i}n\phi(T_{R})}}{\mathrm{i}\,n\,r^{p}(T_{R})\,\omega(T_{R})}\biggl{[}1-x(T_{R})\underbrace{\left(\zeta+\frac{3}{5}\lambda\right)}_{=0}+\mathcal{O}(x^{2})\biggr{]}.$
(24)

Interestingly, the combination $\zeta+\frac{3}{5}\lambda$ vanishes, which
implies that we do not need to take into account the 1PN correction in Eqs.
(22). Thus, the 1PN correction only comes from the one of $r^{p}(T_{R})$ which
leads to the final formula

$\int_{-\infty}^{T_{R}}\mathrm{d}\tau\frac{e^{\mathrm{i}n\phi(\tau)}}{r^{p}(\tau)}=\frac{c^{2p-3}}{(GM)^{p-1}}\frac{e^{\mathrm{i}n\phi(T_{R})}}{\mathrm{i}\,n}x^{p-3/2}\left[1+p\left(1-\frac{\nu}{3}\right)x+\mathcal{O}(x^{2})\right].$
(25)

This formula is valid for $n\neq 0$, but when computing the memory integrals,
we also encounter integrals with $n=0$. The same kind of reasoning allows to
compute

$\int_{-\infty}^{T_{R}}\frac{\mathrm{d}\tau}{r^{p}(\tau)}=\frac{5}{64\nu}\frac{c^{2p-3}}{(GM)^{p-1}}\frac{x^{p-4}}{p-4}\left\\{1+\left[-\kappa+p\left(\frac{\eta}{p-3}+1-\frac{\nu}{3}\right)\right]x+\mathcal{O}(x^{2})\right\\}.$
(26)

Note that in this case, the 1PN correction of $r^{p}(\tau)$ plays a role in
the result. In our computations, the case $p=3$ and $p=4$ do not appear. With
(25) and (26), we have everything in hand to compute the memory integrals
required for the full 3.5PN waveform.

##  Appendix B Explicit quantities of the factorized modes

In this Appendix, we write the explicit expressions for the factorized modes
(see Sec. III.2)

$\displaystyle f_{22}$
$\displaystyle=1+v^{2}\left(-\frac{43}{21}+\frac{55}{42}\nu\right)+v^{4}\left(-\frac{536}{189}-\frac{6745}{1512}\nu+\frac{2047}{1512}\nu^{2}\right)$
$\displaystyle+v^{6}\left(\frac{21428357}{727650}-\frac{856}{105}\left(\gamma_{E}+\ln
4v\right)+\left(-\frac{34625}{3696}+\frac{41}{96}\pi^{2}\right)\nu-\frac{227875}{33264}\nu^{2}+\frac{114635}{99792}\nu^{3}\right)\,,$
(27a) $\displaystyle f_{21}$
$\displaystyle=1+v^{2}\left(-\frac{59}{28}+\frac{23}{42}\nu\right)+v^{4}\left(-\frac{5}{9}-\frac{269}{126}\nu+\frac{85}{252}\nu^{2}\right)$
$\displaystyle+v^{6}\left(\frac{88404893}{11642400}-\frac{214}{105}\left(\gamma_{E}+\ln
2v\right)+\left(\frac{86699}{66528}-\frac{41}{384}\pi^{2}\right)\nu-\frac{37241}{66528}\nu^{2}+\frac{9365}{49896}\nu^{3}\right)\,,$
(27b) $\displaystyle f_{33}$
$\displaystyle=1+v^{2}\left(-\frac{7}{2}+2\nu\right)+v^{4}\left(-\frac{443}{440}-\frac{3401}{330}\nu+\frac{887}{330}\nu^{2}\right)$
$\displaystyle+v^{6}\left(\frac{147471561}{2802800}-\frac{78}{7}\left(\gamma_{E}+\ln
6v\right)+\left(-\frac{17161}{2860}+\frac{41}{64}\pi^{2}\right)\nu-\frac{27409}{1560}\nu^{2}+\frac{8237}{2860}\nu^{3}\right)\,,$
(27c) $\displaystyle f_{32}$
$\displaystyle=1+\frac{v^{2}}{1-3\nu}\left(-\frac{164}{45}+\frac{223}{18}\nu-\frac{32}{9}\nu^{2}\right)$
$\displaystyle+\frac{v^{4}}{1-3\nu}\left(\frac{854}{495}-\frac{23443}{2376}\nu+\frac{31721}{1485}\nu^{2}-\frac{4943}{1485}\nu^{3}\right)\,,$
(27d) $\displaystyle f_{31}$
$\displaystyle=1+v^{2}\left(-\frac{13}{6}-\frac{2}{3}\nu\right)+v^{4}\left(\frac{1273}{792}-\frac{371}{198}\nu-\frac{247}{198}\nu^{2}\right)$
$\displaystyle+v^{6}\left(\frac{400427563}{75675600}-\frac{26}{21}\left(\gamma_{E}+\ln
2v\right)+\left(-\frac{788399}{77220}+\frac{41}{64}\pi^{2}\right)\nu+\frac{311225}{30888}\nu^{2}-\frac{17525}{15444}\nu^{3}\right)\,,$
(27e) $\displaystyle f_{44}$
$\displaystyle=1+\frac{v^{2}}{1-3\nu}\left(-\frac{269}{55}+\frac{587}{33}\nu-\frac{175}{22}\nu^{2}\right)$
$\displaystyle+\frac{v^{4}}{1-3\nu}\left(\frac{63002}{25025}-\frac{9163}{325}\nu+\frac{381541}{6435}\nu^{2}-\frac{226097}{17160}\nu^{3}\right)\,,$
(27f) $\displaystyle f_{43}$
$\displaystyle=1+\frac{v^{2}}{1-2\nu}\left(-\frac{111}{22}+\frac{547}{44}\nu-\frac{40}{11}\nu^{2}\right)$
$\displaystyle+\frac{v^{4}}{1-2\nu}\left(\frac{225543}{40040}-\frac{132529}{5720}\nu+\frac{100003}{3432}\nu^{2}-\frac{3889}{858}\nu^{3}\right)\,,$
(27g) $\displaystyle f_{42}$
$\displaystyle=1+\frac{v^{2}}{1-3\nu}\left(-\frac{191}{55}+\frac{353}{33}\nu-\frac{19}{22}\nu^{2}\right)$
$\displaystyle+\frac{v^{4}}{1-3\nu}\left(\frac{76918}{25025}-\frac{53297}{3575}\nu+\frac{100552}{6435}\nu^{2}+\frac{25783}{17160}\nu^{3}\right)\,,$
(27h) $\displaystyle f_{41}$
$\displaystyle=1+\frac{v^{2}}{1-2\nu}\left(-\frac{301}{66}+\frac{1385}{132}\nu-\frac{24}{11}\nu^{2}\right)$
$\displaystyle+\frac{v^{4}}{1-2\nu}\left(\frac{760181}{120120}-\frac{815329}{51480}\nu+\frac{163313}{10296}\nu^{2}-\frac{1409}{858}\nu^{3}\right)\,,$
(27i) $\displaystyle f_{55}$
$\displaystyle=1+\frac{v^{2}}{1-2\nu}\left(-\frac{487}{78}+\frac{649}{39}\nu-\frac{256}{39}\nu^{2}\right)$
$\displaystyle+\frac{v^{4}}{1-2\nu}\left(\frac{50569}{6552}-\frac{39899}{819}\nu+\frac{4001}{63}\nu^{2}-\frac{10567}{819}\nu^{3}\right)\,,$
(27j) $\displaystyle f_{54}$
$\displaystyle=1+\frac{v^{2}}{1-5\nu+5\nu^{2}}\left(-\frac{2908}{455}+\frac{13717}{390}\nu-\frac{1823}{39}\nu^{2}+\frac{476}{39}\nu^{3}\right)\,,$
(27k) $\displaystyle f_{53}$
$\displaystyle=1+\frac{v^{2}}{1-2\nu}\left(-\frac{125}{26}+\frac{425}{39}\nu-\frac{88}{39}\nu^{2}\right)$
$\displaystyle+\frac{v^{4}}{1-2\nu}\left(\frac{69359}{10920}-\frac{34493}{1365}\nu+\frac{6299}{273}\nu^{2}-\frac{365}{273}\nu^{3}\right)\,,$
(27l) $\displaystyle f_{52}$
$\displaystyle=1+\frac{v^{2}}{1-5\nu+5\nu^{2}}\left(-\frac{2638}{455}+\frac{12097}{390}\nu-\frac{1499}{39}\nu^{2}+\frac{314}{39}\nu^{3}\right)\,,$
(27m) $\displaystyle f_{51}$
$\displaystyle=1+\frac{v^{2}}{1-2\nu}\left(-\frac{319}{78}+\frac{313}{39}\nu-\frac{4}{39}\nu^{2}\right)$
$\displaystyle+\frac{v^{4}}{1-2\nu}\left(\frac{28859}{4680}-\frac{9673}{585}\nu+\frac{917}{117}\nu^{2}+\frac{287}{117}\nu^{3}\right)\,,$
(27n) $\displaystyle f_{66}$
$\displaystyle=1+\frac{v^{2}}{1-5\nu+5\nu^{2}}\left(-\frac{53}{7}+43\nu-\frac{123}{2}\nu^{2}+\frac{39}{2}\nu^{3}\right)\,,$
(27o) $\displaystyle f_{65}$
$\displaystyle=1+\frac{v^{2}}{1-4\nu+3\nu^{2}}\left(-\frac{185}{24}+\frac{419}{12}\nu-\frac{455}{12}\nu^{2}+\frac{55}{6}\nu^{3}\right)\,,$
(27p) $\displaystyle f_{64}$
$\displaystyle=1+\frac{v^{2}}{1-5\nu+5\nu^{2}}\left(-\frac{43}{7}+33\nu-\frac{83}{2}\nu^{2}+\frac{19}{2}\nu^{3}\right)\,,$
(27q) $\displaystyle f_{63}$
$\displaystyle=1+\frac{v^{2}}{1-4\nu+3\nu^{2}}\left(-\frac{169}{24}+\frac{371}{12}\nu-\frac{125}{4}\nu^{2}+\frac{13}{2}\nu^{3}\right)\,,$
(27r) $\displaystyle f_{62}$
$\displaystyle=1+\frac{v^{2}}{1-5\nu+5\nu^{2}}\left(-\frac{37}{7}+27\nu-\frac{59}{2}\nu^{2}+\frac{7}{2}\nu^{3}\right)\,,$
(27s) $\displaystyle f_{61}$
$\displaystyle=1+\frac{v^{2}}{1-4\nu+3\nu^{2}}\left(-\frac{161}{24}+\frac{347}{12}\nu-\frac{335}{12}\nu^{2}+\frac{31}{6}\nu^{3}\right)\,,$
(27t) $\displaystyle f_{77}$
$\displaystyle=1+\frac{v^{2}}{1-4\nu+3\nu^{2}}\left(-\frac{151}{17}+\frac{2123}{51}\nu-\frac{4963}{102}\nu^{2}+\frac{230}{17}\nu^{3}\right)\,,$
(27u) $\displaystyle f_{75}$
$\displaystyle=1+\frac{v^{2}}{1-4\nu+3\nu^{2}}\left(-\frac{127}{17}+\frac{1691}{51}\nu-\frac{3523}{102}\nu^{2}+\frac{134}{17}\nu^{3}\right)\,,$
(27v) $\displaystyle f_{73}$
$\displaystyle=1+\frac{v^{2}}{1-4\nu+3\nu^{2}}\left(-\frac{111}{17}+\frac{1403}{51}\nu-\frac{2563}{102}\nu^{2}+\frac{70}{17}\nu^{3}\right)\,,$
(27w) $\displaystyle f_{71}$
$\displaystyle=1+\frac{v^{2}}{1-4\nu+3\nu^{2}}\left(-\frac{103}{17}+\frac{1259}{51}\nu-\frac{2083}{102}\nu^{2}+\frac{38}{17}\nu^{3}\right)\,,$
(27x)

and at the leading order, $f_{\ell m}=1$. Furthermore, the $\rho_{\ell m}$
read

$\displaystyle\rho_{22}$
$\displaystyle=1+v^{2}\left(-\frac{43}{42}+\frac{55}{84}\nu\right)+v^{4}\left(-\frac{20555}{10584}-\frac{33025}{21168}\nu+\frac{19583}{42336}\nu^{2}\right)$
$\displaystyle+v^{6}\left(\frac{1556919113}{122245200}-\frac{428}{105}\left(\gamma_{E}+\ln
4v\right)+\left(-\frac{48993925}{9779616}+\frac{41}{192}\pi^{2}\right)\nu\right.$
$\displaystyle\left.\qquad\qquad-\frac{6292061}{3259872}\nu^{2}+\frac{10620745}{39118464}\nu^{3}\right)\,,$
(28a) $\displaystyle\rho_{21}$
$\displaystyle=1+v^{2}\left(-\frac{59}{56}+\frac{23}{84}\nu\right)+v^{4}\left(-\frac{47009}{56448}-\frac{10993}{14112}\nu+\frac{617}{4704}\nu^{2}\right)$
$\displaystyle+v^{6}\left(\frac{7613184941}{2607897600}-\frac{107}{105}\left(\gamma_{E}+\ln
2v\right)+\left(\frac{1024181}{17385984}-\frac{41}{768}\pi^{2}\right)\nu\right.$
$\displaystyle\left.\qquad\qquad+\frac{622373}{8692992}\nu^{2}+\frac{2266171}{39118464}\nu^{3}\right)\,,$
(28b) $\displaystyle\rho_{33}$
$\displaystyle=1+v^{2}\left(-\frac{7}{6}+\frac{2}{3}\nu\right)+v^{4}\left(-\frac{6719}{3960}-\frac{1861}{990}\nu+\frac{149}{330}\nu^{2}\right)$
$\displaystyle+v^{6}\left(\frac{3203101567}{227026800}-\frac{26}{7}\left(\gamma_{E}+\ln
6v\right)+\left(-\frac{129509}{25740}+\frac{41}{192}\pi^{2}\right)\nu\right.$
$\displaystyle\left.\qquad\qquad-\frac{274621}{154440}\nu^{2}+\frac{12011}{46332}\nu^{3}\right)\,,$
(28c) $\displaystyle\rho_{32}$
$\displaystyle=1+\frac{v^{2}}{1-3\nu}\left(-\frac{164}{135}+\frac{223}{54}\nu-\frac{32}{27}\nu^{2}\right)$
$\displaystyle+\frac{v^{4}}{(1-3\nu)^{2}}\left(-\frac{180566}{200475}+\frac{1610009}{320760}\nu-\frac{945121}{320760}\nu^{2}-\frac{508474}{40095}\nu^{3}+\frac{77141}{40095}\nu^{4}\right)\,,$
(28d) $\displaystyle\rho_{31}$
$\displaystyle=1+v^{2}\left(-\frac{13}{18}-\frac{2}{9}\nu\right)+v^{4}\left(\frac{101}{7128}-\frac{1685}{1782}\nu-\frac{829}{1782}\nu^{2}\right)$
$\displaystyle+v^{6}\left(\frac{11706720301}{6129723600}-\frac{26}{63}\left(\gamma_{E}+\ln
2v\right)+\left(-\frac{9688441}{2084940}+\frac{41}{192}\pi^{2}\right)\nu\right.$
$\displaystyle\left.\qquad\qquad+\frac{174535}{75816}\nu^{2}-\frac{727247}{1250964}\nu^{3}\right)\,,$
(28e) $\displaystyle\rho_{44}$
$\displaystyle=1+\frac{v^{2}}{1-3\nu}\left(-\frac{269}{220}+\frac{587}{132}\nu-\frac{175}{88}\nu^{2}\right)$
$\displaystyle+\frac{v^{4}}{(1-3\nu)^{2}}\left(-\frac{14210377}{8808800}+\frac{32485357}{4404400}\nu-\frac{1401149}{1415700}\nu^{2}-\frac{801565}{37752}\nu^{3}+\frac{3976393}{1006720}\nu^{4}\right)\,,$
(28f) $\displaystyle\rho_{43}$
$\displaystyle=1+\frac{v^{2}}{1-2\nu}\left(-\frac{111}{88}+\frac{547}{176}\nu-\frac{10}{11}\nu^{2}\right)$
$\displaystyle+\frac{v^{4}}{(1-2\nu)^{2}}\left(-\frac{6894273}{7047040}+\frac{22211989}{7047040}\nu+\frac{1032509}{1098240}\nu^{2}-\frac{181867}{25168}\nu^{3}+\frac{19379}{18876}\nu^{4}\right)\,,$
(28g) $\displaystyle\rho_{42}$
$\displaystyle=1+\frac{v^{2}}{1-3\nu}\left(-\frac{191}{220}+\frac{353}{132}\nu-\frac{19}{88}\nu^{2}\right)$
$\displaystyle+\frac{v^{4}}{(1-3\nu)^{2}}\left(-\frac{3190529}{8808800}+\frac{4108813}{4404400}\nu+\frac{21506941}{5662800}\nu^{2}-\frac{3628549}{377520}\nu^{3}-\frac{1204847}{1006720}\nu^{4}\right)\,,$
(28h) $\displaystyle\rho_{41}$
$\displaystyle=1+\frac{v^{2}}{1-2\nu}\left(-\frac{301}{264}+\frac{1385}{528}\nu-\frac{6}{11}\nu^{2}\right)$
$\displaystyle+\frac{v^{4}}{(1-2\nu)^{2}}\left(-\frac{7775491}{21141120}+\frac{117238909}{63423360}\nu-\frac{332099}{1098240}\nu^{2}-\frac{917167}{226512}\nu^{3}+\frac{7075}{18876}\nu^{4}\right)\,,$
(28i) $\displaystyle\rho_{55}$
$\displaystyle=1+\frac{v^{2}}{1-2\nu}\left(-\frac{487}{390}+\frac{649}{195}\nu-\frac{256}{195}\nu^{2}\right)$
$\displaystyle+\frac{v^{4}}{(1-2\nu)^{2}}\left(-\frac{3353747}{2129400}+\frac{4038803}{1064700}\nu+\frac{925493}{266175}\nu^{2}-\frac{932171}{88725}\nu^{3}+\frac{456206}{266175}\nu^{4}\right)\,,$
(28j) $\displaystyle\rho_{54}$
$\displaystyle=1+\frac{v^{2}}{1-5\nu+5\nu^{2}}\left(-\frac{2908}{2275}+\frac{13717}{1950}\nu-\frac{1823}{195}\nu^{2}+\frac{476}{195}\nu^{3}\right)\,,$
(28k) $\displaystyle\rho_{53}$
$\displaystyle=1+\frac{v^{2}}{1-2\nu}\left(-\frac{25}{26}+\frac{85}{39}\nu-\frac{88}{195}\nu^{2}\right)$
$\displaystyle+\frac{v^{4}}{(1-2\nu)^{2}}\left(-\frac{410833}{709800}+\frac{279697}{354900}\nu+\frac{928009}{266175}\nu^{2}-\frac{296117}{53235}\nu^{3}+\frac{33934}{266175}\nu^{4}\right)\,,$
(28l) $\displaystyle\rho_{52}$
$\displaystyle=1+\frac{v^{2}}{1-5\nu+5\nu^{2}}\left(-\frac{2638}{2275}+\frac{12097}{1950}\nu-\frac{1499}{195}\nu^{2}+\frac{314}{195}\nu^{3}\right)\,,$
(28m) $\displaystyle\rho_{51}$
$\displaystyle=1+\frac{v^{2}}{1-2\nu}\left(-\frac{319}{390}+\frac{313}{195}\nu-\frac{4}{195}\nu^{2}\right)$
$\displaystyle+\frac{v^{4}}{(1-2\nu)^{2}}\left(-\frac{31877}{304200}-\frac{79387}{152100}\nu+\frac{112613}{38025}\nu^{2}-\frac{31849}{12675}\nu^{3}-\frac{37342}{38025}\nu^{4}\right)\,,$
(28n) $\displaystyle\rho_{66}$
$\displaystyle=1+\frac{v^{2}}{1-5\nu+5\nu^{2}}\left(-\frac{53}{42}+\frac{43}{6}\nu-\frac{41}{4}\nu^{2}+\frac{13}{4}\nu^{3}\right)\,,$
(28o) $\displaystyle\rho_{65}$
$\displaystyle=1+\frac{v^{2}}{1-4\nu+3\nu^{2}}\left(-\frac{185}{144}+\frac{419}{72}\nu-\frac{455}{72}\nu^{2}+\frac{55}{36}\nu^{3}\right)\,,$
(28p) $\displaystyle\rho_{64}$
$\displaystyle=1+\frac{v^{2}}{1-5\nu+5\nu^{2}}\left(-\frac{43}{42}+\frac{11}{2}\nu-\frac{83}{12}\nu^{2}+\frac{19}{12}\nu^{3}\right)\,,$
(28q) $\displaystyle\rho_{63}$
$\displaystyle=1+\frac{v^{2}}{1-4\nu+3\nu^{2}}\left(-\frac{169}{144}+\frac{371}{72}\nu-\frac{125}{24}\nu^{2}+\frac{13}{12}\nu^{3}\right)\,,$
(28r) $\displaystyle\rho_{62}$
$\displaystyle=1+\frac{v^{2}}{1-5\nu+5\nu^{2}}\left(-\frac{37}{42}+\frac{9}{2}\nu-\frac{59}{12}\nu^{2}+\frac{7}{12}\nu^{3}\right)\,,$
(28s) $\displaystyle\rho_{61}$
$\displaystyle=1+\frac{v^{2}}{1-4\nu+3\nu^{2}}\left(-\frac{161}{144}+\frac{347}{72}\nu-\frac{335}{72}\nu^{2}+\frac{31}{36}\nu^{3}\right)\,,$
(28t) $\displaystyle\rho_{77}$
$\displaystyle=1+\frac{v^{2}}{1-4\nu+3\nu^{2}}\left(-\frac{151}{119}+\frac{2123}{357}\nu-\frac{709}{102}\nu^{2}+\frac{230}{119}\nu^{3}\right)\,,$
(28u) $\displaystyle\rho_{75}$
$\displaystyle=1+\frac{v^{2}}{1-4\nu+3\nu^{2}}\left(-\frac{127}{119}+\frac{1691}{357}\nu-\frac{3523}{714}\nu^{2}+\frac{134}{119}\nu^{3}\right)\,,$
(28v) $\displaystyle\rho_{73}$
$\displaystyle=1+\frac{v^{2}}{1-4\nu+3\nu^{2}}\left(-\frac{111}{119}+\frac{1403}{357}\nu-\frac{2563}{714}\nu^{2}+\frac{10}{17}\nu^{3}\right)\,,$
(28w) $\displaystyle\rho_{71}$
$\displaystyle=1+\frac{v^{2}}{1-4\nu+3\nu^{2}}\left(-\frac{103}{119}+\frac{1259}{357}\nu-\frac{2083}{714}\nu^{2}+\frac{38}{119}\nu^{3}\right)\,.$
(28x)

As for the $f_{\ell m}$, at leading order $\rho_{\ell m}=1$. Finally, we have

$\displaystyle\delta_{22}=$ $\displaystyle\frac{7}{3}\omega E-24\nu
v^{5}+\frac{428\pi}{105}\left(\omega
E\right)^{2}+v^{7}\nu\left(\frac{30995}{1134}+\frac{3662}{135}\nu\right)\,,$
(29a) $\displaystyle\delta_{21}=$ $\displaystyle\frac{2}{3}\omega
E-\frac{25}{2}\nu v^{5}+\frac{107\pi}{105}\left(\omega E\right)^{2}\,,$ (29b)
$\displaystyle\delta_{33}=$ $\displaystyle\frac{13}{10}\omega
E-\frac{80897}{2430}\nu v^{5}+\frac{39\pi}{7}\left(\omega E\right)^{2}\,,$
(29c) $\displaystyle\delta_{32}=$
$\displaystyle\left(\frac{2}{3}+\frac{11}{5}\nu\right)\frac{\omega
E}{1-3\nu}+\frac{v^{5}\nu}{(1-3\nu)^{2}}\left(-\frac{87893}{5400}+\frac{66871}{600}\nu-\frac{28691}{150}\nu^{2}\right)\,,$
(29d) $\displaystyle\delta_{31}=$ $\displaystyle\frac{13}{30}\omega
E-\frac{17}{10}\nu v^{5}+\frac{13\pi}{21}\left(\omega E\right)^{2}\,,$ (29e)
$\displaystyle\delta_{44}=$
$\displaystyle\left(\frac{14}{15}+\frac{73}{40}\nu\right)\frac{\omega
E}{1-3\nu}+\frac{v^{5}\nu}{(1-3\nu)^{2}}\left(-\frac{5794139}{140800}+\frac{35353757}{140800}\nu-\frac{2740989}{7040}\nu^{2}\right)\,,$
(29f) $\displaystyle\delta_{43}=$
$\displaystyle\left(\frac{3}{5}+\frac{4961}{810}\nu\right)\frac{\omega
E}{1-2\nu}\,,$ (29g) $\displaystyle\delta_{42}=$
$\displaystyle\left(\frac{7}{15}+\frac{14}{5}\nu\right)\frac{\omega
E}{1-3\nu}+\frac{v^{5}\nu}{(1-3\nu)^{2}}\left(-\frac{125569}{4400}+\frac{764879}{4400}\nu-\frac{291003}{1100}\nu^{2}\right)\,,$
(29h) $\displaystyle\delta_{41}=$
$\displaystyle\left(\frac{1}{5}+\frac{507}{10}\nu\right)\frac{\omega
E}{1-2\nu}\,,$ (29i) $\displaystyle\delta_{55}=$
$\displaystyle\left(\frac{31}{42}+\frac{61252}{9375}\nu\right)\frac{\omega
E}{1-2\nu}\,,$ (29j) $\displaystyle\delta_{54}=$
$\displaystyle\left(\frac{8}{15}+\frac{137377}{17920}\nu-\frac{11777}{384}\nu^{2}\right)\frac{\omega
E}{1-5\nu+5\nu^{2}}\,,$ (29k) $\displaystyle\delta_{53}=$
$\displaystyle\left(\frac{31}{70}+\frac{23924}{3645}\nu\right)\frac{\omega
E}{1-2\nu}\,,$ (29l) $\displaystyle\delta_{52}=$
$\displaystyle\left(\frac{4}{15}+\frac{2351}{112}\nu-\frac{3943}{60}\nu^{2}\right)\frac{\omega
E}{1-5\nu+5\nu^{2}}\,,$ (29m) $\displaystyle\delta_{51}=$
$\displaystyle\left(\frac{31}{210}+\frac{1796}{15}\nu\right)\frac{\omega
E}{1-2\nu}\,,$ (29n) $\displaystyle\delta_{66}=$
$\displaystyle\left(\frac{43}{70}+\frac{1756523}{217728}\nu-\frac{85439}{2592}\nu^{2}\right)\frac{\omega
E}{1-5\nu+5\nu^{2}}\,,$ (29o) $\displaystyle\delta_{64}=$
$\displaystyle\left(\frac{43}{105}+\frac{417559}{49152}\nu-\frac{393431}{12288}\nu^{2}\right)\frac{\omega
E}{1-5\nu+5\nu^{2}}\,,$ (29p) $\displaystyle\delta_{62}=$
$\displaystyle\left(\frac{43}{210}+\frac{387629}{13440}\nu-\frac{42859}{480}\nu^{2}\right)\frac{\omega
E}{1-5\nu+5\nu^{2}}\,,$ (29q)

while the values for the others $\delta_{\ell m}$ are 0 at the 3.5PN order.

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