# Spin contributions to the gravitational-waveform modes for spin-aligned
binaries at the 3.5PN order

Quentin Henry<EMAIL_ADDRESS>Max Planck Institute for Gravitational
Physics
(Albert Einstein Institute), D-14476 Potsdam, Germany  Sylvain Marsat
<EMAIL_ADDRESS>Laboratoire des 2 Infinis - Toulouse (L2IT-
IN2P3), Université de Toulouse, CNRS, UPS, F-31062 Toulouse Cedex 9, France
Mohammed Khalil<EMAIL_ADDRESS>Max Planck Institute for
Gravitational Physics
(Albert Einstein Institute), D-14476 Potsdam, Germany

###### Abstract

We complete the post-Newtonian (PN) prediction at the 3.5PN order for the spin
contributions to the gravitational waveforms emitted by inspiraling compact
binaries, in the case of quasi-circular, equatorial orbits, where both spins
are aligned with the orbital angular momentum. Using results from the
multipolar post-Minkowskian wave generation formalism, we extend previous
works that derived the dynamics and gravitational-wave energy flux and
phasing, by computing the full waveform decomposed in spin-weighted spherical
harmonics. This new calculation requires the computation of multipolar moments
of higher multipolar order, new quadratic-in-spin contributions to the
hereditary tail terms entering at the 3.5PN order, as well as other non-linear
interactions between moments. When specialized to the test-mass limit, our
results are equivalent to those obtained in the literature for the waveform
emitted by a test-mass in equatorial, circular orbits around a Kerr black
hole. We also compute the factorized modes for use in effective-one-body
waveform models, correcting the 2.5PN nonspinning and 3PN quadratic-in-spin
terms in the (2,1) mode used in current models.

###### pacs:

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

##  I Introduction

Since the first gravitational-wave (GW) detection in 2015 [1], the LIGO-Virgo-
KAGRA collaboration has observed over 90 GW signals from stellar-mass compact
binary objects [2, 3, 4]. Future GW detectors, such as LISA [5] and the
Einstein Telescope [6], will also widen the range in parameter space of
detectable systems, including signals from extreme-mass-ratio inspirals and
supermassive binary black holes, in addition to improving our knowledge of the
deformability of neutron stars.

The use of accurate waveform templates for data analysis is crucial and
requires constant improvement to match the increasing sensitivity of GW
detectors. The post-Newtonian (PN) scheme is well suited to describe the
inspiral of compact binaries, but its accuracy deteriorates in the strong-
field regime. The gravitational self-force (GSF) framework [7, 8, 9, 10, 11,
12, 13] is valid in the strong field, but for small-mass ratios. However, by
combining different analytical approximation methods and numerical-relativity
results, the effective-one-body (EOB) formalism [14, 15] produces accurate
waveforms over the entire parameter space.

Spin has a significant effect on the binary dynamics; thus, improving the spin
description in waveform models is necessary to obtain accurate parameter
estimations, which helps in improving our understanding of the properties of
compact binaries and their astrophysical formation channels [16, 17, 18]. To
model spin effects in general relativity, we use an effective field theory
(EFT) approach, which constructs an effective action defined on the worldline
of the bodies [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35]. In the conservative sector, many approaches were used to tackle the
problem of the spin contributions in the equations of motion (EOM) which now
reaches the 4.5PN accuracy [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47,
48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 57, 59, 33, 60, 34, 35, 61, 62,
63, 64, 65, 66]. On the other hand, the 3.5PN radiated flux and phase,
including all spin contributions, were derived for quasi-circular orbits and
non-precessing spins [36, 37, 67, 68, 69, 70, 71, 72, 35]. Recently, this
computation has been pushed to 4PN by EFT methods [73]. However, the
polarizations for non-precessing spins are only known to 2.5PN order [74, 58,
75]. In the case of precessing spins, which complicate the computations, the
waveform amplitude is known to 1.5PN order, and the phase to 2PN order [74].

In the present article, we compute the spin effects in the full waveform up to
the 3.5PN order. The dynamics and phase evolution are already known to that
order, and the remaining piece is the computation of the amplitude of the
waveform, decomposed as a sum of spin-weighted spherical harmonics. These
harmonics complete our knowledge of the post-Newtonian waveform, and they are
a crucial ingredient of waveform models such as the EOB approach [76, 14, 77,
78, 79, 80, 81, 82, 83, 84, 85, 86, 87].

At the 3.5PN order, three interactions arise: the spin-orbit (SO), the spin-
induced quadrupole and the spin-induced octupole interactions. We use the
post-Newtonian-multipolar-post-Minkowskian (PN-MPM) formalism [88], where the
observables of the system are parametrized in terms of the so-called radiative
multipole moments, defined in the radiative zone. The PN-MPM formalism allows
linking these moments to another set of moments called the source multipole
moments, which are defined over the source of the system and are linked to the
metric and stress-energy tensor. The PN approximation is an expansion in the
dimensionless quantity $v^{2}/c^{2}\sim GM/c^{2}r\ll 1$, where $v$ is the
relative velocity and $M$ is the total mass of the binary. To make PN
expansions clear, we write $1/c$ explicitly in this paper, and rescale the
physical spin variable $S_{\rm physical}$ as

$S=c\,S_{\rm physical}=Gm^{2}\chi\,,$ (1)

where $\chi$ is the dimensionless spin, whose value is one for an extremal
Kerr black hole.

The paper is organized as follows. In Sec. II, we give some definitions and
conventions, and recall the relation between the spherical modes and the
radiative multipole moments. Sec. III contains the technical aspects regarding
the computation of the radiative multipole moments. First in Sec. III.1, we
recall some aspects of the effective action describing spinning effects up to
the 3.5PN order. In Sec. III.2, we give the expression of the stress-energy
tensor as well as its 3+1 decomposition. Sec. III.3 gives the general
definition of the source multipole moments. In Sec. III.4 we give the PN
metric and compute the potentials that parametrize it. Sec. III.5 is dedicated
to the integration of the source multipole moments using the potentials and
the decomposed stress-energy tensor. In Sec. III.6, we express the source
moments in the center of mass (CM) in the case of aligned spins and quasi-
circular orbits. Finally, for this technical part, in Sec. III.7, we derive
the radiative multipole moments by computing the non-linear interactions in
the GW field. In Sec. IV, we give the results for the spinning contributions
to the amplitude modes, written in the conventional way in Sec. IV.1 and
factorized conveniently for EOB usage in Sec. IV.2. Appendices A and B contain
the lengthy expressions for the source densities and the factorized modes. We
also provide our results for the waveform modes as Mathematica files in the
Supplemental Material [89].

##  II Summary of the formalism

###  II.1 Definitions and conventions

In this paper, we compute the amplitude modes of the GW emitted by a binary
system of spinning compact objects in the inspiral phase. We restrict
ourselves to the non-precessing case which implies that, as for the non-
spinning case, the motion of the system remains planar. Let us define the
radiative coordinate system $X^{\mu}=(cT,\bm{X})$, in which $R=|\bm{X}|$ is
the distance of the observer from the source and $\bm{N}=\bm{X}/R$ is the
direction of propagation of the GW. We denote retarded time $T_{R}=T-R/c$,
where $c$ is the speed of light in vacuum. We define the spatial unitary basis
$(\bm{e}_{X},\bm{e}_{Y},\bm{e}_{Z})$ such that the orbital plane lies within
the $(\bm{e}_{X},\bm{e}_{Y})$ plane and $\bm{N}$ in the
$(\bm{e}_{Y},\bm{e}_{Z})$ plane. The two GW polarizations $h_{+}$ and
$h_{\times}$, defined in Eq. (5), propagate in the plane orthogonal to the
direction of propagation $\bm{N}$. It is convenient to introduce the
orthonormal triad $(\bm{P},\bm{Q},\bm{N})$ by choosing $\bm{P}=\bm{e}_{X}$ and
$\bm{Q}=\bm{N}\times\bm{P}$.

We use the following conventions henceforth: $\mathcal{O}(n)$ means
$\mathcal{O}(1/c^{n})$, i.e. represents a contribution of the order $(n/2)$PN
at least. Greek indices denote spacetime coordinates, i.e. $\mu=0,1,2,3$,
while Latin indices are used for spatial coordinates, i.e. $i=1,2,3$. We use
the multi-index notation $L=i_{1}\dots i_{\ell}$. Symmetrization and anti-
symmetrization are represented by, respectively, parenthesis and brackets
around indices while the symmetric trace-free (STF) projection is denoted by
$\langle\rangle$. We adopt the signature $(-,+,+,+)$ and keep explicit both
Newton’s constant $G$ and the speed of light $c$, unless explicitly specified.
Finally the covariant derivative along the worldline is written as
$\mathrm{D}/(c\,\mathrm{D}\tau)=u^{\mu}\nabla_{\mu}$, where $u^{\mu}$ is the
four-velocity of the particle.

The problem is parametrized using the following notations: we use the
coordinate-time $t$ parametrization. The quantities $m_{A}$, $\bm{y}_{A}$,
$\bm{v}_{A}=\mathrm{d}\bm{y}_{A}/\mathrm{d}t$ and
$\bm{a}_{A}=\mathrm{d}^{2}\bm{y}_{A}/\mathrm{d}t^{2}$ refer respectively to
the mass, position, velocity and acceleration of body $A=1,2$. The notation SO
refers to the spin-orbit interaction, SS to the quadratic-in-spin interaction
and finally SSS to the cubic-in-spin terms.

###  II.2 Spherical harmonics decomposition

The Einstein field equations can be exactly written, by imposing the harmonic
gauge condition $\partial_{\nu}h^{\mu\nu}=0$, as

$\Box h^{\mu\nu}=\frac{16\pi G}{c^{4}}\tau^{\mu\nu},$ (2)

where $\Box$ is the flat d’Alembertian operator defined with respect to the
inverse Minkowski metric $\eta^{\mu\nu}$, $h^{\mu\nu}$ is the deviation to the
gothic metric $h^{\mu\nu}=\sqrt{-g}g^{\mu\nu}-\eta^{\mu\nu}$,
$g=\text{det}(g_{\mu\nu})$ is the determinant of the metric. Finally,
$\tau^{\mu\nu}$ is the stress-energy pseudo tensor

$\tau^{\mu\nu}=|g|T^{\mu\nu}+\frac{c^{4}}{16\pi G}\,\Lambda^{\mu\nu}\,,$ (3)

where $T^{\mu\nu}$ is the stress-energy tensor and $\Lambda^{\mu\nu}$ is a
function of derivatives of at least quadratic terms in the perturbed metric
$h^{\mu\nu}$. Its expression is given in Eq. (24) of Ref. [88].

In the coordinate system $X^{\mu}=(cT,\bm{X})$, the transverse-traceless (TT)
projection $h_{ij}^{\text{TT}}$ of the gravitational field $h^{\mu\nu}$ can
be, at leading order in $1/R$, uniquely decomposed in terms of a set of STF
multipole moments $U_{L}$ and $V_{L}$, called radiative multipole moments, as
[90]

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

where we introduced the TT projection operator
$\mathcal{P}_{ijkl}=\mathcal{P}_{i(k}\mathcal{P}_{l)j}-\frac{1}{2}\mathcal{P}_{ij}\mathcal{P}_{kl}$,
with $\mathcal{P}_{ij}=\delta_{ij}-N_{i}N_{j}$ the projector orthogonal to the
unit direction $\bm{N}$. We can define the usual polarization waveforms in the
orthonormal triad $(\bm{P},\bm{Q},\bm{N})$ as

$\displaystyle h_{+}$
$\displaystyle=\frac{1}{2}(P_{i}P_{j}-Q_{i}Q_{j})h_{ij}^{\text{TT}},$ (5a)
$\displaystyle h_{\times}$
$\displaystyle=\frac{1}{2}(P_{i}Q_{j}+Q_{i}P_{j})h_{ij}^{\text{TT}}.$ (5b)

We can then decompose the quantity $h_{+}-\mathrm{i}h_{\times}$ in a spin-
weighted spherical harmonics basis of weight -2 [91]

$h\equiv
h_{+}-\mathrm{i}h_{\times}=\sum_{l=0}^{\infty}\sum_{m=-\ell}^{\ell}h_{\ell
m}Y^{\ell m}_{-2}(\Theta,\Phi),$ (6)

where the two angles $(\Theta,\Phi)$ characterize the direction of propagation
$\bm{N}$ in the coordinate system $X^{\mu}$. In this paper, we follow the
conventions of Ref. [92]. Notably, the explicit expression of the spin-
weighted spherical harmonics are given in Eqs. (2.4) of that reference. The
amplitude modes $h_{\ell m}$ are then linked to the radiative moments $U_{L}$
and $V_{L}$ through

$h_{\ell m}=-\frac{G}{\sqrt{2}Rc^{\ell+2}}\left(U^{\ell
m}-\frac{\mathrm{i}}{c}V^{\ell m}\right).$ (7)

where $U^{\ell m}$ and $V^{\ell m}$ read111Note that the choice of definitions
in Ref. [91, 79] 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
$(\bm{P},\bm{Q},\bm{N})$.

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

The STF tensorial coefficient $\alpha_{L}^{\ell
m}\equiv\int\mathrm{d}\Omega\,\hat{N}_{L}\,\overline{Y}^{\,\ell m}$ is defined
from the ordinary spherical harmonics $Y^{\ell m}$ (or in fact its complex
conjugate $\overline{Y}^{\,\ell m}$) and its explicit expression is given in
Eq. (4.7) of Ref. [93].

The main result of this paper is the spin contributions in $h$ up to the 3.5PN
order for a non-precessing, quasi-circular motion. In order to derive them, we
have to compute the spin part of the radiative moments to consistent order.
Table 1 shows at which order the different multipoles need to be computed for
each spin interaction. The SO terms in the radiated flux and GW phase have
been derived in the PN-MPM formalism up to 4PN while the SS contributions are
known to 3.5PN [70, 71, 72, 35], however the radiative moments are required to
a higher order for the modes than for the flux. This can be understood by
comparing the expression (4) of the full waveform, with a $1/c$ increment for
each multipolar order, to the expression of the energy flux emitted in
gravitational waves (e.g. (4.2) in Ref. [72]) as sum of squares of radiative
moments with a $1/c^{2}$ increment per multipolar order.

Table 1:  Leading and required (relative to the leading) orders of the spin contributions in the radiative moments for the full gravitational waveform at 3.5PN. The order corresponds to the power in $1/c$ and not the PN order. |  | Leading order | Required relative order
---|---|---|---
$\ell$ | Moment |  SO |  SS |  SSS |  SO |  SS |  SSS
2 | $U_{ij}$ |  3 |  4 |  7 |  4 |  3 |  0
3 | $U_{ijk}$ |  3 |  4 |  7 |  3 |  2 |  -
4 | $U_{ijkl}$ |  3 |  4 |  7 |  2 |  1 |  -
5 | $U_{ijklm}$ |  3 |  4 |  7 |  1 |  0 |  -
6 | $U_{ijklmp}$ |  3 |  4 |  7 |  0 |  - |  -
2 | $V_{ij}$ |  1 |  4 |  5 |  5 |  2 |  1
3 | $V_{ijk}$ |  1 |  4 |  5 |  4 |  1 |  0
4 | $V_{ijkl}$ |  1 |  4 |  5 |  3 |  0 |  -
5 | $V_{ijklm}$ |  1 |  4 |  5 |  2 |  - |  -
6 | $V_{ijklmp}$ |  1 |  4 |  5 |  1 |  - |  -
7 | $V_{ijklmpq}$ |  1 |  4 |  5 |  0 |  - |  -

##  III Computation of the radiative multipole moments

In this paper, we use the PN-MPM formalism, see Ref. [88] for a review. Within
this approach, the procedure to derive the radiative multipole moments is
well-known. For completeness, we give a summarized version of the method,
recalling the different steps: i) we first compute the stress-energy tensor
from the effective skeletonized action modeling spinning particles in Sec.
III.1; ii) in Sec. III.2, we decompose the spatial and temporal indices of the
stress-energy tensor to compute the source densities; iii) in Sec. III.4, we
compute the PN metric parametrized by a set of potentials; iv) in Sec. III.5,
we derive the source multipole moments defined in Sec. III.3; v) we then
express the source moments in the CM frame for aligned spins and quasi-
circular orbits in Sec. III.6; vi) finally, we compute the non-linear effects
to deduce the radiative multipole moments in Sec. III.7.

###  III.1 Effective skeletonized action of spinning particles

In this section, we start by recalling the method of the effective action
approach to the spin-induced effects. We follow here the presentation of [72,
35], but note that a more general formalism in the language of effective field
theories can be found in Ref. [33]. Here, we drop the particle’s label $A$ and
set $c=1$ for more clarity.

The matter action is constructed on the worldline of the individual body
$z^{\mu}$. The affine parameter along this worldline is called $\tau$ and will
be associated after variation of the action to the proper time of the body. We
define the usual 4-velocity of the particle
$u^{\mu}=\mathrm{d}z^{\mu}/\mathrm{d}\tau$ such that $u^{\mu}u_{\mu}=-1$.
Attaching an orthonormal tetrad $\epsilon_{A}^{\phantom{A}\mu}$ to the moving
body, we also introduce antisymmetric rotation coefficients
$\Omega^{\mu\nu}=\epsilon^{A\mu}D\epsilon_{A}^{\phantom{A}\nu}/d\tau$
representing the rotational degrees of freedom. The matter action then reads
symbolically

$S_{\text{M}}=\int\mathrm{d}^{4}x\sqrt{-g}\int\mathrm{d}\tau\,L_{\text{M}}[u^{\mu},\Omega^{\mu\nu},g_{\mu\nu},R_{\mu\nu\rho\sigma},\nabla_{\lambda}R_{\mu\nu\rho\sigma}]\frac{\delta^{(4)}(x^{\mu}-z^{\mu}(\tau))}{\sqrt{-g}},$
(9)

where the matter Lagrangian includes finite-size effects in the form of
couplings to the Riemann tensor and its derivative, necessary to represent
spin-induced effects up to the cubic order in spin. We also introduce the
following quantities

$\displaystyle p_{\mu}$ $\displaystyle\equiv\frac{\partial L}{\partial
u^{\mu}},\qquad\qquad S_{\mu\nu}\equiv 2\frac{\partial
L}{\partial\Omega^{\mu\nu}},$ $\displaystyle J^{\mu\nu\rho\sigma}$
$\displaystyle\equiv-6\frac{\partial L}{\partial R_{\mu\nu\rho\sigma}},\qquad
J^{\lambda\mu\nu\rho\sigma}\equiv-12\frac{\partial
L}{\partial\nabla_{\lambda}R_{\mu\nu\rho\sigma}},$ (10)

that represent the couplings entering the action. The quantities $p^{\mu}$ and
$S^{\mu\nu}$ are the linear momentum and the antisymmetric spin tensor. The
moments $J^{\mu\nu\rho\sigma}$ and $J^{\lambda\mu\nu\rho\sigma}$ are Dixon-
type moments [23, 24, 27] representing the quadrupole and octupole, assumed to
be purely spin-induced in our case (no other finite-size effects such as tidal
effects). The general form of the stress-energy tensor will be expressed in
terms of these quantities in the following section.

Varying the action with respect to the worldine and rotational degrees of
freedom (together with a scalarity condition for the Lagrangian, see (2.8) in
Ref. [35]) yields evolution equations for the 4-momentum as well as the spin
tensor:

$\displaystyle\frac{\mathrm{D}p_{\mu}}{\mathrm{D}\tau}$
$\displaystyle=-\frac{1}{2}R_{\mu\nu\rho\sigma}u^{\nu}S^{\rho\sigma}-\frac{1}{6}J^{\lambda\nu\rho\sigma}\nabla_{\mu}R_{\lambda\nu\rho\sigma}-\frac{1}{12}J^{\tau\lambda\nu\rho\sigma}\nabla_{\mu}\nabla_{\tau}R_{\lambda\nu\rho\sigma}+\mathcal{O}(S^{4})\,,$
(11a) $\displaystyle\frac{\mathrm{D}S^{\mu\nu}}{\mathrm{D}\tau}$
$\displaystyle=2p^{[\mu}u^{\nu]}+\frac{4}{3}R^{[\mu}_{\phantom{\mu}\lambda\rho\sigma}J^{\nu]\lambda\rho\sigma}+\frac{2}{3}\nabla^{\lambda}R^{[\mu}_{\phantom{\mu}\tau\rho\sigma}J_{\lambda}^{\phantom{\lambda}\nu]\tau\rho\sigma}+\frac{1}{6}\nabla^{[\mu}R_{\lambda\tau\rho\sigma}J^{\nu]\lambda\tau\rho\sigma}+\mathcal{O}(S^{4})\,.$
(11b)

However, the introduction of the spin tensor adds 3 degrees of freedom to the
problem. In order to close the system of equations describing the dynamics, we
impose a spin supplementary condition (SSC). In particular we choose to impose
the Tulczyjew-Dixon SSC [22, 27] which reads

$S^{\mu\nu}p_{\nu}=0\,.$ (12)

Note that the mass $p_{\mu}p^{\mu}=-\tilde{m}^{2}$ is not conserved at
$\mathcal{O}(S^{2})$. Together with the equations of motion, the SSC allows us
to find a conserved mass, that we use henceforth in this paper,

$m\equiv\tilde{m}-\frac{1}{6}R_{\mu\nu\rho\sigma}J^{\mu\nu\rho\sigma}\,,$ (13)

and to relate $p^{\mu}$ and the 4-velocity as

$\displaystyle p^{\mu}$
$\displaystyle=mu^{\mu}+\frac{1}{6}u^{\mu}R_{\rho\lambda\mu\nu}J^{\rho\lambda\mu\nu}-\frac{1}{2m}S^{\mu\nu}R_{\nu\lambda\rho\sigma}u^{\lambda}S^{\rho\sigma}+\frac{4}{3}R^{[\mu}_{\phantom{\mu}\lambda\rho\sigma}J^{\nu]\lambda\rho\sigma}u_{\nu}$
$\displaystyle\quad+\frac{2}{3}u_{\nu}\nabla^{\lambda}R^{[\mu}_{\phantom{[\nu}\tau\rho\sigma}J_{\lambda}^{\phantom{\lambda}\nu]\tau\rho\sigma}+\frac{1}{6}u_{\nu}\nabla^{[\mu}R_{\lambda\tau\rho\sigma}J^{\nu]\lambda\tau\rho\sigma}-\frac{1}{6m}S^{\mu\nu}J^{\lambda\tau\rho\sigma}\nabla_{\nu}R_{\lambda\tau\rho\sigma}+\mathcal{O}(S^{4})\,.$
(14)

The norm of the spin tensor

$s^{2}\equiv\frac{1}{2}S_{\mu\nu}S^{\mu\nu},$ (15)

is conserved at the spin-cubic order that we consider:
$\mathrm{d}s/\mathrm{d}\tau=0$.

Imposing that the quadrupole and octupole moments are purely spin-induced with
the appropriate symmetries, we find

$\displaystyle J^{\mu\nu\rho\sigma}$
$\displaystyle=\frac{3\kappa}{m}u^{[\mu}S^{\nu]\lambda}S_{\lambda}^{\phantom{\lambda}[\rho}u^{\sigma]}\,,$
(16a) $\displaystyle J^{\lambda\mu\nu\rho\sigma}$
$\displaystyle=\frac{\lambda}{4m^{2}}\left[\Theta^{\lambda[\mu}u^{\nu]}S^{\rho\sigma}+\Theta^{\lambda[\rho}u^{\sigma]}S^{\mu\nu}\right.$
$\displaystyle\qquad\quad\;-\Theta^{\lambda[\mu}S^{\nu][\rho}u^{\sigma]}-\Theta^{\lambda[\rho}S^{\sigma][\mu}u^{\nu]}$
$\displaystyle\qquad\quad\;\left.-S^{\lambda[\mu}\Theta^{\nu][\rho}u^{\sigma]}-S^{\lambda[\rho}\Theta^{\sigma][\mu}u^{\nu]}\right]\,,$
(16b)

where we used the notation
$\Theta^{\mu\nu}=S^{\mu\lambda}S^{\nu}_{\phantom{\nu}\lambda}$, with
quadrupolar and octupolar parameters $\kappa$, $\lambda$ normalized to unity
for a Kerr black hole.

With the SSC (12), one can construct a spin vector $S^{\mu}$ from the spin
tensor $S^{\mu\nu}$, that also has a conserved norm by construction. They are
linked using the orthogonal unit vector $p^{\mu}/\tilde{m}$ (the mass
correction (13) is irrelevant at our PN order in spin terms) through the
relations

$\displaystyle S^{\mu\nu}$
$\displaystyle=\epsilon^{\mu\nu\rho\sigma}\frac{p_{\rho}}{\tilde{m}}S_{\sigma}\,,$
(17a) $\displaystyle S^{\mu}$
$\displaystyle=-\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}\frac{p_{\nu}}{\tilde{m}}S_{\rho\sigma}\,,$
(17b)

where $\epsilon^{\mu\nu\rho\sigma}$ is the Levi-Civita tensor. One can further
construct a 3-dimensional spin vector with conserved Euclidean norm $\bm{S}$
(see Section II. C in Ref. [72]), which we use in our final results.

For the two-body problem, the equations of motion (EOM) of the individual
bodies as well as the precession evolution of the spins have been derived in
harmonic coordinates in Refs. [51, 72, 35, 33, 60] and known at the considered
PN order of the problem so that we do not need to solve them here. The
precession equation in terms of the 3-dimensional spin vector reads
$\dot{\bm{S}}=\bm{\Omega}\times\bm{S}$ where $\bm{\Omega}$ starts at
$\mathcal{O}(2)$. We recall that in our problem, we have rescaled the spin
variable as in Eq. (1) to explicitly write the $c$ factors.

###  III.2 Effective stress-energy tensor and source densities

Equations (2.23)-(2.25) of Ref. [35] give the general form of the stress-
energy tensor in terms of $p^{\mu}$, $S^{\mu\nu}$, $J^{\mu\nu\rho\sigma}$ and
$J^{\lambda\mu\nu\rho\sigma}$. One can rewrite this expression as

$T^{\mu\nu}=\sum_{A}\left[U^{\mu\nu}_{A}\delta_{A}+\nabla_{\alpha}\left(U_{A}^{\mu\nu\alpha}\delta_{A}\right)+\nabla_{\alpha}\nabla_{\beta}\left(U_{A}^{\mu\nu\alpha\beta}\delta_{A}\right)+\nabla_{\alpha}\nabla_{\beta}\nabla_{\gamma}\left(U_{A}^{\mu\nu\alpha\beta\gamma}\delta_{A}\right)\right]\,,$
(18)

where $\delta_{A}\equiv\delta^{(3)}[\mathbf{x}-\bm{y}_{A}(t)]$ is the usual
three-dimensional Dirac distribution. In our case, the expression of the
$U$’s, up to the octupolar level, are given by

$\displaystyle U^{\mu\nu}_{A}$
$\displaystyle=\dfrac{1}{u_{A}^{0}\sqrt{-g}}\left(p_{A}^{(\mu}u_{A}^{\nu)}+\dfrac{1}{3}R\indices{{}_{A}^{(}{}^{\mu}_{\lambda}{}_{\rho}{}_{\sigma}}J_{A}^{\nu)\lambda\rho\sigma}+\dfrac{1}{6}\nabla_{\lambda}R\indices{{}_{A}^{(}{}^{\mu}_{\xi}{}_{\rho}{}_{\sigma}}J_{A}^{\underline{\lambda}\nu)\xi\rho\sigma}+\dfrac{1}{12}\nabla^{(\mu}R_{A\xi\tau\rho\sigma}J_{A}^{\nu)\xi\tau\rho\sigma}\right)$
$\displaystyle U_{A}^{\mu\nu\alpha}$
$\displaystyle=\dfrac{1}{3u_{A}^{0}\sqrt{-g}}\left(3u_{A}^{(\mu}S_{A}^{\nu)\alpha}-\dfrac{1}{2}R\indices{{}_{A}^{(}{}^{\mu}_{\xi}{}_{\lambda}{}_{\sigma}}J_{A}^{\underline{\alpha}\nu)\xi\lambda\sigma}-R\indices{{}_{A}^{(}{}^{\mu}_{\xi}{}_{\lambda}{}_{\sigma}}J_{A}^{\nu)\alpha\xi\lambda\sigma}+R\indices{{}_{A}^{\alpha}{}_{\xi}{}_{\lambda}{}_{\sigma}}J_{A}^{(\mu\nu)\xi\lambda\sigma}\right),$
(19a) $\displaystyle U_{A}^{\mu\nu\alpha\beta}$
$\displaystyle=-\dfrac{2}{3u_{A}^{0}\sqrt{-g}}J_{A}^{\alpha(\mu\nu)\beta},$
(19b) $\displaystyle U_{A}^{\mu\nu\alpha\beta\gamma}$
$\displaystyle=-\dfrac{1}{3u_{A}^{0}\sqrt{-g}}J_{A}^{\gamma\beta(\mu\nu)\alpha},$
(19c)

where $u_{A}^{0}=\mathrm{d}t/\mathrm{d}\tau_{A}$. The goal of this section is
to decompose the spatial and temporal indices of the stress-energy tensor in
order to make the factors in $c$ explicit. After reinstoring the $c$ powers in
the expressions, we first express the covariant derivatives in terms of the
partial derivatives and the Christoffel symbols, then separate spatial and
temporal indices. The practical formulas to perform this decomposition are
given in Sec. II. B. of Ref. [94]. Then, the source densities are defined by

$\sigma=\frac{T^{00}+T^{ii}}{c^{2}},\qquad\sigma_{i}=\frac{T^{0i}}{c},\qquad\sigma_{ij}=T^{ij}.$
(19t)

These quantities source the PN potentials as well as the source multipole
moments. We provide in Eqs. (19bo) the values of their spin contribution. As
we will see later on, $\sigma$ is required up to $\mathcal{O}(7)$,
$\sigma_{i}$ to $\mathcal{O}(6)$ and $\sigma_{ij}$ to $\mathcal{O}(4)$.

###  III.3 Source multipole moments

In the PN-MPM approach, a crucial step to compute the radiative moments is the
derivation of the so-called source multipole moments, which are expressed as
integrals over the source. The mass and current source moments are given by
[88]

$\displaystyle I_{L}(t)$
$\displaystyle=\mathop{\mathrm{FP}}_{B=0}\,\int\mathrm{d}^{3}\mathbf{x}\,\left(\frac{r}{r_{0}}\right)^{B}\int^{1}_{-1}\mathrm{d}z\left\\{\delta_{\ell}\,\hat{x}_{L}\,\Sigma-\frac{4(2\ell+1)}{c^{2}(\ell+1)(2\ell+3)}\,\delta_{\ell+1}\,\hat{x}_{iL}\,\Sigma_{i}^{(1)}\right.$
$\displaystyle\qquad\quad\left.+\frac{2(2\ell+1)}{c^{4}(\ell+1)(\ell+2)(2\ell+5)}\,\delta_{\ell+2}\,\hat{x}_{ijL}\Sigma_{ij}^{(2)}\right\\}(\mathbf{x},t+z\,r/c)\,,$
(19ua) $\displaystyle J_{L}(t)$
$\displaystyle=\mathop{\mathrm{FP}}_{B=0}\,\varepsilon_{ab<i_{\ell}}\int\mathrm{d}^{3}\mathbf{x}\,\left(\frac{r}{r_{0}}\right)^{B}\int^{1}_{-1}\mathrm{d}z\left\\{\vphantom{\frac{1}{1}}\delta_{\ell}\,\hat{x}_{L-1>a}\,\Sigma_{b}\right.$
$\displaystyle\qquad\quad\left.-\frac{2\ell+1}{c^{2}(\ell+2)(2\ell+3)}\,\delta_{\ell+1}\,\hat{x}_{L-1>ac}\,\Sigma_{bc}^{(1)}\right\\}(\mathbf{x},t+z\,r/c)\,,$
(19ub)

where we have $\delta_{\ell}(z)=a_{\ell}(1-z^{2})^{\ell}$ and
$a_{\ell}=(2\ell+1)!!/2^{\ell+1}\ell!$ and

$\Sigma=\frac{\tau^{00}+\tau^{ii}}{c^{2}},\qquad\Sigma_{i}=\frac{\tau^{0i}}{c},\qquad\Sigma_{ij}=\tau^{ij}.$
(19v)

The explicit expressions of the source moments in terms of the source
densities and the PN metric are given in Eqs. (4.7) of Ref. [95]. Furthermore,
the computation of the general expressions of the source moments requires the
introduction of a regularization. In this paper, we use the Hadamard partie
finie regularization, denoted by FP in which we introduce the scale constant
$r_{0}$ [96]. The integral over $z$ can be easily integrated after performing
a PN expansion

$\int^{1}_{-1}dz~{}\delta_{\ell}(z)\,\Sigma(\mathbf{x},t+z\,r/c)=\sum_{k=0}^{+\infty}\,\frac{(2\ell+1)!!}{(2k)!!(2\ell+2k+1)!!}\,\left(\frac{r}{c}\right)^{2k}\\!\Sigma^{(2k)}(\mathbf{x},t)\,.$
(19w)

For the computation of the radiative moments at the order we aim at, we also
need to compute another set of moments called gauge moments
$(W_{L},X_{L},Y_{L},Z_{L})$. They are required to low order and admit similar
definitions that can be found, e.g., in Eqs. (125) of Ref. [88].

From Eq. (3), we see for example that
$\Sigma_{ij}=|g|\sigma_{ij}+\tfrac{c^{4}}{16\pi G}\Lambda_{ij}$. This means
that the multipoles are sourced by the densities derived in the previous
section, as well as the non-linearities of the gravitational field
$\Lambda^{\mu\nu}$ through the PN metric. The relation between the source and
radiative moments, given in Sec. III.7, shows that the source moments are to
be computed at the same orders as the ones displayed in Table 1. This implies
that, looking at (19u), we require $\Sigma$ up to $\mathcal{O}(7)$,
$\Sigma_{i}$ to $\mathcal{O}(6)$ and $\Sigma_{ij}$ to $\mathcal{O}(4)$, which
is why the source densities were required to these orders.

###  III.4 Post-Newtonian metric and potentials

In the post-Newtonian framework, the metric is parametrized by a set of
elementary retarded-type potentials that satisfy sourced wave equations. In
this problem, we need the 3PN metric, which is given by [97]

$\displaystyle g_{00}$
$\displaystyle=-1+\frac{2}{c^{2}}V-\frac{2}{c^{4}}V^{2}+\frac{8}{c^{6}}\left(\hat{X}+V_{k}V_{k}+\frac{V^{3}}{6}\right)$
$\displaystyle\qquad+\frac{16}{c^{8}}\left(-\frac{V^{4}}{24}-VV_{k}V_{k}-V\hat{X}+2\hat{R}_{k}V_{k}+2\hat{T}\right)+\mathcal{O}(10)\,,$
(19xa) $\displaystyle g_{0i}$
$\displaystyle=-\frac{4}{c^{3}}V_{i}-\frac{8}{c^{5}}\hat{R}_{i}-\frac{8}{c^{7}}\left(V^{2}V_{i}+V_{k}\hat{W}_{ik}+2\hat{Y}_{i}\right)+\mathcal{O}(9)\,,$
(19xb) $\displaystyle g_{ij}$
$\displaystyle=\delta_{ij}\left[1+\frac{2}{c^{2}}V+\frac{2}{c^{4}}V^{2}+\frac{8}{c^{6}}\left(\hat{X}+V_{k}V_{k}+\frac{V^{3}}{6}\right)\right]+\frac{4}{c^{4}}\hat{W}_{ij}$
$\displaystyle\qquad+\frac{8}{c^{6}}\left(-2V_{i}V_{j}+V\hat{W}_{ij}+2\hat{Z}_{ij}\right)+\mathcal{O}(8)\,,$
(19xc)

where the potentials are defined below. Note that an extension of this
parametrization is known at 4PN together with the definition of the potentials
required to this order. They are displayed in Appendix A of Ref. [98].

However, in the present paper, we do not need to use the full metric. Instead,
we need to know the source of the multipole moments to consistent order. It
turns out that the only potentials that have a non-zero contribution to the
spin effects in the source multipole moments are the following

$\displaystyle V$ $\displaystyle=\Box_{\mathcal{R}}^{-1}[-4\pi G\,\sigma]\;,$
(19ya) $\displaystyle V_{i}$ $\displaystyle=\Box_{\mathcal{R}}^{-1}[-4\pi
G\,\sigma_{i}]\,,$ (19yb) $\displaystyle\hat{W}_{ij}$
$\displaystyle=\Box_{{\cal R}}^{-1}\left[-4\pi
G\,(\sigma_{ij}-\delta_{ij}\sigma_{kk})-\partial_{i}V\partial_{j}V\right]\,,$
(19yc) $\displaystyle\hat{R}_{i}$
$\displaystyle=\Box_{\mathcal{R}}^{-1}\bigg{[}-4\pi
G\,(V\sigma_{i}-V_{i}\sigma)-2\partial_{k}V\partial_{i}V_{k}-\frac{3}{2}\partial_{t}V\partial_{i}V\bigg{]}\,,$
(19yd) $\displaystyle\hat{X}$
$\displaystyle=\Box_{\mathcal{R}}^{-1}\bigg{[}\vphantom{\frac{1}{2}}-4\pi
G\,V\sigma_{ii}+\hat{W}_{ij}\partial_{ij}V+2V_{i}\partial_{t}\partial_{i}V+V\partial_{t}^{2}V+\frac{3}{2}(\partial_{t}V)^{2}-2\partial_{i}V_{j}\partial_{j}V_{i}\bigg{]}\,,$
(19ye) $\displaystyle\hat{Z}_{ij}$
$\displaystyle=\Box^{-1}_{\mathcal{R}}\bigg{[}-4\pi
G(\sigma_{ij}-\delta_{ij}\sigma_{kk})V-2\partial_{(i}V\partial_{t}V_{j)}+\partial_{i}V_{k}\partial_{j}V_{k}+\partial_{k}V_{i}\partial_{k}V_{j}-2\partial_{(i}V_{k}\partial_{k}V_{j)}$
$\displaystyle\qquad~{}-\delta_{ij}\partial_{k}V_{l}(\partial_{k}V_{l}-\partial_{l}V_{k})-\frac{3}{4}\delta_{ij}(\partial_{t}V)^{2}\bigg{]}\,,$
(19yf)

where $\Box^{-1}_{\mathcal{R}}$ refers to the retarded flat d’Alembertian.
More specifically, we are interested in the spin contributions to these
potentials as their point-particle part are already known. As we can see, the
potentials are sourced by the source densities derived in the previous section
as well as other simpler potentials.

To find the specific orders at which we require the potentials, one has to
look at the order at which the potentials enter in $\Sigma$, $\Sigma_{i}$ and
$\Sigma_{ij}$ including also the order of appearance in the source densities
$\sigma$, $\sigma_{i}$ and $\sigma_{ij}$. The orders required for each
interaction are displayed in Table 2.

Table 2:  Orders at which the potentials are required for the spin contributions in the full gravitational waveform at 3.5PN. LO refers to the leading order of the contribution of the interaction while RO refers to the relative order required.  Potential | SO | SS
---|---|---
|  LO |  RO |  LO |  RO
$V$ |  3 |  2 |  0 |  0
$V_{i}$ |  1 |  3 |  0 |  0
$\hat{W}_{ij}$ |  1 |  2 |  - |  -
$\hat{R}_{i}$ |  1 |  1 |  - |  -
$\hat{X}$ |  1 |  0 |  - |  -
$\hat{Z}_{ij}$ |  1 |  0 |  - |  -

Note that the SSS interaction does not appear in Table 2 because it will play
no role in the expressions of the source moments. More details regarding this
affirmation are given in the following section. The regularization scheme used
to compute the potentials is the Hadamard regularization [99]. The methods
used to compute the spin part of the potentials to these orders are detailed
in, e.g., Sec. IV of Ref. [50]. An interesting feature of the SS interaction
is that, as explained in Ref. [72], the potentials $V$ and $V_{i}$ contain
distributional terms which are crucial to take into account for the
computation of the source multipole moments.

###  III.5 Integration of the source multipole moments

This step in the overall project is the most technical one. However, no new
methods were required and we used the same computational techniques as, in
e.g. Refs. [98, 50, 72, 100]. Thus, we refer to these articles for more
details.

As discussed above, the $\Sigma$’s are composed of the source densities and
the non-linearities of the gravitational field whose dependency comes through
the potentials. Thus, the multipole moments are integrals sourced by three
types of terms: the compact terms, the non-compact terms and the surface
terms. Note that the sources of the moments can be written in different ways.
For example a term of the form
$\int\mathrm{d}^{3}\mathbf{x}\,\hat{x}_{L}\partial_{k}V\partial_{k}V$ can be
re-written, dropping here the FP notation, as

$\int\mathrm{d}^{3}\mathbf{x}\,\hat{x}_{L}\partial_{k}V\partial_{k}V=-4\pi
G\int\mathrm{d}^{3}\mathbf{x}\,\hat{x}_{L}\sigma
V+\frac{1}{2}\int\mathrm{d}^{3}\mathbf{x}\,\hat{x}_{L}\Delta(V^{2})+\mathcal{O}(2).$
(19ah)

We see that the left-hand side, which is a non-compact term, can be turned
into a sum of a compact and a so-called surface term222The second term of the
right-hand side is called a surface term because we only need to know the
expansion in $r$ of $V^{2}$ when $r\rightarrow\infty$ to compute it.. The
multipole moments have been independently computed and double checked using
different formulations for their sources. It is crucial to take into account
the distributional derivatives induced in the left-hand side of (19ah) in
order to recover the value of the right-hand side.

At this stage, after integrating Eqs. (19u) for required $\ell$ to consistent
order, we obtain the expressions of the source moments in a general frame for
arbitrary orbits. In particular, we assumed neither the aligned-spin nor the
quasi-circular orbits conditions. The lengthy expressions of the source
moments are not displayed in this paper, however we recall here their leading
PN order for the spin contributions [35]

$\displaystyle\left(I_{L}\right)_{\rm NS}$
$\displaystyle=m_{1}y_{1}^{<L>}+1\leftrightarrow 2+\mathcal{O}(2)\,,$ (19aia)
$\displaystyle\left(J_{L}\right)_{\rm NS}$
$\displaystyle=y_{1}^{a}v_{1}^{b}\varepsilon^{ab<i_{\ell}}y_{1}^{L-1>}+1\leftrightarrow
2+\mathcal{O}(2)\,,$ (19aib) $\displaystyle\left(I_{L}\right)_{\rm SO}$
$\displaystyle=\frac{2\ell}{c^{3}(\ell+1)}\left[\ell
v_{1}^{a}S_{1}^{b}\varepsilon^{ab<i_{\ell}}y_{1}^{L-1>}-(\ell-1)y_{1}^{a}S_{1}^{b}\varepsilon^{ab<i_{\ell}}v_{1}^{i_{\ell-1}}y_{1}^{L-2>}\right]+1\leftrightarrow
2+\mathcal{O}(5)\,,$ (19aic) $\displaystyle\left(J_{L}\right)_{\rm SO}$
$\displaystyle=\frac{\ell+1}{2c}S_{1}^{<i_{\ell}}y_{1}^{L-1>}+1\leftrightarrow
2+\mathcal{O}(3)\,,$ (19aid) $\displaystyle\left(I_{L}\right)_{\rm SS}$
$\displaystyle=-\frac{\ell(\ell-1)\kappa_{1}}{2m_{1}c^{4}}S_{1}^{<i_{\ell}}S_{1}^{i_{\ell-1}}y_{1}^{L-2>}+1\leftrightarrow
2+\mathcal{O}(6)\,,$ (19aie) $\displaystyle\left(J_{L}\right)_{\rm SS}$
$\displaystyle=\frac{(\ell-1)\kappa_{1}}{2m_{1}c^{4}}\left[2v_{1}^{a}S_{1}^{b}\varepsilon^{ab<i_{\ell}}S_{1}^{i_{\ell-1}}y_{1}^{L-2>}-(\ell-2)y_{1}^{a}v_{1}^{b}\varepsilon^{ab<i_{\ell}}S_{1}^{i_{\ell-1}}S_{1}^{i_{\ell-2}}y_{1}^{L-3>}\right]$
$\displaystyle\quad+1\leftrightarrow 2+\mathcal{O}(6)\,,$ (19aif)
$\displaystyle\left(I_{L}\right)_{\rm SSS}$
$\displaystyle=\frac{\ell(\ell-1)(\ell-2)\lambda_{1}}{3(\ell+1)m_{1}^{2}c^{7}}\left[-\ell
v_{1}^{a}S_{1}^{b}\varepsilon^{ab<i_{\ell}}S_{1}^{i_{\ell-1}}S_{1}^{i_{\ell-2}}y_{1}^{L-3>}\right.$
$\displaystyle\qquad\qquad\left.+(\ell-3)y_{1}^{a}S_{1}^{b}\varepsilon^{ab<i_{\ell}}v_{1}^{i_{\ell-1}}S_{1}^{i_{\ell-1}}S_{1}^{i_{\ell-2}}v_{1}^{i_{\ell-3}}y_{1}^{L-4>}\right]+1\leftrightarrow
2+\mathcal{O}(8)\,,$ (19aig) $\displaystyle\left(J_{L}\right)_{\rm SSS}$
$\displaystyle=-\frac{(\ell+1)(\ell-1)(\ell-2)\lambda_{1}}{12m_{1}^{2}c^{5}}S_{1}^{<i_{\ell}}S_{1}^{i_{\ell-1}}S_{1}^{i_{\ell-2}}y_{1}^{L-3>}+1\leftrightarrow
2+\mathcal{O}(6)\,,$ (19aih)

where NS refers to the non-spinning contributions. As stated in the previous
section, the SSS interaction plays no role at the level of the radiative
moments. Indeed, the only mass-type moment in which the SSS appears is for
$\ell=2$ which vanishes due to the $\ell-2$ factor and similarly for the
current quadrupole. The source current octupole has a non-zero contribution.
However, to obtain the radiative current octupole, one has to perform a time
derivative which operates on the spin vector. Since
$\dot{\bm{S}}_{A}=\mathcal{O}(2)$, the SSS contribution to the radiative
moment is higher order.

###  III.6 Reduction to quasi-circular orbits in the CM frame for aligned
spins

In this section, we reduce the expressions of the source moments in the CM
frame in the quasi-circular orbits approximation without precession. The first
step is to express them in the CM frame. We define as usual the quantities
$M=m_{1}+m_{2}$, $\delta=(m_{1}-m_{2})/M$, $\nu=m_{1}m_{2}/M^{2}$,
$r=|\bm{y}_{1}-\bm{y}_{2}|$, $\bm{n}=(\bm{y}_{1}-\bm{y}_{2})/r$,
$\bm{v}=\bm{v}_{1}-\bm{v}_{2}$, $\kappa_{\pm}=\kappa_{1}\pm\kappa_{2}$ and
$\lambda_{\pm}=\lambda_{1}\pm\lambda_{2}$. The CM frame is defined as the
frame in which the CM position of the system $G^{i}$ vanishes. It allows us to
express the positions and velocities of the two compact objects in terms of
the dynamical variables of the system

$\displaystyle\bm{y}_{1}^{\text{CM}}$
$\displaystyle=\frac{m_{2}}{M}r\bm{n}+\bm{z}\,,$ (19aja)
$\displaystyle\bm{y}_{2}^{\text{CM}}$
$\displaystyle=-\frac{m_{1}}{M}r\bm{n}+\bm{z}\,,$ (19ajb)

and similarly for the velocities. The function $\bm{z}$ is a higher order
quantity known from previous works [51, 72, 35]. We also introduce the
following combinations of the individual spins

$\displaystyle\bm{S}$ $\displaystyle=\bm{S}_{1}+\bm{S}_{2}\,,$ (19aka)
$\displaystyle\bm{\Sigma}$
$\displaystyle=\frac{M}{m_{2}}\bm{S}_{2}-\frac{M}{m_{1}}\bm{S}_{1}\,.$ (19akb)

With these relations in hand, we obtain the source multipoles in the CM frame.

The next step is to impose the aligned-spins condition which drastically
simplifies the computations. In particular, this implies that the orbital
motion remains planar and that the two individual spins are aligned with the
total angular momentum of the system. We can define the unitary vector
$\bm{\ell}=\bm{n}\times\bm{v}/|\bm{n}\times\bm{v}|$. In the absence of
precession, $\bm{\ell}$ is constant and coincides with $\bm{e}_{Z}$ defined in
Sec. II.1. The spin combinations $\bm{S}$ and $\bm{\Sigma}$ are also directed
along $\bm{\ell}$ such that

$\displaystyle\bm{S}=S_{\ell}\bm{\ell}\,,$ (19ala)
$\displaystyle\bm{\Sigma}=\Sigma_{\ell}\bm{\ell}\,.$ (19alb)

The last step is the reduction to quasi-circular orbits. As for the non-
spinning case, the acceleration $\bm{a}=\bm{a}_{1}-\bm{a}_{2}$ is directed
along $\bm{n}$ as

$\bm{a}=-r\omega^{2}\bm{n}\,,$ (19am)

which defines the orbital frequency $\omega$. Note that we neglect the
radiation reaction force in the EOM because we are interested in the spin
contributions to the acceleration. The spin contributions to the radiation
reaction term is at least of order $\mathcal{O}(8)$, which is a higher order
than required. The EOM (19am) are known from previous works [72, 35]. In this
approximation, $\bm{v}=-r\omega\bm{\lambda}$ where $\bm{\lambda}$ completes
the time-dependent orthonormal basis $(\bm{n},\bm{\lambda},\bm{\ell})$ as
$\bm{\lambda}=\bm{\ell}\times\bm{n}$. With this parametrization,

$\displaystyle\bm{n}(t)$
$\displaystyle=\cos\phi(t)\,\bm{e}_{X}+\sin\phi(t)\,\bm{e}_{Y}\,,$ (19ana)
$\displaystyle\bm{\lambda}(t)$
$\displaystyle=-\sin\phi(t)\,\bm{e}_{X}+\cos\phi(t)\,\bm{e}_{Y}\,,$ (19anb)
$\displaystyle\bm{\ell}(t)$ $\displaystyle=\bm{e}_{Z}\,,$ (19anc)

where $\phi$ is the phase and is given by $\phi=\int\mathrm{d}t\,\omega$.
After defining the usual PN quantities

$\gamma=\frac{GM}{rc^{2}}\,,\qquad
x=\left(\frac{GM\omega}{c^{3}}\right)^{2/3},$ (19ao)

we can read the expression of $\omega^{2}$ in terms of $\gamma$. Then, we
invert this relation to obtain $\gamma$ in terms of $\omega$ and thus $x$
which leads to

$\gamma=x\left[1+xg_{\text{NS}}+x^{3/2}\frac{g_{\text{SO}}}{GM^{2}}+x^{2}\frac{g_{\text{SS}}}{G^{2}M^{4}}+x^{7/2}\frac{g_{\text{SSS}}}{G^{3}M^{6}}+\mathcal{O}(8)\right]\,,$
(19ap)

where $g_{\text{NS}}$ and $g_{\text{SO}}$ are given in Eq. (4.3) of Ref. [51],
$g_{\text{SS}}$ in Eq. (3.32) of Ref. [72] and $g_{\text{SSS}}$ in Eq. (6.15)
of Ref. [35] at consistent order. This allows us to express the source
multipoles in terms of the orbital frequency and thus $x$.

###  III.7 Non-linear contributions to the radiative moments

Once the source multipoles are known in the CM, we follow rigorously the
procedure in Ref. [92] to compute the radiative moments. Without precession,
there are no additional technicalities and the problem is equivalent to point-
particle.

To link the radiative to the source moments, one has to introduce the
canonical moments $(M_{L},S_{L})$ which are related to the set of source and
gauge moments $(I_{L},J_{L},W_{L},X_{L},Y_{L},Z_{L})$ (see [92] for an account
of the procedure). The computation of the canonical moments is detailed in
Sec. III.7.1. Once the canonical moments are known, we can deduce the
radiative moments through the following relations

$\displaystyle U_{L}$ $\displaystyle=M_{L}^{(\ell)}+(\text{non-linear
terms}),$ (19aqa) $\displaystyle V_{L}$
$\displaystyle=S_{L}^{(\ell)}+(\text{non-linear terms}),$ (19aqb)

where the non-linear terms are at least of order $\mathcal{O}(3)$, so for
radiative multipoles that are required at a low order, their contribution
vanish. They are composed of three different types of contributions: the
instantaneous terms, the tail terms and the memory terms. The computation of
these different terms are detailed below.

####  III.7.1 Link between source and canonical moments

The relations between the canonical and source moments take the form

$\displaystyle M_{L}$ $\displaystyle=I_{L}+\delta I_{L}\,,$ (19ara)
$\displaystyle S_{L}$ $\displaystyle=J_{L}+\delta J_{L}\,,$ (19arb)

where $\delta I_{L}$ and $\delta J_{L}$ are non-linear corrections made of
products of source and gauge moments, and starting at the 2.5PN order. The
full expressions of $\delta I_{L}$ and $\delta J_{L}$ up to the 3.5PN order
are displayed in Sec. III. B. of Ref. [101]. Recently, the expression for
$\delta I_{ij}$ has been derived up to the 4PN order [102]. For the spin
contributions, we have explicitly

$\displaystyle\left[\delta I_{ij}\right]_{S}$
$\displaystyle=\mathcal{O}(8)\,,$ (19asa) $\displaystyle\left[\delta
I_{ijk}\right]_{S}$ $\displaystyle={12G\over c^{5}}\left[I_{\langle
ij}Y_{k\rangle}^{(1)}\right]_{S}+\mathcal{O}(8)\,,$ (19asb)
$\displaystyle\left[\delta J_{ij}\right]_{S}$ $\displaystyle={2G\over
c^{5}}\left[\epsilon_{ab\langle i}\left(-I^{(3)}_{j\rangle
b}W_{a}-2I_{j\rangle b}Y_{a}^{(2)}+I_{j\rangle
b}^{(1)}Y_{a}^{(1)}\right)+3J_{\langle
i}W_{j\rangle}^{(1)}\right]_{S}+\mathcal{O}(8)\,,$ (19asc)

where the spin parts of $W_{i}$, $Y_{i}$ and $J_{i}$ for aligned spins are
given by $[J_{i}]_{S}=S_{\ell}\ell^{i}/c$, $[W_{i}]_{S}=\nu
r\Sigma_{\ell}\lambda^{i}/4c$ and $[Y_{i}]_{S}=\nu
r\omega\Sigma_{\ell}n^{i}/4c$. With these relations in hand, it is very
straightforward to compute the canonical moments. They have been computed at
the same orders as those displayed in Table 1. After obtaining their
expression, one has to perform a time differentiation to obtain the linear
part of the radiative moments.

####  III.7.2 Instantaneous terms

At the 3.5PN order, the instantaneous terms are quadratic interactions of the
canonical moments and their complete expressions are displayed in Sec.
III.A.1. of Ref. [92]. The relevant spin contributions of these terms are the
following

$\displaystyle\left[U_{ij}^{\rm inst}\right]_{S}$
$\displaystyle=\frac{2G}{c^{5}}\left[\frac{1}{3}\varepsilon_{ab\langle
i}M_{j\rangle a}^{(4)}S_{b}\right]_{S}+\mathcal{O}(8)\,,$ (19ata)
$\displaystyle\left[U_{ijk}^{\rm inst}\right]_{S}$
$\displaystyle=\frac{G}{c^{5}}\left[{1\over 5}\epsilon_{ab\langle
i}\left(-12S^{(2)}_{j\underline{a}}M^{(3)}_{k\rangle
b}-8M^{(2)}_{j\underline{a}}S^{(3)}_{k\rangle
b}-3S^{(1)}_{j\underline{a}}M^{(4)}_{k\rangle
b}-27M^{(1)}_{j\underline{a}}S^{(4)}_{k\rangle
b}-S_{j\underline{a}}M^{(5)}_{k\rangle b}\vphantom{\frac{1}{2}}\right.\right.$
$\displaystyle\qquad\qquad\qquad\left.\left.-9M_{j\underline{a}}S^{(5)}_{k\rangle
b}-{9\over 4}S_{\underline{a}}M^{(5)}_{jk\rangle b}\right)+{12\over
5}S_{\langle i}S^{(4)}_{jk\rangle}\right]_{S}+\mathcal{O}(7)\,,$ (19atb)
$\displaystyle\left[V_{ij}^{\rm inst}\right]_{S}$ $\displaystyle={G\over
7\,c^{5}}\left[4S^{(2)}_{a\langle i}M^{(3)}_{j\rangle a}+8M^{(2)}_{a\langle
i}S^{(3)}_{j\rangle a}+17S^{(1)}_{a\langle i}M^{(4)}_{j\rangle
a}-3M^{(1)}_{a\langle i}S^{(4)}_{j\rangle a}+9S_{a\langle i}M^{(5)}_{j\rangle
a}\vphantom{\frac{1}{2}}\right.$ $\displaystyle\qquad\qquad\left.-3M_{a\langle
i}S^{(5)}_{j\rangle a}-{1\over 4}S_{a}M^{(5)}_{ija}-7\epsilon_{ab\langle
i}S_{\underline{a}}S^{(4)}_{j\rangle b}\right]_{S}+\mathcal{O}(7)\,,$ (19atc)
$\displaystyle\left[V_{ijk}^{\rm inst}\right]_{S}$
$\displaystyle=-\frac{2G}{c^{3}}\left[\mathrm{S}_{\langle
i}M^{(4)}_{jk\rangle}\right]_{S}+\mathcal{O}(6)\,,$ (19atd)
$\displaystyle\left[V_{ijkl}^{\rm inst}\right]_{S}$
$\displaystyle=\frac{G}{c^{3}}\left[-\frac{35}{3}S^{(2)}_{\langle
ij}M^{(3)}_{kl\rangle}-\frac{25}{3}M^{(2)}_{\langle
ij}S^{(3)}_{kl\rangle}-\frac{65}{6}S^{(1)}_{\langle
ij}M^{(4)}_{kl\rangle}-\frac{25}{6}M^{(1)}_{\langle
ij}S^{(4)}_{kl\rangle}-\frac{19}{6}S_{\langle ij}M^{(5)}_{kl\rangle}\right.$
$\displaystyle\left.\qquad\qquad-\frac{11}{6}M_{\langle
ij}S^{(5)}_{kl\rangle}-\frac{11}{12}S_{\langle
i}M^{(5)}_{jkl\rangle}\right]_{S}+\mathcal{O}(5)\,.$ (19ate)

They are treated as $\delta I_{L}$ and $\delta J_{L}$. In these equations,
$S_{i}$ and $S_{ij}$ correspond to the current canonical moments and not the
spin vector or tensor of the individual bodies. Note that these interactions
are to be evaluated at the retarded time $T_{R}=T-R/c$.

####  III.7.3 Tail terms

The tails correspond to time integrals over the past of the source. Their
expressions are known for each $\ell$. For the spin effects at the order
considered, we only need to compute the following ones

$\displaystyle\left[U_{ij}^{\rm tail}\right]_{S}$
$\displaystyle=\frac{2G\mathcal{M}}{c^{3}}\int^{+\infty}_{0}\mathrm{d}\tau\left[\ln\left(\frac{\tau}{2b}\right)+\frac{11}{12}\right]\left[M^{(4)}_{ij}(T_{R}-\tau)\right]_{S}\,,$
(19aua) $\displaystyle\left[U_{ijk}^{\rm tail}\right]_{S}$
$\displaystyle={2G\mathcal{M}\over
c^{3}}\int^{+\infty}_{0}\mathrm{d}\tau\left[\ln\left(\frac{\tau}{2b}\right)+{97\over
60}\right]\left[M^{(5)}_{ijk}(T_{R}-\tau)\right]_{S}\,,$ (19aub)
$\displaystyle\left[V_{ij}^{\rm tail}\right]_{S}$
$\displaystyle={2G\mathcal{M}\over
c^{3}}\int^{+\infty}_{0}\mathrm{d}\tau\left[\ln\left(\frac{\tau}{2b}\right)+{7\over
6}\right]\left[S^{(4)}_{ij}(T_{R}-\tau)\right]_{S}\,,$ (19auc)
$\displaystyle\left[V_{ijk}^{\rm tail}\right]_{S}$
$\displaystyle={2G\mathcal{M}\over
c^{3}}\int^{+\infty}_{0}\mathrm{d}\tau\left[\ln\left(\frac{\tau}{2b}\right)+{5\over
3}\right]\left[S^{(5)}_{ijk}(T_{R}-\tau)\right]_{S}\,,$ (19aud)
$\displaystyle\left[V_{ijkl}^{\rm tail}\right]_{S}$
$\displaystyle={2G\mathcal{M}\over
c^{3}}\int^{+\infty}_{0}\mathrm{d}\tau\left[\ln\left(\frac{\tau}{2b}\right)+{119\over
60}\right]\left[S^{(6)}_{ijkl}(T_{R}-\tau)\right]_{S}\,,$ (19aue)

where we introduced an arbitrary time-scale constant $b$. The mass monopole
$\mathcal{M}$, or ADM mass, differs from the total constant mass $M$ through
the relation $\mathcal{M}=M+\bar{E}/c^{2}$ where $\bar{E}$ is the conservative
binding energy of the system. This implies that the spin contributions in
$\mathcal{M}$ start at $\mathcal{O}(5)$ and thus do not need to be taken into
account in the computation of the tail terms as they are of higher order.

To compute the tail integrals, we consider the aligned-spin case. There are no
precession effects to consider here and the evolution of the dynamics of the
binary is qualitatively the same as for the usual quasi-circular orbits, with
the aligned conserved norm spins acting simply as constant vectors. This is to
be contrasted with the more general case of binaries on quasi-circular but
precessing orbits (as defined for instance in Ref. [74]), where one must solve
analytically the dynamics consistently with the order at which the analysis is
carried, to be able to compute these integrals. The idea to compute these
integrals in the non-precessing case is to project the moments in the spatial
basis $(\bm{e}_{X},\bm{e}_{Y},\bm{e}_{Z})$ defined in Sec. II.1 using the
relations (19an) and assuming that the separation $r$ is constant over
time333It has been shown [103, 104] that for the tail integrals, such an
approximation is valid because the remote past of the source is negligible
when compared to its recent past. On this time scale, $r$ does not have the
time to vary significantly and can be assumed constant.. By doing so, we find
one-dimensional integrals of the type
$\int_{0}^{\infty}\mathrm{d}y\ln(y)e^{\mathrm{i}\alpha y}$ that are computable
analytically [105].

####  III.7.4 Memory terms

The memory terms, as well as the tail terms, are called hereditary effects in
the sense that they are integrals over the past of the source. They are
integrals of quadratic interactions of canonical moments and only appear in
the mass-type multipoles [106]. In our case, only the mass octupole moment
contains a spin contribution to the memory terms according to

$\left[U_{ijk}^{\rm mem}\right]_{S}=-{4G\over
5c^{5}}\int^{+\infty}_{0}\mathrm{d}\tau\left[\epsilon_{ab\langle
i}\mathrm{M}^{(3)}_{j\underline{a}}(T_{R}-\tau)\mathrm{S}^{(3)}_{k\rangle
b}(T_{R}-\tau)\right]_{S}+\mathcal{O}(7)\,.$ (19av)

For the mass quadrupole, the spin contribution to the memory effects are of
order $\mathcal{O}(8)$ and thus do not need to be considered.

The computation of the memory terms differ from the one of the tails because
one cannot assume that the separation $r$ is constant. We have to take into
account the radiation reaction. As for the tails, we project the values of the
moments on the basis $(\bm{e}_{X},\bm{e}_{Y},\bm{e}_{Z})$. By doing so, we
encounter integrals of the type

$\int_{-\infty}^{T_{R}}\mathrm{d}\tau\frac{e^{\mathrm{i}n\phi(\tau)}}{r^{p}(\tau)},$
(19aw)

where $n$ is a non-zero integer, $p$ is a half-integer, and $\phi$ is the
phase variable defined in Sec. III.6, which satisfies $\dot{\phi}=\omega$. As
we can see, if we assumed the separation to be constant over time, this
integral would diverge. At leading order, the separation scales as
$r(\tau)\sim(-\tau)^{1/4}$ and the phase as $\phi(\tau)\sim(-\tau)^{5/8}$,
which allows to compute this integral, as detailed in Refs. [104, 92].

After combining all the previous intermediate results, we derived the
radiative multipole moments in the quasi-circular, spin-aligned approximations
at the orders displayed in Table 1. To obtain the amplitude spherical modes,
we insert the obtained expressions in Eq. (8) and then in Eq. (7).

##  IV Results

We now present the waveform modes. In Sec. IV.1, we express them in a PN
expansion following, e.g. [92, 101], whereas in Sec. IV.2, we factorize them
in a way that is suitable for the EOB approach and notably template building.

###  IV.1 Spin-weighted spherical modes

The amplitude modes defined in Eq. (7) can be written in terms of the phase
variable $\phi$ defined in Sec. III.6. However it is convenient to introduce a
new phase variable $\psi$ that allows factoring out the logarithm dependency
on the orbital frequency induced by the tail terms in the radiative moments.
The new phase variable reads

$\psi\equiv\phi-\frac{2G\mathcal{M}\omega}{c^{3}}\ln\left(\frac{\omega}{\omega_{0}}\right),$
(19ax)

where the constant $\omega_{0}$ is linked to the time scale constant $b$
introduced in Eqs. (19au) through
$\omega_{0}=\tfrac{1}{4b}\text{exp}[\tfrac{11}{12}-\gamma_{\text{E}}]$ and we
recall that $\mathcal{M}$ is the ADM mass. The amplitude modes then read

$\displaystyle h_{\ell
m}=\frac{2GM\,\nu\,x}{R\,c^{2}}\,\sqrt{\frac{16\pi}{5}}\,\hat{H}_{\ell
m}\,e^{-\mathrm{i}m\psi}\,,$ (19ay)

where we recall that

$x=\left(\frac{GM\omega}{c^{3}}\right)^{2/3}\,.$ (19az)

The spin part of the $\hat{H}_{\ell m}$ are given by

$\displaystyle\hat{H}_{22}^{\text{S}}$
$\displaystyle=\frac{x^{3/2}}{GM^{2}}\left[-2S_{\ell}-\tfrac{2}{3}\Sigma_{\ell}\delta+\bigl{(}S_{\ell}(-\tfrac{163}{63}-\tfrac{92}{63}\nu)+\Sigma_{\ell}\delta(-\tfrac{1}{21}+\tfrac{20}{63}\nu)\bigr{)}x\right.$
$\displaystyle\left.\qquad\qquad+\bigl{(}(-\tfrac{4}{3}\mathrm{i}-4\pi)S_{\ell}-\tfrac{4}{3}\pi\Sigma_{\ell}\delta\bigr{)}x^{3/2}\right.$
$\displaystyle\left.\qquad\qquad+\bigl{(}S_{\ell}(\tfrac{1061}{84}+\tfrac{4043}{84}\nu+\tfrac{499}{84}\nu^{2})+\Sigma_{\ell}\delta(\tfrac{3931}{756}+\tfrac{7813}{378}\nu+\tfrac{1025}{252}\nu^{2})\bigr{)}x^{2}\right]$
$\displaystyle+\frac{x^{2}}{G^{2}M^{4}}\left[S_{\ell}^{2}(2+\kappa_{+})+S_{\ell}\Sigma_{\ell}(2\delta-\kappa_{-}+\delta\kappa_{+})+\Sigma_{\ell}^{2}(-\tfrac{1}{2}\delta\kappa_{-}+\tfrac{1}{2}\kappa_{+}-2\nu-\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad+\bigl{(}S_{\ell}^{2}(-\tfrac{404}{63}+\tfrac{55}{42}\delta\kappa_{-}-\tfrac{31}{42}\kappa_{+}+\tfrac{68}{21}\nu+\tfrac{34}{21}\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad+S_{\ell}\Sigma_{\ell}(-\tfrac{481}{63}\delta+\tfrac{43}{21}\kappa_{-}-\tfrac{43}{21}\delta\kappa_{+}+\tfrac{68}{21}\delta\nu-\tfrac{48}{7}\kappa_{-}\nu+\tfrac{34}{21}\delta\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad+\Sigma_{\ell}^{2}(-\tfrac{5}{3}+\tfrac{43}{42}\delta\kappa_{-}-\tfrac{43}{42}\kappa_{+}+\tfrac{172}{21}\nu-\tfrac{89}{42}\delta\kappa_{-}\nu+\tfrac{25}{6}\kappa_{+}\nu-\tfrac{68}{21}\nu^{2}-\tfrac{34}{21}\kappa_{+}\nu^{2})\bigr{)}x\right.$
$\displaystyle\left.\qquad\qquad+\pi\bigl{(}S_{\ell}^{2}(4+2\kappa_{+})+S_{\ell}\Sigma_{\ell}(4\delta-2\kappa_{-}+2\delta\kappa_{+})\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Sigma_{\ell}^{2}(-\delta\kappa_{-}+\kappa_{+}-4\nu-2\kappa_{+}\nu)\bigr{)}x^{3/2}\right]$
$\displaystyle+\frac{x^{7/2}}{G^{3}M^{6}}\left[S_{\ell}^{3}(\tfrac{32}{3}-\tfrac{2}{3}\kappa_{+}-2\lambda_{+})+S_{\ell}^{2}\Sigma_{\ell}(\tfrac{52}{3}\delta-\tfrac{7}{3}\kappa_{-}-\tfrac{1}{3}\delta\kappa_{+}+3\lambda_{-}-3\delta\lambda_{+})\right.$
$\displaystyle\left.\qquad\qquad+S_{\ell}\Sigma_{\ell}^{2}(\tfrac{20}{3}-3\delta\kappa_{-}+3\kappa_{+}+3\delta\lambda_{-}-3\lambda_{+}-\tfrac{112}{3}\nu-\tfrac{2}{3}\kappa_{+}\nu+6\lambda_{+}\nu)\right.$
(19baa)
$\displaystyle\left.\qquad\qquad+\Sigma_{\ell}^{3}(-\tfrac{5}{3}\kappa_{-}+\tfrac{5}{3}\delta\kappa_{+}+\lambda_{-}-\delta\lambda_{+}-\tfrac{20}{3}\delta\nu+\tfrac{11}{3}\kappa_{-}\nu-\tfrac{1}{3}\delta\kappa_{+}\nu-3\lambda_{-}\nu+\delta\lambda_{+}\nu)\right]\,,$
$\displaystyle\hat{H}_{21}^{\text{S}}$
$\displaystyle=\frac{\mathrm{i}x}{2GM^{2}}\left[\Sigma_{\ell}+\bigl{(}-\tfrac{86}{21}S_{\ell}\delta+\Sigma_{\ell}(-\tfrac{79}{21}+\tfrac{139}{21}\nu)\bigr{)}x+\Sigma_{\ell}\bigl{(}-\tfrac{\mathrm{i}}{2}+\pi-2\mathrm{i}\ln(2)\bigr{)}x^{3/2}\right.$
$\displaystyle\left.\qquad\qquad+\bigl{(}S_{\ell}\delta(-\tfrac{331}{378}+\tfrac{772}{189}\nu)+\Sigma_{\ell}(\tfrac{293}{378}-\tfrac{2615}{756}\nu-\tfrac{1723}{189}\nu^{2})\bigr{)}x^{2}\right.$
$\displaystyle\left.\qquad\qquad+\Bigl{(}S_{\ell}\delta\bigl{(}\tfrac{181}{105}\mathrm{i}-\tfrac{86}{21}\pi+\tfrac{172}{21}\mathrm{i}\ln(2)\bigr{)}+\Sigma_{\ell}\bigl{(}\tfrac{79}{42}\mathrm{i}-\tfrac{79}{21}\pi-\tfrac{1951}{140}\mathrm{i}\nu\right.$
$\displaystyle\left.\qquad\qquad\qquad+\tfrac{257}{42}\pi\nu+\tfrac{158}{21}\mathrm{i}\ln(2)-\tfrac{257}{21}\mathrm{i}\nu\ln(2)\bigr{)}\Bigr{)}x^{5/2}\right]$
$\displaystyle+\frac{\mathrm{i}x^{5/2}}{G^{2}M^{4}}\left[S_{\ell}^{2}(\delta-\tfrac{1}{3}\kappa_{-}+\tfrac{1}{2}\delta\kappa_{+})+S_{\ell}\Sigma_{\ell}(-\tfrac{1}{3}-\tfrac{5}{6}\delta\kappa_{-}+\tfrac{5}{6}\kappa_{+}-4\nu-2\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad+\Sigma_{\ell}^{2}(-\tfrac{1}{2}\delta-\tfrac{5}{12}\kappa_{-}+\tfrac{5}{12}\delta\kappa_{+}-\delta\nu+\tfrac{4}{3}\kappa_{-}\nu-\tfrac{1}{2}\delta\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad+\bigl{(}S_{\ell}^{2}(\tfrac{41}{42}\delta+\tfrac{23}{48}\kappa_{-}+\tfrac{47}{336}\delta\kappa_{+}-\tfrac{1}{7}\delta\nu-\tfrac{191}{72}\kappa_{-}\nu-\tfrac{1}{14}\delta\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad+S_{\ell}\Sigma_{\ell}(-\tfrac{29}{21}+\tfrac{19}{56}\delta\kappa_{-}-\tfrac{19}{56}\kappa_{+}+\tfrac{100}{21}\nu-\tfrac{1301}{504}\delta\kappa_{-}\nu+\tfrac{1019}{504}\kappa_{+}\nu+\tfrac{4}{7}\nu^{2}+\tfrac{2}{7}\kappa_{+}\nu^{2})\right.$
$\displaystyle\left.\qquad\qquad+\Sigma_{\ell}^{2}(-\tfrac{6}{7}\delta+\tfrac{19}{112}\kappa_{-}-\tfrac{19}{112}\delta\kappa_{+}+\tfrac{59}{21}\delta\nu-\tfrac{751}{504}\kappa_{-}\nu+\tfrac{145}{126}\delta\kappa_{+}\nu+\tfrac{1}{7}\delta\nu^{2}\right.$
$\displaystyle\left.\qquad\qquad\qquad+\tfrac{1265}{504}\kappa_{-}\nu^{2}+\tfrac{1}{14}\delta\kappa_{+}\nu^{2})\bigr{)}x\right]$
(19bab)
$\displaystyle+\frac{\mathrm{i}x^{3}}{2G^{3}M^{6}}\left[S_{\ell}^{2}\Sigma_{\ell}(1+\tfrac{1}{2}\kappa_{+})+S_{\ell}\Sigma_{\ell}^{2}(\delta-\tfrac{1}{2}\kappa_{-}+\tfrac{1}{2}\delta\kappa_{+})+\Sigma_{\ell}^{3}(-\tfrac{1}{4}\delta\kappa_{-}+\tfrac{1}{4}\kappa_{+}-\nu-\tfrac{1}{2}\kappa_{+}\nu)\right]\,,$
$\displaystyle\hat{H}_{33}^{\text{S}}$
$\displaystyle=\frac{3\mathrm{i}\sqrt{15}x^{2}}{8\sqrt{14}GM^{2}}\left[7S_{\ell}\delta+\Sigma_{\ell}(3-9\nu)+\bigl{(}S_{\ell}\delta(-\tfrac{139}{15}+\tfrac{83}{15}\nu)+\Sigma_{\ell}(-\tfrac{43}{5}+24\nu+5\nu^{2})\bigr{)}x\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Bigl{(}S_{\ell}\delta\bigl{(}-\tfrac{213}{10}\mathrm{i}+21\pi-42\mathrm{i}\ln(2)+42\mathrm{i}\ln(3)\bigr{)}\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Sigma_{\ell}\bigl{(}-\tfrac{63}{5}\mathrm{i}+9\pi+\tfrac{8797}{270}\mathrm{i}\nu-27\pi\nu-18\mathrm{i}\ln(2)+54i\nu\ln(2)\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad+18\mathrm{i}\ln(3)-54\mathrm{i}\nu\ln(3)\bigr{)}\Bigr{)}x^{3/2}\right]$
$\displaystyle+\frac{3\mathrm{i}\sqrt{15}x^{5/2}}{8\sqrt{14}G^{2}M^{4}}\left[S_{\ell}^{2}\delta(-6-3\kappa_{+})+S_{\ell}\Sigma_{\ell}(-6+3\delta\kappa_{-}-3\kappa_{+}+24\nu+12\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Sigma_{\ell}^{2}(\tfrac{3}{2}\kappa_{-}-\tfrac{3}{2}\delta\kappa_{+}+6\delta\nu-6\kappa_{-}\nu+3\delta\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+\bigl{(}S_{\ell}^{2}(23\delta-\tfrac{7}{2}\kappa_{-}+\tfrac{15}{2}\delta\kappa_{+}-12\delta\nu+16\kappa_{-}\nu-6\delta\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+S_{\ell}\Sigma_{\ell}(26-11\delta\kappa_{-}+11\kappa_{+}-128\nu+22\delta\kappa_{-}\nu-52\kappa_{+}\nu+48\nu^{2}+24\kappa_{+}\nu^{2})\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Sigma_{\ell}^{2}(3\delta-\tfrac{11}{2}\kappa_{-}+\tfrac{11}{2}\delta\kappa_{+}-32\delta\nu+\tfrac{59}{2}\kappa_{-}\nu-\tfrac{37}{2}\delta\kappa_{+}\nu+12\delta\nu^{2}\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad-28\kappa_{-}\nu^{2}+6\delta\kappa_{+}\nu^{2})\bigr{)}x\right]\,,$
(19bac) $\displaystyle\hat{H}_{32}^{\text{S}}$
$\displaystyle=\frac{2\sqrt{5}x^{3/2}}{3\sqrt{7}GM^{2}}\left[S_{\ell}+\Sigma_{\ell}\delta+\bigl{(}S_{\ell}(-\tfrac{13}{2}+\tfrac{73}{6}\nu)+\Sigma_{\ell}\delta(-\tfrac{31}{6}+5\nu)\bigr{)}x\right.$
$\displaystyle\left.\qquad\qquad\qquad+\bigl{(}S_{\ell}(-\mathrm{i}+2\pi)+\Sigma_{\ell}\delta(-3\mathrm{i}+2\pi)\bigr{)}x^{3/2}\right.$
$\displaystyle\left.\qquad\qquad\qquad+\bigl{(}S_{\ell}(\tfrac{4859}{1320}-\tfrac{15413}{792}\nu-\tfrac{419}{88}\nu^{2})+\Sigma_{\ell}\delta(\tfrac{19241}{3960}-\tfrac{808}{55}\nu-\tfrac{16153}{3960}\nu^{2})\bigr{)}x^{2}\right]$
$\displaystyle+\frac{8\sqrt{5}x^{3}}{9\sqrt{7}G^{2}M^{4}}\left[S_{\ell}^{2}(-1-\tfrac{3}{8}\delta\kappa_{-}+\tfrac{9}{8}\kappa_{+}-\tfrac{9}{2}\nu-\tfrac{9}{4}\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+S_{\ell}\Sigma_{\ell}(-\tfrac{5}{2}\delta-\tfrac{3}{2}\kappa_{-}+\tfrac{3}{2}\delta\kappa_{+}-\tfrac{9}{2}\delta\nu+\tfrac{15}{4}\kappa_{-}\nu-\tfrac{9}{4}\delta\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Sigma_{\ell}^{2}(-\tfrac{3}{2}-\tfrac{3}{4}\delta\kappa_{-}+\tfrac{3}{4}\kappa_{+}+3\nu+\tfrac{3}{2}\delta\kappa_{-}\nu-3\kappa_{+}\nu+\tfrac{9}{2}\nu^{2}+\tfrac{9}{4}\kappa_{+}\nu^{2})\right]$
$\displaystyle+\frac{4\sqrt{5}x^{7/2}}{3\sqrt{7}G^{3}M^{6}}\left[S_{\ell}^{3}(1+\tfrac{1}{2}\kappa_{+})+S_{\ell}^{2}\Sigma_{\ell}(2\delta-\tfrac{1}{2}\kappa_{-}+\delta\kappa_{+})\right.$
$\displaystyle\left.\qquad\qquad\qquad+S_{\ell}\Sigma_{\ell}^{2}(1-\tfrac{3}{4}\delta\kappa_{-}+\tfrac{3}{4}\kappa_{+}-5\nu-\tfrac{5}{2}\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Sigma_{\ell}^{3}(-\tfrac{1}{4}\kappa_{-}+\tfrac{1}{4}\delta\kappa_{+}-\delta\nu+\kappa_{-}\nu-\tfrac{1}{2}\delta\kappa_{+}\nu)\right]\,,$
(19bad) $\displaystyle\hat{H}_{31}^{\text{S}}$
$\displaystyle=\frac{\mathrm{i}x^{2}}{24\sqrt{14}GM^{2}}\left[S_{\ell}\delta+\Sigma_{\ell}(5-15\nu)+\bigl{(}S_{\ell}\delta(-\tfrac{79}{9}+\tfrac{443}{9}\nu)+\Sigma_{\ell}(-\tfrac{149}{9}+\tfrac{700}{9}\nu-\tfrac{841}{9}\nu^{2})\bigr{)}x\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Bigl{(}S_{\ell}\delta\bigl{(}\tfrac{47}{10}\mathrm{i}+\pi-2\mathrm{i}\ln(2)\bigr{)}+\Sigma_{\ell}\bigl{(}-7\mathrm{i}+5\pi+\tfrac{11}{10}\mathrm{i}\nu-15\pi\nu\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad-10\mathrm{i}\ln(2)+30\mathrm{i}\nu\ln(2)\bigr{)}\Bigr{)}x^{3/2}\right]$
$\displaystyle+\frac{\mathrm{i}x^{5/2}}{4\sqrt{14}G^{2}M^{4}}\left[S_{\ell}^{2}(\delta-\tfrac{4}{3}\kappa_{-}+\tfrac{1}{2}\delta\kappa_{+})+S_{\ell}\Sigma_{\ell}(1-\tfrac{11}{6}\delta\kappa_{-}+\tfrac{11}{6}\kappa_{+}-4\nu-2\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Sigma_{\ell}^{2}(-\tfrac{11}{12}\kappa_{-}+\tfrac{11}{12}\delta\kappa_{+}-\delta\nu+\tfrac{7}{3}\kappa_{-}\nu-\tfrac{1}{2}\delta\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+\bigl{(}S_{\ell}^{2}(-\tfrac{149}{18}\delta+\tfrac{13}{4}\kappa_{-}-\tfrac{53}{36}\delta\kappa_{+}-\tfrac{22}{9}\delta\nu-\tfrac{16}{9}\kappa_{-}\nu-\tfrac{11}{9}\delta\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+S_{\ell}\Sigma_{\ell}(-\tfrac{115}{9}+\tfrac{85}{18}\delta\kappa_{-}-\tfrac{85}{18}\kappa_{+}+40\nu-\tfrac{5}{9}\delta\kappa_{-}\nu+\tfrac{58}{9}\kappa_{+}\nu+\tfrac{88}{9}\nu^{2}+\tfrac{44}{9}\kappa_{+}\nu^{2})\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Sigma_{\ell}^{2}(-\tfrac{9}{2}\delta+\tfrac{85}{36}\kappa_{-}-\tfrac{85}{36}\delta\kappa_{+}+\tfrac{100}{9}\delta\nu-\tfrac{233}{36}\kappa_{-}\nu+\tfrac{7}{4}\delta\kappa_{+}\nu+\tfrac{22}{9}\delta\nu^{2}\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad-\tfrac{2}{3}\kappa_{-}\nu^{2}+\tfrac{11}{9}\delta\kappa_{+}\nu^{2})\bigr{)}x\right]\,,$
(19bae) $\displaystyle\hat{H}_{44}^{\text{S}}$
$\displaystyle=\frac{32x^{5/2}}{9\sqrt{35}GM^{2}}\left[S_{\ell}(\tfrac{19}{3}-19\nu)+\Sigma_{\ell}\delta(3-6\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+\bigl{(}S_{\ell}(-\tfrac{437}{22}+\tfrac{10063}{132}\nu-\tfrac{971}{44}\nu^{2})+\Sigma_{\ell}\delta(-\tfrac{153}{11}+\tfrac{2165}{66}\nu+\tfrac{67}{44}\nu^{2})\bigr{)}x\right]$
$\displaystyle+\frac{8\sqrt{5}x^{3}}{9\sqrt{7}G^{2}M^{4}}\left[S_{\ell}^{2}(-4-2\kappa_{+}+12\nu+6\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+S_{\ell}\Sigma_{\ell}(-4\delta+2\kappa_{-}-2\delta\kappa_{+}+12\delta\nu-6\kappa_{-}\nu+6\delta\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Sigma_{\ell}^{2}(\delta\kappa_{-}-\kappa_{+}+4\nu-3\delta\kappa_{-}\nu+5\kappa_{+}\nu-12\nu^{2}-6\kappa_{+}\nu^{2})\right]\,,$
(19baf) $\displaystyle\hat{H}_{43}^{\text{S}}$
$\displaystyle=\frac{9\mathrm{i}\sqrt{5}x^{2}}{8\sqrt{14}GM^{2}}\left[-S_{\ell}\delta+\Sigma_{\ell}(-1+3\nu)+\bigl{(}S_{\ell}\delta(\tfrac{1303}{165}-\tfrac{1451}{165}\nu)+\Sigma_{\ell}(\tfrac{361}{55}-\tfrac{284}{11}\nu+\tfrac{163}{11}\nu^{2})\bigr{)}x\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Bigl{(}S_{\ell}\delta\bigl{(}\tfrac{53}{10}\mathrm{i}-3\pi+6\mathrm{i}\ln(2)-6\mathrm{i}\ln(3)\bigr{)}\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Sigma_{\ell}\bigl{(}\tfrac{32}{5}\mathrm{i}-3\pi-\tfrac{6007}{270}\mathrm{i}\nu+9\pi\nu+6\mathrm{i}\ln(2)-18\mathrm{i}\nu\ln(2)\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad-6\mathrm{i}\ln(3)+18\mathrm{i}\nu\ln(3)\bigr{)}\Bigr{)}x^{3/2}\right]$
$\displaystyle+\frac{9\mathrm{i}\sqrt{5}x^{7/2}}{8\sqrt{14}G^{2}M^{4}}\left[S_{\ell}^{2}(3\delta+\tfrac{3}{5}\kappa_{-}-\tfrac{8}{5}\delta\kappa_{+}+4\delta\nu-\tfrac{6}{5}\kappa_{-}\nu+2\delta\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+S_{\ell}\Sigma_{\ell}(6+\tfrac{11}{5}\delta\kappa_{-}-\tfrac{11}{5}\kappa_{+}-12\nu-\tfrac{16}{5}\delta\kappa_{-}\nu+\tfrac{48}{5}\kappa_{+}\nu-16\nu^{2}-8\kappa_{+}\nu^{2})\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Sigma_{\ell}^{2}(3\delta+\tfrac{11}{10}\kappa_{-}-\tfrac{11}{10}\delta\kappa_{+}-4\delta\nu-\tfrac{27}{5}\kappa_{-}\nu+\tfrac{16}{5}\delta\kappa_{+}\nu-4\delta\nu^{2}\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad+\tfrac{26}{5}\kappa_{-}\nu^{2}-2\delta\kappa_{+}\nu^{2})\right]\,,$
(19bag) $\displaystyle\hat{H}_{42}^{\text{S}}$
$\displaystyle=\frac{4x^{5/2}}{21\sqrt{5}GM^{2}}\left[S_{\ell}(-\tfrac{1}{9}+\tfrac{1}{3}\nu)+\Sigma_{\ell}\delta(1-2\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+\bigl{(}S_{\ell}(-\tfrac{43}{22}+\tfrac{6653}{396}\nu-\tfrac{1387}{44}\nu^{2})+\Sigma_{\ell}\delta(-\tfrac{313}{66}+\tfrac{3349}{198}\nu-\tfrac{725}{44}\nu^{2})\bigr{)}x\right]$
$\displaystyle+\frac{4x^{3}}{21\sqrt{5}G^{2}M^{4}}\left[S_{\ell}^{2}(\tfrac{5}{3}-\tfrac{5}{4}\delta\kappa_{-}+\tfrac{25}{12}\kappa_{+}-5\nu-\tfrac{5}{2}\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+S_{\ell}\Sigma_{\ell}(\tfrac{5}{3}\delta-\tfrac{10}{3}\kappa_{-}+\tfrac{10}{3}\delta\kappa_{+}-5\delta\nu+\tfrac{15}{2}\kappa_{-}\nu-\tfrac{5}{2}\delta\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Sigma_{\ell}^{2}(-\tfrac{5}{3}\delta\kappa_{-}+\tfrac{5}{3}\kappa_{+}-\tfrac{5}{3}\nu+\tfrac{5}{2}\delta\kappa_{-}\nu-\tfrac{35}{6}\kappa_{+}\nu+5\nu^{2}+\tfrac{5}{2}\kappa_{+}\nu^{2})\right]\,,$
(19bah) $\displaystyle\hat{H}_{41}^{\text{S}}$
$\displaystyle=\frac{\mathrm{i}\sqrt{5}x^{2}}{168\sqrt{2}GM^{2}}\left[S_{\ell}\delta+\Sigma_{\ell}(1-3\nu)+\bigl{(}S_{\ell}\delta(-\tfrac{1147}{165}+\tfrac{1139}{165}\nu)+\Sigma_{\ell}(-\tfrac{309}{55}+\tfrac{232}{11}\nu-\tfrac{111}{11}\nu^{2})\bigr{)}x\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Bigl{(}S_{\ell}\delta\bigl{(}-\tfrac{53}{30}\mathrm{i}+\pi-2\mathrm{i}\ln(2)\bigr{)}\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Sigma_{\ell}\bigl{(}-\tfrac{32}{15}\mathrm{i}+\pi+\tfrac{181}{30}\mathrm{i}\nu-3\pi\nu-2\mathrm{i}\ln(2)+6\mathrm{i}\nu\ln(2)\bigr{)}\Bigr{)}x^{3/2}\right]$
$\displaystyle+\frac{\mathrm{i}\sqrt{5}x^{7/2}}{168\sqrt{2}G^{2}M^{4}}\left[S_{\ell}^{2}(-3\delta-\tfrac{7}{5}\kappa_{-}+\tfrac{12}{5}\delta\kappa_{+}-4\delta\nu+\tfrac{14}{5}\kappa_{-}\nu-2\delta\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+S_{\ell}\Sigma_{\ell}(-6-\tfrac{19}{5}\delta\kappa_{-}+\tfrac{19}{5}\kappa_{+}+12\nu+\tfrac{24}{5}\delta\kappa_{-}\nu-\tfrac{72}{5}\kappa_{+}\nu+16\nu^{2}+8\kappa_{+}\nu^{2})\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Sigma_{\ell}^{2}(-3\delta-\tfrac{19}{10}\kappa_{-}+\tfrac{19}{10}\delta\kappa_{+}+4\delta\nu+\tfrac{43}{5}\kappa_{-}\nu-\tfrac{24}{5}\delta\kappa_{+}\nu+4\delta\nu^{2}\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad-\tfrac{34}{5}\kappa_{-}\nu^{2}+2\delta\kappa_{+}\nu^{2})\right]\,,$
(19bai) $\displaystyle\hat{H}_{55}^{\text{S}}$
$\displaystyle=\frac{3125\mathrm{i}x^{3}}{36\sqrt{66}GM^{2}}\left[S_{\ell}\delta(-\tfrac{1}{2}+\nu)+\Sigma_{\ell}(-\tfrac{1}{4}+\tfrac{5}{4}\nu-\tfrac{5}{4}\nu^{2})\right]$
$\displaystyle+\frac{3125\mathrm{i}x^{7/2}}{96\sqrt{66}G^{2}M^{4}}\left[S_{\ell}^{2}\delta(1+\tfrac{1}{2}\kappa_{+}-2\nu-\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+S_{\ell}\Sigma_{\ell}(1-\tfrac{1}{2}\delta\kappa_{-}+\tfrac{1}{2}\kappa_{+}-6\nu+\delta\kappa_{-}\nu-3\kappa_{+}\nu+8\nu^{2}+4\kappa_{+}\nu^{2})\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Sigma_{\ell}^{2}(-\tfrac{1}{4}\kappa_{-}+\tfrac{1}{4}\delta\kappa_{+}-\delta\nu+\tfrac{3}{2}\kappa_{-}\nu-\delta\kappa_{+}\nu+2\delta\nu^{2}-2\kappa_{-}\nu^{2}+\delta\kappa_{+}\nu^{2})\right]\,,$
(19baj) $\displaystyle\hat{H}_{54}^{\text{S}}$
$\displaystyle=\frac{32x^{5/2}}{3\sqrt{165}GM^{2}}\left[S_{\ell}(-1+3\nu)+\Sigma_{\ell}\delta(-1+2\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+\bigl{(}S_{\ell}(\tfrac{241}{26}-\tfrac{2939}{78}\nu+\tfrac{633}{26}\nu^{2})+\Sigma_{\ell}\delta(\tfrac{619}{78}-\tfrac{300}{13}\nu+\tfrac{817}{78}\nu^{2})\bigr{)}x\right]\,,$
(19bak) $\displaystyle\hat{H}_{53}^{\text{S}}$
$\displaystyle=\frac{3\mathrm{i}\sqrt{3}x^{3}}{4\sqrt{110}GM^{2}}\left[S_{\ell}\delta(\tfrac{1}{2}-\nu)+\Sigma_{\ell}(-\tfrac{3}{4}+\tfrac{15}{4}\nu-\tfrac{15}{4}\nu^{2})\right]$
$\displaystyle+\frac{9\mathrm{i}\sqrt{15}x^{7/2}}{32\sqrt{22}G^{2}M^{4}}\left[S_{\ell}^{2}(-\delta+\tfrac{4}{5}\kappa_{-}-\tfrac{13}{10}\delta\kappa_{+}+2\delta\nu-\tfrac{8}{5}\kappa_{-}\nu+\delta\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+S_{\ell}\Sigma_{\ell}(-1+\tfrac{21}{10}\delta\kappa_{-}-\tfrac{21}{10}\kappa_{+}+6\nu-\tfrac{13}{5}\delta\kappa_{-}\nu+\tfrac{39}{5}\kappa_{+}\nu-8\nu^{2}-4\kappa_{+}\nu^{2})\right.$
$\displaystyle\left.\qquad\qquad\qquad+\Sigma_{\ell}^{2}(\tfrac{21}{20}\kappa_{-}-\tfrac{21}{20}\delta\kappa_{+}+\delta\nu-\tfrac{47}{10}\kappa_{-}\nu+\tfrac{13}{5}\delta\kappa_{+}\nu-2\delta\nu^{2}+\tfrac{18}{5}\kappa_{-}\nu^{2}-\delta\kappa_{+}\nu^{2})\right]\,,$
(19bal) $\displaystyle\hat{H}_{52}^{\text{S}}$
$\displaystyle=\frac{2x^{5/2}}{9\sqrt{55}GM^{2}}\left[S_{\ell}(1-3\nu)+\Sigma_{\ell}\delta(1-2\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad+(S_{\ell}(-\tfrac{213}{26}+\tfrac{2519}{78}\nu-\tfrac{493}{26}\nu^{2})+\Sigma_{\ell}\delta(-\tfrac{535}{78}+\tfrac{244}{13}\nu-\tfrac{565}{78}\nu^{2})\bigr{)}x\right]\,,$
(19bam) $\displaystyle\hat{H}_{51}^{\text{S}}$
$\displaystyle=\frac{\mathrm{i}x^{3}}{216\sqrt{385}GM^{2}}\left[S_{\ell}\delta(1-2\nu)+\Sigma_{\ell}(\tfrac{7}{2}-\tfrac{35}{2}\nu+\tfrac{35}{2}\nu^{2})\right]$
$\displaystyle+\frac{\mathrm{i}\sqrt{5}x^{7/2}}{288\sqrt{77}G^{2}M^{4}}\left[S_{\ell}^{2}(\delta-\tfrac{6}{5}\kappa_{-}+\tfrac{17}{10}\delta\kappa_{+}-2\delta\nu+\tfrac{12}{5}\kappa_{-}\nu-\delta\kappa_{+}\nu)\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad+S_{\ell}\Sigma_{\ell}(1-\tfrac{29}{10}\delta\kappa_{-}+\tfrac{29}{10}\kappa_{+}-6\nu+\tfrac{17}{5}\delta\kappa_{-}\nu-\tfrac{51}{5}\kappa_{+}\nu+8\nu^{2}+4\kappa_{+}\nu^{2})\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad+\Sigma_{\ell}^{2}(-\tfrac{29}{20}\kappa_{-}+\tfrac{29}{20}\delta\kappa_{+}-\delta\nu+\tfrac{63}{10}\kappa_{-}\nu-\tfrac{17}{5}\delta\kappa_{+}\nu+2\delta\nu^{2}\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad-\tfrac{22}{5}\kappa_{-}\nu^{2}+\delta\kappa_{+}\nu^{2})\right]\,,$
(19ban) $\displaystyle\hat{H}_{66}^{\text{S}}$
$\displaystyle=-\frac{3132x^{7/2}}{35\sqrt{143}GM^{2}}\left[S_{\ell}(1-5\nu+5\nu^{2})+\Sigma_{\ell}\delta(\tfrac{15}{29}-\tfrac{60}{29}\nu+\tfrac{45}{29}\nu^{2})\right]\,,$
(19bao) $\displaystyle\hat{H}_{65}$
$\displaystyle=\frac{3125\mathrm{i}x^{3}}{144\sqrt{429}GM^{2}}\left[S_{\ell}\delta(1-2\nu)+\Sigma_{\ell}(1-5\nu+5\nu^{2})\right]\,,$
(19bap) $\displaystyle\hat{H}_{64}^{\text{S}}$
$\displaystyle=\frac{256\sqrt{2}x^{7/2}}{385\sqrt{39}GM^{2}}\left[S_{\ell}(1-5\nu+5\nu^{2})+\Sigma_{\ell}\delta(-\tfrac{5}{9}+\tfrac{20}{9}\nu-\tfrac{5}{3}\nu^{2})\right]\,,$
(19baq) $\displaystyle\hat{H}_{63}^{\text{S}}$
$\displaystyle=-\frac{81\mathrm{i}x^{3}}{176\sqrt{65}GM^{2}}\left[S_{\ell}\delta(1-2\nu)+\Sigma_{\ell}(1-5\nu+5\nu^{2})\right]\,,$
(19bar) $\displaystyle\hat{H}_{62}^{\text{S}}$
$\displaystyle=\frac{4x^{7/2}}{693\sqrt{65}GM^{2}}\left[S_{\ell}(1-5\nu+5\nu^{2})+\Sigma_{\ell}\delta\left(\tfrac{17}{3}-\tfrac{68}{3}\nu+17\nu^{2}\right)\right]\,,$
(19bas) $\displaystyle\hat{H}_{61}^{\text{S}}$
$\displaystyle=\frac{\mathrm{i}x^{3}}{2376\sqrt{26}GM^{2}}\left[S_{\ell}\delta(1-2\nu)+\Sigma_{\ell}(1-5\nu+5\nu^{2})\right]\,,$
(19bat) $\displaystyle\hat{H}_{77}^{\text{S}}$ $\displaystyle=0\,,$ (19bau)
$\displaystyle\hat{H}_{76}^{\text{S}}$
$\displaystyle=\frac{324\sqrt{3}x^{7/2}}{35\sqrt{143}GM^{2}}\left[S_{\ell}(1-5\nu+5\nu^{2})+\Sigma_{\ell}\delta(1-4\nu+3\nu^{2})\right]\,,$
(19bav) $\displaystyle\hat{H}_{75}^{\text{S}}$ $\displaystyle=0\,,$ (19baw)
$\displaystyle\hat{H}_{74}^{\text{S}}$
$\displaystyle=-\frac{512\sqrt{2}x^{7/2}}{1365\sqrt{33}GM^{2}}\left[S_{\ell}(1-5\nu+5\nu^{2})+\Sigma_{\ell}\delta(1-4\nu+3\nu^{2})\right]\,,$
(19bax) $\displaystyle\hat{H}_{73}^{\text{S}}$ $\displaystyle=0\,,$ (19bay)
$\displaystyle\hat{H}_{72}^{\text{S}}$
$\displaystyle=\frac{4x^{7/2}}{3003\sqrt{3}GM^{2}}\left[S_{\ell}(1-5\nu+5\nu^{2})+\Sigma_{\ell}\delta(1-4\nu+3\nu^{2})\right]\,,$
(19baz) $\displaystyle\hat{H}_{71}^{\text{S}}$ $\displaystyle=0\,.$ (19baaa)

Furthermore, for all $\ell$, we have $\hat{H}^{\text{S}}_{\ell 0}=0$. The
modes are in agreement with the literature, and notably with the modes derived
for a test particle around a Kerr black hole [107, 79].

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

Conveniently for EOB waveform models, we write the PN-expanded waveform given
by Eqs. (19ay)–(19ba) in a factorized, resummed form as [77, 78, 79, 80]

$h_{\ell m}^{\text{F}}=h_{\ell
m}^{(N,\epsilon_{p})}\hat{S}_{\text{eff}}^{(\epsilon_{p})}T_{\ell
m}e^{\mathrm{i}\delta_{\ell m}}f_{\ell m}\,,$ (19bb)

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. The first term $h_{\ell
m}^{(N,\epsilon_{p})}$ is the leading (Newtonian) order waveform, which is
known for any ($\ell,m$) [90, 91], and its explicit expression is given in,
e.g., Eq. (3) of Ref. [79]. Note that the convention for the definition of the
waveform modes differs by a global minus sign between this paper and Ref.
[79]. However this difference only affects $h_{\ell m}^{(N,\epsilon_{p})}$,
while the other factors in Eq. (19bb) are not altered.

The (dimensionless) effective source term $\hat{S}_{\text{eff}}$ is given by
either the effective energy $E_{\text{eff}}$ or the orbital angular momentum
$p_{\phi}$, both expressed as functions of
$v\equiv\left(M\omega\right)^{1/3}=\sqrt{x}$, such that444In this section, we
use units in which $c=G=1$ to simplify the notation.

$\hat{S}_{\text{eff}}=\left\\{\begin{array}[]{ll}\frac{E_{\text{eff}}(v)}{\mu},&\quad\ell+m\text{
even}\\\ v\,\frac{p_{\phi}(v)}{\mu M},&\quad\ell+m\text{
odd}\end{array}\right.,$ (19bc)

where $\mu\equiv m_{1}m_{2}/M$ is the reduced mass, and $E_{\text{eff}}$ is
related to the total energy $E$ via the EOB energy map
$E=M\sqrt{1+2\nu\left(E_{\text{eff}}/\mu-1\right)}$. This source term is
motivated by the Regge-Wheeler-Zerilli equation [108, 109], whose source
depends on the stress-energy tensor for a test body in a Schwarzschild
background

The factor $T_{\ell m}$ resums the infinite number of “leading logarithms”
entering the tail effects [110, 111, 112], and is given by

$T_{\ell
m}=\frac{\Gamma\left(\ell+1-2i\hat{k}\right)}{\Gamma(\ell+1)}e^{\pi\hat{k}}e^{2i\hat{k}\ln(2m\omega
r_{0})},$ (19bf)

where $\Gamma(...)$ is the Euler gamma function, $\hat{k}\equiv m\omega E$,
$\omega$ is the orbital frequency, and the constant $r_{0}$ takes the value
$2M/\sqrt{e}$ to give agreement with waveforms computed in the test-body limit
[79].

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.
(19ay). To improve the agreement with numerical-relativity waveforms, $f_{\ell
m}$ is further resummed as [80, 79] $f_{\ell m}=(\rho_{\ell m})^{\ell}$ to
reduce the magnitude of the 1PN non-spinning coefficient, which grows linearly
with $\ell$. For spinning binaries, the non-spinning and spin contributions
are separated for the odd $m$ modes, such that

$\displaystyle f_{\ell m}=\left\\{\begin{array}[]{ll}\rho_{\ell
m}^{\ell},&\quad m\text{ even}\\\ (\rho_{\ell m}^{\text{NS}})^{\ell}+f_{\ell
m}^{\text{S}},&\quad m\text{ odd}\end{array}\right.,$ (19bi)

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

To simplify the expressions for the factorized modes, and to be consistent
with the notation used in the literature, we introduce the dimensionless
symmetric and antisymmetric spin parameters

$\displaystyle\chi_{S}$
$\displaystyle\equiv\frac{1}{2}\left(\chi_{1}+\chi_{2}\right)=\frac{S_{\ell}+\delta\Sigma_{\ell}}{2\nu
GM^{2}}\,,$ (19bja) $\displaystyle\chi_{A}$
$\displaystyle\equiv\frac{1}{2}\left(\chi_{1}-\chi_{2}\right)=-\frac{\delta
S_{\ell}+(1-2\nu)\Sigma_{\ell}}{2\nu GM^{2}},$ (19bjb)

and define the following combinations of the spin-multipole constants and
spins:

$\displaystyle\tilde{\kappa}_{S}$
$\displaystyle\equiv\frac{1}{2}\left[\chi_{1}^{2}(\kappa_{1}-1)+\chi_{2}^{2}(\kappa_{2}-1)\right],$
(19bka) $\displaystyle\tilde{\kappa}_{A}$
$\displaystyle\equiv\frac{1}{2}\left[\chi_{1}^{2}(\kappa_{1}-1)-\chi_{2}^{2}(\kappa_{2}-1)\right],$
(19bkb) $\displaystyle\tilde{\lambda}_{S}$
$\displaystyle\equiv\frac{1}{2}\left[\chi_{1}^{3}(\lambda_{1}-1)+\chi_{2}^{3}(\lambda_{2}-1)\right],$
(19bkc) $\displaystyle\tilde{\lambda}_{A}$
$\displaystyle\equiv\frac{1}{2}\left[\chi_{1}^{3}(\lambda_{1}-1)-\chi_{2}^{3}(\lambda_{2}-1)\right],$
(19bkd)

which equal zero for black holes.

For the $(2,2)$ mode, we obtain

$\displaystyle\rho_{22}$
$\displaystyle=1+v^{2}\left(\tfrac{55}{84}\nu-\tfrac{43}{42}\right)+v^{3}\left[\left(\tfrac{2}{3}\nu-\tfrac{2}{3}\right)\chi_{S}-\tfrac{2}{3}\delta\chi_{A}\right]+v^{4}\Big{[}\tfrac{19583}{42336}\nu^{2}-\tfrac{33025}{21168}\nu-\tfrac{20555}{10584}$
$\displaystyle\qquad+\left(\tfrac{1}{2}-2\nu\right)\chi_{A}^{2}+\delta\chi_{A}\chi_{S}+\tfrac{1}{2}\chi_{S}^{2}+\tfrac{1}{2}\delta\kappa_{A}+\kappa_{S}\left(\tfrac{1}{2}-\nu\right)\Big{]}$
$\displaystyle\quad+v^{5}\left[\delta\left(-\tfrac{19}{42}\nu-\tfrac{34}{21}\right)\chi_{A}+\left(\tfrac{209}{126}\nu^{2}+\tfrac{49}{18}\nu-\tfrac{34}{21}\right)\chi_{S}\right]$
$\displaystyle\quad+v^{6}\Big{[}\delta\left(\tfrac{89}{126}-\tfrac{781}{252}\nu\right)\chi_{A}\chi_{S}+\left(-\tfrac{27}{14}\nu^{2}-\tfrac{457}{504}\nu+\tfrac{89}{252}\right)\chi_{A}^{2}+\left(\tfrac{10}{9}\nu^{2}-\tfrac{1817}{504}\nu+\tfrac{89}{252}\right)\chi_{S}^{2}$
$\displaystyle\qquad+\left(\tfrac{67}{84}-\tfrac{139}{168}\nu\right)\delta\tilde{\kappa}_{A}+\left(-\tfrac{27}{28}\nu^{2}-\tfrac{407}{168}\nu+\tfrac{67}{84}\right)\tilde{\kappa}_{S}\Big{]}$
$\displaystyle\quad+v^{7}\Big{[}\delta\left(\tfrac{97865}{63504}\nu^{2}+\tfrac{50140}{3969}\nu+\tfrac{18733}{15876}\right)\chi_{A}+\left(\tfrac{50803}{63504}\nu^{3}-\tfrac{245717}{63504}\nu^{2}+\tfrac{74749}{5292}\nu+\tfrac{18733}{15876}\right)\chi_{S}$
$\displaystyle\qquad+\delta\chi_{A}^{3}\left(\tfrac{1}{3}-\tfrac{4}{3}\nu\right)+\delta(2\nu+1)\chi_{A}\chi_{S}^{2}+\left(-4\nu^{2}-3\nu+1\right)\chi_{A}^{2}\chi_{S}+\left(\nu+\tfrac{1}{3}\right)\chi_{S}^{3}$
$\displaystyle\qquad+\tilde{\kappa}_{S}\left[\left(\tfrac{1}{3}\nu+\tfrac{4}{3}\right)\delta\chi_{A}+\left(-2\nu^{2}-\tfrac{14}{3}\nu+\tfrac{4}{3}\right)\chi_{S}\right]+\tilde{\kappa}_{A}\left[\left(\tfrac{4}{3}-\tfrac{7}{3}\nu\right)\chi_{A}+\left(\tfrac{4}{3}-2\nu\right)\delta\chi_{S}\right]$
$\displaystyle\qquad+(\nu-1)\delta\tilde{\lambda}_{A}+(3\nu-1)\tilde{\lambda}_{S}\Big{]},$
(19bla) $\displaystyle\delta_{22}$ $\displaystyle=\tfrac{7}{3}\omega
E+\big{(}\omega
E\big{)}^{2}\left[\left(\tfrac{8}{3}\nu-\tfrac{4}{3}\right)\chi_{S}-\tfrac{4}{3}\delta\chi_{A}\right],$
(19blb)

where we only write the non-spinning part to the order needed for the 3.5PN
spin contributions. The energy $E$ in $\delta_{\ell m}$ is replaced by the
Hamiltonian in EOB waveform models.

For the $(2,1)$ mode, we obtain

$\displaystyle\rho_{21}^{\text{NS}}$
$\displaystyle=1+\left(\tfrac{23}{84}\nu-\tfrac{59}{56}\right)v^{2}+\left(\tfrac{617}{4704}\nu^{2}-\tfrac{10993}{14112}\nu-\tfrac{47009}{56448}\right)v^{4},$
(19bma) $\displaystyle f_{21}^{\text{S}}$
$\displaystyle=v\left(-\tfrac{3}{2}\tfrac{\chi_{A}}{\delta}-\tfrac{3}{2}\chi_{S}\right)+v^{3}\Big{[}\left(\tfrac{131}{84}\nu+\tfrac{61}{12}\right)\tfrac{\chi_{A}}{\delta}+\left(\tfrac{79}{84}\nu+\tfrac{61}{12}\right)\chi_{S}\Big{]}$
$\displaystyle\quad+v^{4}\left[(-2\nu-3)\chi_{A}^{2}+\left(\tfrac{21}{2}\nu-6\right)\frac{\chi_{A}\chi_{S}}{\delta}+\left(\tfrac{1}{2}\nu-3\right)\chi_{S}^{2}+\left(-\nu-\tfrac{1}{2}\right)\tilde{\kappa}_{S}-\frac{\tilde{\kappa}_{A}}{2\delta}\right]$
$\displaystyle\quad+v^{5}\Big{[}\left(-\tfrac{703}{112}\nu^{2}+\tfrac{8797}{1008}\nu-\tfrac{81}{16}\right)\frac{\chi_{A}}{\delta}+\left(\tfrac{613}{1008}\nu^{2}+\tfrac{1709}{1008}\nu-\tfrac{81}{16}\right)\chi_{S}$
$\displaystyle\qquad+\left(\tfrac{3}{4}-3\nu\right)\frac{\chi_{A}^{3}}{\delta}+\left(\tfrac{9}{4}-6\nu\right)\frac{\chi_{A}\chi_{S}^{2}}{\delta}+\left(\tfrac{9}{4}-3\nu\right)\chi_{A}^{2}\chi_{S}+\tfrac{3}{4}\chi_{S}^{3}$
$\displaystyle\qquad+\tfrac{3}{4}\tilde{\kappa}_{A}\chi_{A}+\left(\tfrac{3}{4}-3\nu\right)\frac{\tilde{\kappa}_{A}\chi_{S}}{\delta}+\left(\tfrac{3}{4}-\tfrac{3}{2}\nu\right)\frac{\tilde{\kappa}_{S}\chi_{A}}{\delta}+\left(\tfrac{3}{4}-\tfrac{3}{2}\nu\right)\tilde{\kappa}_{S}\chi_{S}\Big{]}$
$\displaystyle\quad+v^{6}\Big{[}\left(\tfrac{5}{7}\nu^{2}-\tfrac{9287}{1008}\nu+\tfrac{4163}{252}\right)\chi_{A}^{2}+\left(\tfrac{139}{72}\nu^{2}-\tfrac{2633}{1008}\nu+\tfrac{4163}{252}\right)\chi_{S}^{2}$
$\displaystyle\qquad+\left(\tfrac{9487}{504}\nu^{2}-\tfrac{1636}{21}\nu+\tfrac{4163}{126}\right)\frac{\chi_{A}\chi_{S}}{\delta}+\left(\tfrac{473}{84}\nu^{2}+\tfrac{8}{21}\nu+\tfrac{1}{21}\right)\frac{\tilde{\kappa}_{A}}{\delta}$
$\displaystyle\qquad+\left(\tfrac{5}{14}\nu^{2}+\tfrac{10}{21}\nu+\tfrac{1}{21}\right)\tilde{\kappa}_{S}\Big{]},$
(19bmb) $\displaystyle\delta_{21}$ $\displaystyle=\tfrac{2}{3}\omega
E-\tfrac{25}{2}\nu v^{5}+\big{(}\omega
E\big{)}^{2}\left[\left(-\tfrac{69}{140}\nu-\tfrac{17}{35}\right)\frac{\chi_{A}}{\delta}+\left(-\tfrac{41}{28}\nu-\tfrac{17}{35}\right)\chi_{S}\right].$
(19bmc)

We note that the $\mathcal{O}(v^{6}\chi^{2}\nu^{2})$ terms in the $(2,1)$ mode
disagree with those used in the SEOBNRv4HM model [87].555 The difference, for
black holes, is given by  $f_{21}^{\text{S,(this
paper)}}-f_{21}^{\text{S,({SEOBNRv4HM})}}=v^{6}\nu^{2}\left(\tfrac{165}{112}\chi_{A}^{2}+\tfrac{87}{56}\frac{\chi_{A}\chi_{S}}{\delta}+\tfrac{165}{112}\chi_{S}^{2}\right).$
(19bn) Those terms were used in SEOBNRv4HM based on unpublished results by one
of the authors of this paper. However, we checked that the SS contributions to
the (2,1) mode should be given by Eq. (19bm).

Interestingly, we also find a discrepancy with literature [80, 113] in the NS
part of $\delta_{21}$, which was required to derive the spin terms in the
factorized waveform. The difference is in the radiation reaction term
$\mathcal{O}(\nu v^{5})$ which, in these papers, has a coefficient $-493/42$.
After investigation, we found out that this value came from a wrong expression
for the (2,1) mode in Ref. [92], which was later corrected in an erratum, but
the factorized mode was never corrected. The coefficient -493/42 should read
-25/2 as we see in Eq. (19bmc).

Explicit expressions for the other modes are given in Appendix B. We also
provide all expressions as a Mathematica file in the Supplemental Materials
[89].

##  V Summary

We computed the spin contributions to the spherical-harmonics modes of the GW
polarizations to the 3.5PN order, for non-precessing spins in quasi-circular
orbits. We used the PN-MPM formalism to tackle the computation of the
radiative multipole moments, which were required to a higher multipolar order
than what was known in the literature. We also derived the spin contributions
to the hereditary tail terms, as well as other non-linear interactions between
the moments.

Our results include all spin terms, i.e., all SO, SS and SSS terms to that
order. We wrote the waveform modes in two forms: in the conventional PN-
expanded form, as well as a factorized form convenient for the EOB approach.
The factorized modes we obtained are in agreement with Refs. [79, 87] except
for the three terms $\mathcal{O}(v^{6}\chi^{2}\nu^{2})$ in the (2,1) mode used
in SEOBNRv4HM. As stated in Ref. [87], these terms came from private
communications and are now corrected. We also corrected a NS term in the
quantity $\delta_{21}$.

The results derived in this paper can be useful in improving analytical
waveform models. Interestingly, preliminary implementation of some of our
results in an EOB waveform model showed a good improvement when compared to
numerical relativity [114].

Future work will focus on relaxing the non-precessing approximation as well as
the quasi-circular-orbits condition, which is important to improve waveform
models for eccentric orbits and precessing spins.

## Acknowledgments

We are grateful to Luc Blanchet, Alejandro Bohé, Alessandra Buonanno, François
Larrouturou, Guillaume Faye, Maarten van de Meent, Jan Steinhoff and Justin
Vines for interesting discussions.

##  Appendix A Source densities

The explicit expressions of the spin part of source densities are given by

$\displaystyle\sigma^{\text{SO}}$
$\displaystyle=-\frac{4}{c^{7}\sqrt{-g}}\partial_{t}\left(\delta_{1}S_{1}^{ab}v_{1}^{a}V^{b}\right)+\frac{1}{\sqrt{-g}}\partial_{i}\left\\{\delta_{1}\left[-\frac{2}{c^{3}}S_{1}^{ia}v_{1}^{a}+\frac{4}{c^{5}}\left(S_{1}^{ia}V^{a}-S_{1}^{ia}Vv_{1}^{a}\right)\right.\right.$
$\displaystyle\left.\left.\qquad\qquad+\frac{4}{c^{7}}\left(2\hat{R}^{a}S_{1}^{ia}+S_{1}^{ia}(v_{1}^{k}V^{k})v_{1}^{a}-2S_{1}^{ia}V^{2}v_{1}^{a}+2S_{1}^{ia}VV^{a}-S_{1}^{ia}v_{1}^{b}\hat{W}_{ab}\right)\right]\right\\}$
$\displaystyle+\delta_{1}\left\\{\frac{2}{c^{5}}\left[S_{1}^{ab}v_{1}^{a}\partial^{b}V+2S_{1}^{ab}\partial^{a}V^{b}\right]+\frac{4}{c^{7}}\left[S_{1}^{ab}v_{1}^{a}\partial_{t}V^{b}+S_{1}^{bk}v_{1}^{a}v_{1}^{b}\partial_{a}V^{k}\right.\right.$
$\displaystyle\left.\left.+S_{1}^{ab}v_{1}^{k}\partial_{b}\hat{W}_{ka}-S_{1}^{ab}V^{a}\partial^{b}V-2S_{1}^{ab}\partial^{b}\hat{R}^{a}-2S_{1}^{ab}v_{1}^{a}V\partial^{b}V+4S_{1}^{ab}V\partial^{b}V^{a}\right]\right\\}$
$\displaystyle+1\leftrightarrow 2+\mathcal{O}(8),$ (19boa)
$\displaystyle\sigma^{\text{SS}}$
$\displaystyle=\frac{\kappa_{1}}{m_{1}\sqrt{-g}}\partial_{kl}\left\\{\delta_{1}\left[\frac{S_{1}^{ka}S_{1}^{la}}{2c^{4}}+\frac{1}{c^{6}}\left(S_{1}^{ka}S_{1}^{la}\left(\frac{3}{4}v_{1}^{2}+\frac{3}{2}V\right)-\frac{1}{2}S_{1}^{ka}S_{1}^{lb}v_{1}^{a}v_{1}^{b}\right.\right.\right.$
$\displaystyle\left.\left.\left.+\frac{1}{2}(S_{1}^{ab}S_{1}^{ab})v_{1}^{k}v_{1}^{l}+S_{1}^{ab}\left(-S_{1}^{lb}v_{1}^{a}v_{1}^{k}-S_{1}^{kb}v_{1}^{a}v_{1}^{l}\right)\right)\right]\right\\}$
$\displaystyle-\frac{\kappa_{1}}{2m_{1}}S_{1}^{ba}S_{1}^{bi}\partial_{ia}V\delta_{1}+\frac{\kappa_{1}}{m_{1}c^{6}\sqrt{-g}}\partial_{k}\left[\delta_{1}\left(S_{1}^{ab}S_{1}^{kb}\partial_{a}V-(S_{1}^{ab}S_{1}^{ab})\partial_{k}V\right)\right]$
$\displaystyle+\frac{\kappa_{1}}{2m_{1}c^{6}\sqrt{-g}}\partial_{t}^{2}(\delta_{1}S_{1}^{ab}S_{1}^{ab})+\frac{\kappa_{1}}{m_{1}c^{6}\sqrt{-g}}\partial_{t}\partial_{k}\left[\delta_{1}\left(-S_{1}^{ab}S_{1}^{kb}v_{1}^{a}+(S_{1}^{ab}S_{1}^{ab})v_{1}^{k}\right)\right]$
$\displaystyle+1\leftrightarrow 2+\mathcal{O}(8),$ (19bob)
$\displaystyle\sigma^{\text{SSS}}$
$\displaystyle=\frac{\lambda_{1}}{3m_{1}^{2}c^{7}}v_{1}^{m}S_{1}^{mi}S_{1}^{jl}S_{1}^{kl}\partial_{ijk}\delta_{1}+1\leftrightarrow
2+\mathcal{O}(8),$ (19boc) $\displaystyle\sigma_{i}^{\text{SO}}$
$\displaystyle=\frac{1}{\sqrt{-g}}\partial_{k}\left\\{\delta_{1}\left[\frac{S_{1}^{ik}}{2c}+\frac{S_{1}^{ak}v_{1}^{a}v_{1}^{i}}{2c^{3}}+\frac{2}{c^{5}}(S_{1}^{ka}v_{1}^{i}V^{a}-S_{1}^{ka}v_{1}^{a}V)\right]\right\\}$
$\displaystyle+\frac{1}{\sqrt{-g}}\partial_{t}\left\\{\delta_{1}\left[\frac{S_{1}^{ia}v_{1}^{a}}{2c^{3}}+\frac{2}{c^{5}}(S_{1}^{ia}v_{1}^{a}V-S_{1}^{ia}V^{a})\right]\right\\}$
$\displaystyle+\delta_{1}\left\\{-\frac{S_{1}^{ia}\partial^{a}V}{2c^{3}}+\frac{1}{c^{5}}\left[-S_{1}^{ia}v_{1}^{k}\partial^{k}Vv_{1}^{a}+\frac{1}{2}v_{1}^{i}S_{1}^{ab}v_{1}^{a}\partial^{b}V-\frac{1}{2}S_{1}^{ia}v_{1}^{a}\partial_{t}V+S_{1}^{ib}v_{1}^{a}\partial^{a}V^{b}\right.\right.$
$\displaystyle\left.\left.\qquad\qquad+S_{1}^{ia}V\partial^{a}V+S_{1}^{ib}v_{1}^{a}\partial^{b}V^{a}-S_{1}^{ab}v_{1}^{a}\partial^{b}V^{i}+S_{1}^{ab}v_{1}^{a}\partial^{i}V^{b}\right]\right\\}+1\leftrightarrow
2+\mathcal{O}(8),$ (19bod) $\displaystyle\sigma_{i}^{\text{SS}}$
$\displaystyle=\frac{\delta_{1}}{2m_{1}c^{6}}\left[2\kappa_{1}S_{1}^{ab}S_{1}^{bj}\partial^{j(a}Vv_{1}^{i)}-S_{1}^{ab}S_{1}^{ib}\partial_{t}\partial_{a}V+S_{1}^{bj}S_{1}^{ib}v_{1}^{a}\partial_{aj}V+2S_{1}^{bj}S_{1}^{ia}\partial_{aj}V^{b}\right.$
$\displaystyle\left.-S_{1}^{aj}S_{1}^{ib}v_{1}^{a}\partial_{bj}V\right]+\frac{\kappa_{1}}{m_{1}c^{6}\sqrt{-g}}\partial_{t}\left[\delta_{1}\left(\frac{1}{2}S_{1}^{ab}S_{1}^{ab}\partial^{i}V-S_{1}^{ab}S_{1}^{ib}\partial^{a}V\right)\right]$
$\displaystyle+\frac{\kappa_{1}}{c^{4}m_{1}\sqrt{-g}}\partial_{kl}\left\\{v_{1}^{[i}S_{1}^{k]a}S_{1}^{la}\delta_{1}+\frac{v_{1}^{[i}S_{1}^{k]a}}{c^{2}}\delta_{1}\left[\frac{1}{2}S_{1}^{la}v_{1}^{2}+3S_{1}^{la}V-S_{1}^{lb}v_{1}^{a}v_{1}^{b}+S_{1}^{ab}v_{1}^{b}v_{1}^{l}\right]\right\\}$
$\displaystyle-\frac{\kappa_{1}}{2m_{1}c^{4}\sqrt{-g}}\partial_{t}\partial_{k}\left\\{S_{1}^{ia}S_{1}^{ka}\delta_{1}+\frac{\delta_{1}}{c^{2}}\left[S_{1}^{ia}S_{1}^{ka}\left(\frac{v_{1}^{2}}{2}+3V\right)-2v_{1}^{(i}S_{1}^{k)b}S_{1}^{ab}v_{1}^{a}-S_{1}^{ia}S_{1}^{kb}v_{1}^{a}v_{1}^{b}\right]\right\\}$
$\displaystyle+\frac{\kappa_{1}}{m_{1}c^{6}\sqrt{-g}}\partial_{k}\left\\{\delta_{1}\left[S_{1}^{ia}S_{1}^{ka}v_{1}^{l}\partial^{l}V+\frac{1}{2}S_{1}^{ia}S_{1}^{ka}\partial_{t}V+S_{1}^{ab}S_{1}^{kb}v_{1}^{i}\partial^{a}V-2S_{1}^{ab}S_{1}^{ib}v_{1}^{k}\partial^{a}V\right.\right.$
$\displaystyle\left.\left.-S_{1}^{ab}S_{1}^{kb}\partial^{a}V^{i}+S_{1}^{ab}S_{1}^{ib}\partial^{a}V^{k}-\frac{1}{2}S_{1}^{ab}S_{1}^{kb}v_{1}^{a}\partial^{i}V+\frac{1}{2}S_{1}^{ab}S_{1}^{ab}v_{1}^{k}\partial^{i}V+S_{1}^{ab}S_{1}^{kb}\partial^{i}V^{a}\right.\right.$
$\displaystyle\left.\left.+\frac{1}{2}S_{1}^{ab}S_{1}^{ib}v_{1}^{a}\partial^{k}V-\frac{1}{2}S_{1}^{ab}S_{1}^{ab}v_{1}^{i}\partial^{k}V-S_{1}^{ab}S_{1}^{ib}\partial^{k}V^{a}\right]\right\\}+1\leftrightarrow
2+\mathcal{O}(7),$ (19boe) $\displaystyle\sigma_{i}^{\text{SSS}}$
$\displaystyle=\frac{\lambda_{1}}{12m_{1}^{2}c^{5}}S_{1}^{ij}S_{1}^{km}S_{1}^{lm}\partial_{jkl}\delta_{1}+1\leftrightarrow
2+\mathcal{O}(7),$ (19bof) $\displaystyle\sigma_{ij}^{\text{SO}}$
$\displaystyle=\frac{1}{c\sqrt{-g}}\partial_{k}\left(\delta_{1}v_{1}^{(i}S_{1}^{j)k}\right)+\frac{1}{c^{3}\sqrt{-g}}\partial_{t}\left(\delta_{1}v_{1}^{(i}S_{1}^{j)k}v_{1}^{k}\right)$
$\displaystyle+2\frac{\delta_{1}}{c^{3}}S_{1}^{k(i}\left(\partial_{k}V^{j)}+\partial^{j)}Vv_{1}^{k}-\partial^{j)}V^{k}-\frac{1}{2}v_{1}^{j)}\partial_{k}V\right)+1\leftrightarrow
2+\mathcal{O}(5)\,,$ (19bog) $\displaystyle\sigma_{ij}^{\text{SS}}$
$\displaystyle=\frac{\kappa_{1}}{m_{1}c^{4}\sqrt{-g}}\left\\{-\delta_{1}\sqrt{-g}S_{1}^{ab}S_{1}^{a(i}\partial^{j)b}V+\partial_{t}\partial_{k}\left[\delta_{1}\left(S_{1}^{ia}S_{1}^{ja}v_{1}^{k}+S_{1}^{ka}S_{1}^{a(i}v_{1}^{j)}\right)\right]\right.$
$\displaystyle\left.+\frac{1}{2}\partial_{t}^{2}\left(\delta_{1}S_{1}^{ia}S_{1}^{ja}\right)-\partial_{k}\left[\delta_{1}\left(S_{1}^{ka}S_{1}^{a(i}\partial^{j)}V+\frac{1}{2}S_{1}^{ia}S_{1}^{ja}\partial^{k}V\right)\right]\right.$
$\displaystyle\left.+\frac{1}{2}\partial_{kl}\left[\delta_{1}\left(S_{1}^{ka}S_{1}^{la}v_{1}^{i}v_{1}^{j}-2v_{1}^{(i}S_{1}^{j)a}S_{1}^{ka}v_{1}^{l}+S_{1}^{ia}S_{1}^{ja}v_{1}^{k}v_{1}^{l}\right)\right]\right\\}+1\leftrightarrow
2+\mathcal{O}(5)\,,$ (19boh) $\displaystyle\sigma_{ij}^{\text{SSS}}$
$\displaystyle=\mathcal{O}(5),$ (19boi)

where $v_{1}^{2}=v_{1}^{k}v_{1}^{k}$.

##  Appendix B Explicit quantities of the factorized modes

In this Appendix, we write the explicit expressions for the factorized modes
(see Sec. IV.2). The $(2,2)$ mode is given by Eq. (19bl), the $(2,1)$ mode by
Eq. (19bm), while the other subdominant modes read

$\displaystyle\rho_{33}^{\text{NS}}$
$\displaystyle=1+\left(\tfrac{2}{3}\nu-\tfrac{7}{6}\right)v^{2},$ (19bpa)
$\displaystyle f_{33}^{\text{S}}$
$\displaystyle=v^{3}\left[\left(\tfrac{19}{2}\nu-2\right)\frac{\chi_{A}}{\delta}+\left(\tfrac{5}{2}\nu-2\right)\chi_{S}\right]+v^{4}\bigg{[}\left(\tfrac{3}{2}-6\nu\right)\chi_{A}^{2}+(3-12\nu)\frac{\chi_{A}\chi_{S}}{\delta}+\tfrac{3}{2}\chi_{S}^{2}$
$\displaystyle\qquad+\left(\tfrac{3}{2}-6\nu\right)\frac{\tilde{\kappa}_{A}}{\delta}+\left(\tfrac{3}{2}-3\nu\right)\tilde{\kappa}_{S}\\!\bigg{]}+v^{5}\left[\left(\tfrac{407}{30}\nu^{2}-\tfrac{593}{60}\nu+\tfrac{2}{3}\right)\frac{\chi_{A}}{\delta}+\left(\tfrac{241}{30}\nu^{2}+\tfrac{11}{20}\nu+\tfrac{2}{3}\right)\chi_{S}\right]$
$\displaystyle\quad+v^{6}\bigg{[}\left(-12\nu^{2}+\tfrac{11}{2}\nu-\tfrac{7}{4}\right)\chi_{A}^{2}+\left(44\nu^{2}-\nu-\tfrac{7}{2}\right)\frac{\chi_{A}\chi_{S}}{\delta}+\left(6\nu^{2}-\tfrac{27}{2}\nu-\tfrac{7}{4}\right)\chi_{S}^{2}$
$\displaystyle\qquad+\left(-12\nu^{2}+\tfrac{11}{2}\nu-\tfrac{7}{4}\right)\chi_{A}^{2}+\left(44\nu^{2}-\nu-\tfrac{7}{2}\right)\frac{\chi_{A}\chi_{S}}{\delta}+\left(6\nu^{2}-\tfrac{27}{2}\nu-\tfrac{7}{4}\right)\chi_{S}^{2}\bigg{]},$
(19bpb) $\displaystyle\delta_{33}$ $\displaystyle=\tfrac{13}{10}\omega
E+\big{(}\omega
E\big{)}^{2}\left[\left(\tfrac{7339}{540}\nu-\tfrac{81}{20}\right)\frac{\chi_{A}}{\delta}+\left(\tfrac{593}{108}\nu-\tfrac{81}{20}\right)\chi_{S}\right],$
(19bpc)

$\displaystyle\rho_{32}$
$\displaystyle=1+v\frac{4\nu\chi_{S}}{3(1-3\nu)}+v^{2}\left[\frac{-\tfrac{32}{27}\nu^{2}+\tfrac{223}{54}\nu-\tfrac{164}{135}}{1-3\nu}-\frac{16\nu^{2}\chi_{S}^{2}}{9(1-3\nu)^{2}}\right]$
$\displaystyle\quad+v^{3}\left[\left(\tfrac{13}{9}\nu+\tfrac{2}{9}\right)\frac{\delta\chi_{A}}{1-3\nu}+\left(\tfrac{607}{81}\nu^{3}+\tfrac{503}{81}\nu^{2}-\tfrac{1478}{405}\nu+\tfrac{2}{9}\right)\frac{\chi_{S}}{(1-3\nu)^{2}}+\frac{320\nu^{3}\chi_{S}^{3}}{81(1-3\nu)^{3}}\right]$
$\displaystyle\quad+v^{4}\bigg{[}\frac{\tfrac{77141}{40095}\nu^{4}-\tfrac{508474}{40095}\nu^{3}-\tfrac{945121}{320760}\nu^{2}+\tfrac{1610009\nu}{320760}-\tfrac{180566}{200475}}{(1-3\nu)^{2}}+\left(4\nu^{2}-3\nu+\tfrac{1}{3}\right)\frac{\chi_{A}^{2}}{1-3\nu}$
$\displaystyle\qquad+\left(-\tfrac{50}{27}\nu^{2}-\tfrac{88}{27}\nu+\tfrac{2}{3}\right)\frac{\delta\chi_{A}\chi_{S}}{(1-3\nu)^{2}}+\left(-\tfrac{2452}{243}\nu^{4}-\tfrac{1997}{243}\nu^{3}+\tfrac{1435}{243}\nu^{2}-\tfrac{43}{27}\nu+\tfrac{1}{3}\right)\frac{\chi_{S}^{2}}{(1-3\nu)^{3}}$
$\displaystyle\qquad+\left(\tfrac{1}{3}-\tfrac{1}{3}\nu\right)\frac{\delta\tilde{\kappa}_{A}}{1-3\nu}+\left(2\nu^{2}-\nu+\tfrac{1}{3}\right)\frac{\tilde{\kappa}_{S}}{1-3\nu}\bigg{]}$
$\displaystyle\quad+v^{5}\bigg{[}\left(-\tfrac{1184225}{96228}\nu^{5}-\tfrac{40204523}{962280}\nu^{4}+\tfrac{101706029}{962280}\nu^{3}-\tfrac{14103833}{192456}\nu^{2}+\tfrac{20471053}{962280}\nu-\tfrac{2788}{1215}\right)\frac{\chi_{S}}{(1-3\nu)^{3}}$
$\displaystyle\qquad+\left(\tfrac{889673}{106920}\nu^{3}-\tfrac{75737}{5346}\nu^{2}+\tfrac{376177}{35640}\nu-\tfrac{2788}{1215}\right)\frac{\delta\chi_{A}}{(1-3\nu)^{2}}+\left(\tfrac{608}{81}\nu^{3}+\tfrac{736}{81}\nu^{2}-\tfrac{16}{9}\nu\right)\frac{\delta\chi_{A}\chi_{S}^{2}}{(1-3\nu)^{3}}$
$\displaystyle\qquad+\left(\tfrac{96176}{2187}\nu^{5}+\tfrac{43528}{2187}\nu^{4}-\tfrac{40232}{2187}\nu^{3}+\tfrac{376}{81}\nu^{2}-\tfrac{8\nu}{9}\right)\frac{\chi_{S}^{3}}{(1-3\nu)^{4}}$
$\displaystyle\qquad+\left(-\tfrac{32}{3}\nu^{3}+8\nu^{2}-\tfrac{8}{9}\nu\right)\frac{\chi_{A}^{2}\chi_{S}}{(1-3\nu)^{2}}+\left(\tfrac{8}{9}\delta\nu^{2}-\tfrac{8}{9}\delta\nu\right)\frac{\chi_{S}\tilde{\kappa}_{A}}{(1-3\nu)^{2}}$
$\displaystyle\qquad+\left(-\tfrac{16}{3}\nu^{3}+\tfrac{8}{3}\nu^{2}-\tfrac{8}{9}\nu\right)\frac{\chi_{S}\tilde{\kappa}_{S}}{(1-3\nu)^{2}}\bigg{]},$
(19bqa) $\displaystyle\delta_{32}$
$\displaystyle=\left(\tfrac{11}{5}\nu+\tfrac{2}{3}\right)\frac{\omega
E}{1-3\nu}+v^{4}\left[\frac{4\delta\chi_{A}}{1-3\nu}+\left(\tfrac{36}{5}\nu^{2}-20\nu+4\right)\frac{\chi_{S}}{(1-3\nu)^{2}}\right]$
$\displaystyle\quad+v^{5}\left[\left(-\tfrac{144}{5}\nu^{3}+80\nu^{2}-16\nu\right)\frac{\chi_{S}^{2}}{(1-3\nu)^{3}}-\frac{16\delta\nu\chi_{A}\chi_{S}}{(1-3\nu)^{2}}\right],$
(19bqb)

$\displaystyle\rho_{31}^{\text{NS}}$
$\displaystyle=1+v^{2}\left(-\tfrac{2}{9}\nu-\tfrac{13}{18}\right),$ (19bra)
$\displaystyle f_{31}^{\text{S}}$
$\displaystyle=v^{3}\left[\left(\tfrac{11}{2}\nu-2\right)\frac{\chi_{A}}{\delta}+\left(\tfrac{13}{2}\nu-2\right)\chi_{S}\right]$
$\displaystyle\quad+v^{4}\left[\left(-6\nu-\tfrac{5}{2}\right)\chi_{A}^{2}+(4\nu-5)\frac{\chi_{A}\chi_{S}}{\delta}-\tfrac{5}{2}\chi_{S}^{2}+\left(2\nu-\tfrac{5}{2}\right)\frac{\tilde{\kappa}_{A}}{\delta}+\left(-3\nu-\tfrac{5}{2}\right)\tilde{\kappa}_{S}\right]$
$\displaystyle\quad+v^{5}\left[\left(-\tfrac{931}{18}\nu^{2}+\tfrac{25}{36}\nu+\tfrac{38}{9}\right)\frac{\chi_{A}}{\delta}+\left(-\tfrac{5}{2}\nu^{2}-\tfrac{35}{12}\nu+\tfrac{38}{9}\right)\chi_{S}\right]$
$\displaystyle\quad+v^{6}\bigg{[}\left(\tfrac{44}{3}\nu^{2}+\tfrac{25}{6}\nu+\tfrac{43}{12}\right)\chi_{A}^{2}+\left(\tfrac{452}{3}\nu^{2}-\tfrac{83}{3}\nu+\tfrac{43}{6}\right)\frac{\chi_{A}\chi_{S}}{\delta}+\left(22\nu^{2}-\tfrac{35}{2}\nu+\tfrac{43}{12}\right)\chi_{S}^{2}$
$\displaystyle\qquad+\left(\tfrac{76}{3}\nu^{2}-\tfrac{47}{6}\nu+\tfrac{43}{12}\right)\frac{\tilde{\kappa}_{A}}{\delta}+\left(\tfrac{22}{3}\nu^{2}-\tfrac{2}{3}\nu+\tfrac{43}{12}\right)\tilde{\kappa}_{S}\bigg{]},$
(19brb) $\displaystyle\delta_{31}$ $\displaystyle=\tfrac{13}{30}\omega
E+\big{(}\omega
E\big{)}^{2}\left[\left(-\tfrac{9}{4}\nu+\tfrac{61}{20}\right)\frac{\chi_{A}}{\delta}+\left(\tfrac{77}{20}\nu+\tfrac{61}{20}\right)\chi_{S}\right],$
(19brc)

$\displaystyle\rho_{44}$
$\displaystyle=1+\frac{v^{2}}{1-3\nu}\left(-\tfrac{175}{88}\nu^{2}+\tfrac{587}{132}\nu-\tfrac{269}{220}\right)+\frac{v^{3}}{1-3\nu}\left[\left(\tfrac{13}{5}\nu-\tfrac{2}{3}\right)\delta\chi_{A}+\left(-\tfrac{14}{5}\nu^{2}+\tfrac{41}{15}\nu-\tfrac{2}{3}\right)\chi_{S}\right]$
$\displaystyle\quad+v^{4}\Big{[}\tfrac{1}{2}\delta\tilde{\kappa}_{A}+\left(\tfrac{1}{2}-\nu\right)\tilde{\kappa}_{S}+\left(\tfrac{1}{2}-2\nu\right)\chi_{A}^{2}+\delta\chi_{A}\chi_{S}+\tfrac{1}{2}\chi_{S}^{2}\Big{]}$
$\displaystyle\quad+\frac{v^{5}}{(1-3\nu)^{2}}\Big{[}\delta\left(\tfrac{597}{440}\nu^{3}-\tfrac{1933}{220}\nu^{2}+\tfrac{10679}{1650}\nu-\tfrac{69}{55}\right)\chi_{A}$
$\displaystyle\qquad+\left(\tfrac{591}{44}\nu^{4}+\tfrac{8539}{440}\nu^{3}-\tfrac{2673}{100}\nu^{2}+\tfrac{16571}{1650}\nu-\tfrac{69}{55}\right)\chi_{S}\Big{]},$
(19bs)

$\displaystyle\rho_{43}^{\text{NS}}$
$\displaystyle=1+\frac{v^{2}}{1-2\nu}\left(-\tfrac{10}{11}\nu^{2}+\tfrac{547}{176}\nu-\tfrac{111}{88}\right),$
(19bta) $\displaystyle f_{43}^{\text{S}}$
$\displaystyle=\frac{v}{1-2\nu}\left(\tfrac{5}{2}\nu\chi_{S}-\tfrac{5}{2}\nu\frac{\chi_{A}}{\delta}\right)+\frac{v^{3}}{1-2\nu}\left[\left(\tfrac{887}{44}\nu-\tfrac{3143}{132}\nu^{2}\right)\frac{\chi_{A}}{\delta}+\left(-\tfrac{529}{132}\nu^{2}-\tfrac{667}{44}\nu\right)\chi_{S}\right]$
$\displaystyle\quad+\frac{v^{4}}{1-2\nu}\Big{[}\left(12\nu^{2}-\tfrac{37}{3}\nu+\tfrac{3}{2}\right)\chi_{A}^{2}+\left(\tfrac{137}{6}\nu^{2}-18\nu+3\right)\frac{\chi_{A}\chi_{S}}{\delta}+\left(\tfrac{35}{6}\nu^{2}+\tfrac{1}{3}\nu+\tfrac{3}{2}\right)\chi_{S}^{2}$
$\displaystyle\qquad+\left(6\nu^{2}-9\nu+\tfrac{3}{2}\right)\frac{\tilde{\kappa}_{A}}{\delta}+\left(6\nu^{2}-6\nu+\tfrac{3}{2}\right)\tilde{\kappa}_{S}\Big{]},$
(19btb) $\displaystyle\delta_{43}$
$\displaystyle=\left(\tfrac{4961}{810}\nu+\tfrac{3}{5}\right)\frac{\omega
E}{1-2\nu}$
$\displaystyle\quad+\frac{v^{4}}{(1-2\nu)^{2}}\left[\left(\tfrac{17999}{324}\nu^{2}-\tfrac{2605}{108}\nu+\tfrac{11}{4}\right)\frac{\chi_{A}}{\delta}+\left(\tfrac{2569}{324}\nu^{2}-\tfrac{2011}{108}\nu+\tfrac{11}{4}\right)\chi_{S}\right],$
(19btc)

$\displaystyle\rho_{42}$
$\displaystyle=1+\frac{v^{2}}{1-3\nu}\left(-\tfrac{19}{88}\nu^{2}+\tfrac{353}{132}\nu-\tfrac{191}{220}\right)+\frac{v^{3}}{1-3\nu}\left[\delta\left(\tfrac{7}{5}\nu-\tfrac{2}{3}\right)\chi_{A}+\left(-\tfrac{26}{5}\nu^{2}+\tfrac{59}{15}\nu-\tfrac{2}{3}\right)\chi_{S}\right]$
$\displaystyle\quad+\frac{v^{4}}{1-3\nu}\left[\left(6\nu^{2}-2\nu+\tfrac{1}{2}\right)\chi_{A}^{2}+\delta\chi_{A}\chi_{S}+\tfrac{1}{2}\chi_{S}^{2}+\tfrac{1}{2}\delta\tilde{\kappa}_{A}+3\nu^{2}\tilde{\kappa}_{S}-\nu\tilde{\kappa}_{S}+\tfrac{1}{2}\tilde{\kappa}_{S}+\tfrac{1}{2}\chi_{S}^{2}\right]$
$\displaystyle\quad+\frac{v^{5}}{(1-3\nu)^{2}}\Big{[}\left(\tfrac{2463}{88}\nu^{3}-\tfrac{4309}{220}\nu^{2}+\tfrac{2977}{825}\nu-\tfrac{7}{110}\right)\delta\chi_{A}$
$\displaystyle\qquad+\left(\tfrac{393}{220}\nu^{4}+\tfrac{6437}{440}\nu^{3}-\tfrac{12443}{1100}\nu^{2}+\tfrac{1873}{825}\nu-\tfrac{7}{110}\right)\chi_{S}\Big{]},$
(19bu)

$\displaystyle\rho_{41}^{\text{NS}}$
$\displaystyle=1+\frac{v^{2}}{1-2\nu}\left(-\tfrac{6}{11}\nu^{2}+\tfrac{1385}{528}\nu-\tfrac{301}{264}\right),$
(19bva) $\displaystyle f_{41}^{\text{S}}$
$\displaystyle=\frac{v}{1-2\nu}\left(\tfrac{5}{2}\nu\chi_{S}-\tfrac{5}{2}\nu\frac{\chi_{A}}{\delta}\right)+\frac{v^{3}}{1-2\nu}\left[\left(\tfrac{783}{44}\nu-\tfrac{2207}{132}\nu^{2}\right)\frac{\chi_{A}}{\delta}+\left(-\tfrac{841}{132}\nu^{2}-\tfrac{563}{44}\nu\right)\chi_{S}\right]$
$\displaystyle\quad+\frac{v^{4}}{1-2\nu}\Big{[}\left(12\nu^{2}-\tfrac{37}{3}\nu+\tfrac{3}{2}\right)\chi_{A}^{2}+\left(\frac{41\nu^{2}}{6}-18\nu+3\right)\frac{\chi_{A}\chi_{S}}{\delta}+\left(\tfrac{35}{6}+\tfrac{1}{3}\nu+\tfrac{3}{2}\right)\chi_{S}^{2}$
$\displaystyle\qquad+\left(-2\nu^{2}-9\nu+\tfrac{3}{2}\right)\frac{\tilde{\kappa}_{A}}{\delta}+\left(6\nu^{2}-6\nu+\tfrac{3}{2}\right)\tilde{\kappa}_{S}\Big{]},$
(19bvb) $\displaystyle\delta_{41}$
$\displaystyle=\left(\tfrac{507}{10}\nu+\tfrac{1}{5}\right)\frac{\omega
E}{1-2\nu}+\frac{v^{4}}{(1-2\nu)^{2}}\left[\left(\tfrac{533}{4}\nu^{2}-\tfrac{55}{12}\nu+\tfrac{11}{12}\right)\frac{\chi_{A}}{\delta}+\left(-\tfrac{1511}{12}\nu^{2}-\tfrac{11}{4}\nu+\tfrac{11}{12}\right)\chi_{S}\right],$
(19bvc)

$\displaystyle\rho_{55}^{\text{NS}}$ $\displaystyle=1,$ (19bwa) $\displaystyle
f_{55}^{\text{S}}$
$\displaystyle=\frac{v^{3}}{1-2\nu}\left[\left(-\tfrac{110}{3}\nu^{2}+\tfrac{70}{3}\nu-\tfrac{10}{3}\right)\frac{\chi_{A}}{\delta}+\left(-10\nu^{2}+10\nu-\tfrac{10}{3}\right)\chi_{S}\right]$
$\displaystyle\quad+v^{4}\left[\left(\tfrac{5}{2}-10\nu\right)\chi_{A}^{2}+(5-20\nu)\frac{\chi_{A}\chi_{S}}{\delta}+\tfrac{5}{2}\chi_{S}^{2}+\left(\tfrac{5}{2}-10\nu\right)\frac{\tilde{\kappa}_{A}}{\delta}+\left(\tfrac{5}{2}-5\nu\right)\tilde{\kappa}_{S}\right],$
(19bwb)

$\displaystyle\rho_{54}$ $\displaystyle=1+\frac{3\nu
v\left(-\delta\chi_{A}-2\nu\chi_{S}+\chi_{S}\right)}{5\left(5\nu^{2}-5\nu+1\right)}$
$\displaystyle\quad+v^{2}\bigg{\\{}\frac{33320\nu^{3}-127610\nu^{2}+96019\nu-17448}{13650\left(5\nu^{2}-5\nu+1\right)}$
$\displaystyle\qquad-\frac{18\nu^{2}\left[(1-4\nu)\chi_{A}^{2}+2\delta(2\nu-1)\chi_{A}\chi_{S}+(1-2\nu)^{2}\chi_{S}^{2}\right]}{25\left(5\nu^{2}-5\nu+1\right)^{2}}\bigg{\\}}$
$\displaystyle\quad+v^{3}\bigg{\\{}\frac{\delta\left(-1268785\nu^{4}+2296805\nu^{3}-1386497\nu^{2}+268024\nu-9100\right)\chi_{A}}{68250\left(5\nu^{2}-5\nu+1\right)^{2}}$
$\displaystyle\qquad+\frac{\left(1690430\nu^{5}-1436855\nu^{4}-174699\nu^{3}+239045\nu^{2}+426\nu-9100\right)\chi_{S}}{68250\left(5\nu^{2}-5\nu+1\right)^{2}}$
$\displaystyle\qquad-\frac{162\nu^{3}}{125\left(5\nu^{2}-5\nu+1\right)^{3}}\Big{[}(1-4\nu)\delta\chi_{A}^{3}+3(1-2\nu)^{2}\delta\chi_{A}\chi_{S}^{2}$
$\displaystyle\qquad\qquad+(-24\nu^{2}+18\nu-3)\chi_{S}\chi_{A}^{2}-(1-2\nu)^{3}\chi_{S}^{3}\Big{]}\bigg{\\}},$
(19bx)

$\displaystyle\rho_{53}^{\text{NS}}$ $\displaystyle=1,$ (19bya) $\displaystyle
f_{53}^{\text{S}}$
$\displaystyle=\frac{v^{3}}{1-2\nu}\left[\left(-\tfrac{62}{3}\nu^{2}+18\nu-\tfrac{10}{3}\right)\frac{\chi_{A}}{\delta}+\left(-\tfrac{46}{3}\nu^{2}+\tfrac{46}{3}\nu-\tfrac{10}{3}\right)\chi_{S}\right]$
$\displaystyle\quad+\frac{v^{4}}{1-2\nu}\Big{[}\left(20\nu^{2}-15\nu+\tfrac{5}{2}\right)\chi_{A}^{2}+\left(8\nu^{2}-30\nu+5\right)\frac{\chi_{A}\chi_{S}}{\delta}+\left(\tfrac{5}{2}-5\nu\right)\chi_{S}^{2}$
$\displaystyle\qquad+\left(4\nu^{2}-15\nu+\tfrac{5}{2}\right)\frac{\tilde{\kappa}_{A}}{\delta}+\left(10\nu^{2}-10\nu+\tfrac{5}{2}\right)\tilde{\kappa}_{S}\Big{]},$
(19byb)

$\displaystyle\rho_{52}$ $\displaystyle=1+\frac{3\nu
v\left(-\delta\chi_{A}-2\nu\chi_{S}+\chi_{S}\right)}{5\left(5\nu^{2}-5\nu+1\right)}$
$\displaystyle\quad+v^{2}\bigg{\\{}\frac{21980\nu^{3}-104930\nu^{2}+84679\nu-15828}{13650\left(5\nu^{2}-5\nu+1\right)}$
$\displaystyle\qquad-\frac{18\nu^{2}\left[(1-4\nu)\chi_{A}^{2}+2\delta(2\nu-1)\chi_{A}\chi_{S}+(1-2\nu)^{2}\chi_{S}^{2}\right]}{25\left(5\nu^{2}-5\nu+1\right)^{2}}\bigg{\\}}$
$\displaystyle\quad+v^{3}\bigg{\\{}\frac{\left(-963865\nu^{4}+1907465\nu^{3}-1213877\nu^{2}+243364\nu-9100\right)\delta\chi_{A}}{68250\left(5\nu^{2}-5\nu+1\right)^{2}}$
$\displaystyle\qquad+\frac{\left(1859270\nu^{5}-2079455\nu^{4}+471681\nu^{3}+17105\nu^{2}+25086\nu-9100\right)\chi_{S}}{68250\left(5\nu^{2}-5\nu+1\right)^{2}}$
$\displaystyle\qquad-\frac{162\nu^{3}}{125\left(5\nu^{2}-5\nu+1\right)^{3}}\Big{[}(1-4\nu)\delta\chi_{A}^{3}+3(1-2\nu)^{2}\delta\chi_{A}\chi_{S}^{2}$
$\displaystyle\qquad\qquad+(-24\nu^{2}+18\nu-3)\chi_{S}\chi_{A}^{2}-(1-2\nu)^{3}\chi_{S}^{3}\Big{]}\bigg{\\}},$
(19bz)

$\displaystyle\rho_{51}^{\text{NS}}$ $\displaystyle=1,$ (19caa) $\displaystyle
f_{51}^{\text{S}}$
$\displaystyle=\frac{v^{3}}{1-2\nu}\left[\left(-\tfrac{38}{3}\nu^{2}+\tfrac{46}{3}\nu-\tfrac{10}{3}\right)\frac{\chi_{A}}{\delta}+\left(-18\nu^{2}+18\nu-\tfrac{10}{3}\right)\chi_{S}\right]$
$\displaystyle\quad+\frac{v^{4}}{1-2\nu}\Big{[}\left(20\nu^{2}-15\nu+\tfrac{5}{2}\right)\chi_{A}^{2}+\left(-8\nu^{2}-30\nu+5\right)\frac{\chi_{A}\chi_{S}}{\delta}+\left(\tfrac{5}{2}-5\nu\right)\chi_{S}^{2}$
$\displaystyle\qquad+\left(-4\nu^{2}-15\nu+\tfrac{5}{2}\right)\frac{\tilde{\kappa}_{A}}{\delta}+\left(10\nu^{2}-10\nu+\tfrac{5}{2}\right)\tilde{\kappa}_{S}\Big{]},$
(19cab)

$\displaystyle\rho_{66}$
$\displaystyle=1+v^{3}\frac{\delta\left(-100\nu^{2}+85\nu-14\right)\chi_{A}+\left(110\nu^{3}-150\nu^{2}+83\nu-14\right)\chi_{S}}{21\left(5\nu^{2}-5\nu+1\right)},$
(19cb)

$\displaystyle\rho_{65}^{\text{NS}}$ $\displaystyle=1,$ (19cca) $\displaystyle
f_{65}^{\text{S}}$ $\displaystyle=-\frac{7\nu
v\left[(1-3\nu)\chi_{A}+\delta(\nu-1)\chi_{S}\right]}{2\delta(\nu-1)(3\nu-1)},$
(19ccb)

$\displaystyle\rho_{64}$
$\displaystyle=1+v^{3}\frac{\delta\left(-60\nu^{2}+65\nu-14\right)\chi_{A}+\left(150\nu^{3}-230\nu^{2}+103\nu-14\right)\chi_{S}}{21\left(5\nu^{2}-5\nu+1\right)},$
(19cd)

$\displaystyle\rho_{63}^{\text{NS}}$ $\displaystyle=1,$ (19cea) $\displaystyle
f_{63}^{\text{S}}$ $\displaystyle=-\frac{7\nu
v\left[(1-3\nu)\chi_{A}+\delta(\nu-1)\chi_{S}\right]}{2\delta(\nu-1)(3\nu-1)},$
(19ceb)

$\displaystyle\rho_{62}$
$\displaystyle=1+v^{3}\frac{\delta\left(-36\nu^{2}+53\nu-14\right)\chi_{A}+\left(174\nu^{3}-278\nu^{2}+115\nu-14\right)\chi_{S}}{21\left(5\nu^{2}-5\nu+1\right)},$
(19cf)

$\displaystyle\rho_{61}^{\text{NS}}$ $\displaystyle=1,$ (19cga) $\displaystyle
f_{61}^{\text{S}}$ $\displaystyle=-\frac{7\nu
v\left[(1-3\nu)\chi_{A}+\delta(\nu-1)\chi_{S}\right]}{2\delta(\nu-1)(3\nu-1)},$
(19cgb)

$\rho_{76}=1-\frac{4\nu
v\left[\delta(2\nu-1)\chi_{A}+\left(2\nu^{2}-4\nu+1\right)\chi_{S}\right]}{7\left(7\nu^{3}-14\nu^{2}+7\nu-1\right)},$
(19ch)

$\rho_{74}=1-\frac{4\nu
v\left[\delta(2\nu-1)\chi_{A}+\left(2\nu^{2}-4\nu+1\right)\chi_{S}\right]}{7\left(7\nu^{3}-14\nu^{2}+7\nu-1\right)},$
(19ci)

$\rho_{72}=1-\frac{4\nu
v\left[\delta(2\nu-1)\chi_{A}+\left(2\nu^{2}-4\nu+1\right)\chi_{S}\right]}{7\left(7\nu^{3}-14\nu^{2}+7\nu-1\right)}.$
(19cj)

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