# Spin effects in spherical harmonic modes of gravitational waves from
eccentric compact binary inspirals

Kaushik Paul<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Madras, Chennai 600036, India  Centre for Strings, Gravitation and
Cosmology, Department of Physics, Indian Institute of Technology Madras,
Chennai 600036, India  Chandra Kant Mishra<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology Madras, Chennai 600036, India  Centre
for Strings, Gravitation and Cosmology, Department of Physics, Indian
Institute of Technology Madras, Chennai 600036, India

(August 27, 2024)

###### Abstract

We compute the leading and subleading spin effects through the second post-
Newtonian order (2PN) in spherical harmonic modes of gravitational waveforms
from inspiralling compact binaries in noncircular orbits with nonprecessing
components. The two spin couplings, linear-in-spin (spin-orbit) and quadratic-
in-spin (spin-spin), that appear in 2PN waveforms are computed with desired
accuracy and explicit expressions for relevant modes are derived. The modes
that have spin corrections through 2PN include
$(\ell,|m|)$=$((2,2),\,(2,1),\,(3,3),\,(3,2),\,(3,1),\,(4,3),\,(4,1))$ modes.
Additionally, two $m=0$ modes—$(2,0)$ and $(3,0)$—also contribute to the 2PN
order. Closed-form expressions for these modes for compact binaries in general
orbits as well as in elliptical orbits are provided. While the general orbit
results can be used to study signals from binaries in orbits of arbitrary
shape and nature, elliptical orbit results are applicable to systems with
arbitrary eccentricities. We also express the elliptical orbit results as
leading eccentric corrections to the circular results. Our prescription
represents, the first, fully analytical treatment that combines spins,
eccentricity and higher modes together and completes computation of spin
effects through 2PN order. These should find immediate applications in
inspiral-merger-ringdown modeling for eccentric mergers including the effect
of nonprecessing spins and higher modes as well as in parameter estimation
analyses employing an inspiral waveform.

##  I Introduction

The ground-based network of gravitational-wave (GW) detectors involving the
LIGO Aasi  _et al._ (2015) and Virgo Acernese  _et al._ (2015) detectors has
detected so far nearly 90 compact binary mergers (Abbott  _et al._ , 2019a,
2021a, 2021b, 2021c). While the majority of these events are inferred to be
binary black hole (BBH) mergers, two binary neutron star (Abbott  _et al._ ,
2017, 2020a) and two neutron star-black hole (Abbott  _et al._ , 2021d)
mergers have also been observed. While compact binaries involving neutron
stars and stellar mass black holes (BHs) will continue to dominate the
frequency bands observable from ground $[$above 1-10Hz; observable by planned
(LIGO A+ Shoemaker (2019); McClelland  _et al._ (2016), Voyager McClelland
_et al._ (2016)) and proposed future $[$Cosmic Explorer (CE) Dwyer  _et al._
(2015), Einstein Telescope (ET) Punturo  _et al._ (2010) detectors
configurations, intermediate mass BHs and supermassive BHs will dominate the
observation bands accessible only from space $[$in deci-Hz band (DECIGO Sato
_et al._ (2017)) and in milli-Hz band (LISA Amaro-Seoane  _et al._ (2017))$]$.
Detection of GWs from these sources involves cross-correlating the detector
data with a collection of simulated copies (called templates) of expected
signals. This template-based search method is referred to as matched filtering
Cutler and Flanagan (1994); Poisson and Will (1995); Krolak  _et al._ (1995)
and its reliability critically depends on the closeness of the templates to
the true signal buried in the detector’s noisy data. The requirements on the
accuracy of these templates is quite stringent for detection (mismatches with
the true signal should be no larger than $\sim 3$%) and for subsequent
parameter estimation analyses (no larger than $\sim$1%). (See for instance
Ref. Abbott  _et al._ (2016) for details on search methods currently in use
for analyzing data from LIGO and Virgo detectors.) Templates that are
currently used in performing GW searches assume binaries in inspiralling
circular orbits owing to the expected circularisation of the binary Peters
(1964); Peters and Mathews (1963). Although, partly it is also due to the
absence of appropriate signal models (and of pertinent search
strategies111That optimally manage the cost of adding another direction
(orbital eccentricity) to the search parameter space.) including the effects
of orbital eccentricity. Moreover, compact binaries formed via dynamical
interactions in dense stellar environments or (if the binary is part of a
stable hierarchical triple system) through Kozai-Lidov processes Kozai (1962);
Lidov (1962) are expected to be observed in ground-based detectors with
residual eccentricities ($e_{0}$) $\gtrsim 0.1$ Abbott  _et al._ (2019b). In
fact, some of the recent analyses Abbott  _et al._ (2020b); Kimball  _et al._
(2021); Romero-Shaw  _et al._ (2021, 2022); O’Shea and Kumar (2021) show
support for the presence of eccentricity in the observed data for BBH events.
While it may be still possible to detect systems with residual eccentricities,
$e_{0}\lesssim 0.1$ (at 10Hz), in ground-based detectors by employing
quasicircular templates, systems with larger eccentricities will require
constructing templates including the effect of eccentricity Brown and
Zimmerman (2010); Huerta and Brown (2013). Additionally, as argued in Ref.
Favata  _et al._ (2022), the presence of even smaller eccentricities
($e_{0}\sim$ 0.01-0.05) may induce systematic biases in parameter estimation
analyses focusing on extracting source parameters. Moreover, the next
generation ground based detectors, CE Dwyer  _et al._ (2015) and ET Punturo
_et al._ (2010), with improved low frequency sensitivity compared to the
current generation detectors, should make confident as well as frequent
observations of eccentric systems Lower  _et al._ (2018); Kumar  _et al._ . It
may be worth noting that regardless of the availability of eccentric waveforms
for compact binary searches, current search strategies will need thorough
revisions in order to perform template based eccentric searches. In fact,
current template-based searches employing circular templates too ignore a
number of important physical effects such as those due to precessing spins and
nonquadrupole modes to manage the cost of the analysis Abbott  _et al._
(2016). More complete models are used in subsequent analyses to assess the
impact of different physical effects on measurements of important source
parameters such as masses and spins and/or to measure of the said effect in a
parameter estimation analyses. And hence, even though eccentric models being
developed today may not be used in conducting eccentric searches immediately,
these may be used, in injection analyses, to check the suitability of circular
templates in analyzing the data, or in a parameter estimation analysis
employing eccentric waveforms, to measure eccentricity. The latter, in the
context of ground based detectors, has important astrophysical implications,
as eccentricity can be used as a discriminator for the binary’s formation
channels (see Abbott  _et al._ (2019b) for a discussion). These considerations
have greatly motivated the (ongoing) efforts worldwide in the direction of
waveform development including the effect of eccentricity. In general, a
compact binary merger is characterized by a set of intrinsic (component masses
and spins) and extrinsic parameters (binary’s location and orientation).
Additionally, if the binary is in a noncircular orbit or has at least one
neutron star, eccentricity and tidal deformability parameters may also become
important. While the tidal effects only become important during late stages of
binary evolution for systems with at least one neutron star, eccentricity may
contribute through the entire inspiral stage depending upon its value when the
signal enters the detector’s sensitive band. Further, the presence of
eccentricity induces modulations in signal waveforms typical of eccentric
systems (see for instance, Chattaraj  _et al._ (2022); Hinder  _et al._
(2018)). The larger the eccentricity is, the larger these modulations become
and hence their impact on detection and parameter estimation. Additionally,
contributions from nonquadrupole modes become relevant for systems with
nonequal mass components and/or if the binary’s orbit is not face-on.222Face-
on systems are those which have orbital plane perpendicular to our line of
sight.Each of these modes are further subject to eccentricity induced
modulations indicative of the fact that two effects (of orbital eccentricity
and nonquadrupole modes) are inseparable when modeling eccentric systems
Chattaraj  _et al._ (2022) and neglecting one or both may lead to significant
biases or worse nondetection Rebei  _et al._ (2019). There have been numerous
efforts in the past focusing on including eccentricity in inspiral waveforms
from compact binary mergers (Mishra  _et al._ , 2015; Moore  _et al._ , 2016;
Tanay  _et al._ , 2016; Boetzel  _et al._ , 2019; Ebersold  _et al._ , 2019;
Konigsdorffer and Gopakumar, 2006; Moore and Yunes, 2019). While these efforts
model eccentricity together with the effect of nonquadrupole modes, these
assume binary components are nonspinning. Even though the effect of spins
(including the effect of spin precession) has been modeled within the post-
Newtonian formalism (Arun  _et al._ , 2009; Hartung and Steinhoff, 2011;
Marsat  _et al._ , 2013; Levi and Steinhoff, 2016; Bohe  _et al._ , 2013; Bohé
_et al._ , 2013; Marsat  _et al._ , 2014; Bohé  _et al._ , 2015; Blanchet  _et
al._ , 2011; Klein  _et al._ , 2013; Chatziioannou  _et al._ , 2013; Henry
_et al._ , 2022), a combined treatment including spins and eccentricity is
largely absent. Although, there have been efforts such as those of (Kidder,
1995; Buonanno  _et al._ , 2013; Majar and Vasuth, 2008; Klein and Jetzer,
2010; Klein  _et al._ , 2018; Vasuth and Majar, 2007; Klein, 2021) that
attempt to address this concern to some extent. For instance, spin
contributions to the 2PN order in a radiation field ($h_{ij}$) for binaries in
general orbits are available and can be connected to GW polarization and
subsequently to the spherical harmonic modes of gravitational waveforms using
standard formulas (see for instance Ref. Blanchet  _et al._ (2008)). Spherical
harmonic modes can also be directly computed using the inputs from the PN
theory following the approach of Kidder (2008). Typically it is this form of
gravitational waveforms that is extracted in numerical relativity (NR)
simulations. Additionally, they can readily be combined using the usual basis
of spherical harmonics to obtain GW polarizations which are typically useful
for data-analysis purposes. In this work, we compute spin contributions
through 2PN order in spherical harmonic modes of gravitational waveforms
following the approach of Kidder (2008) using the inputs developed within the
framework of the post-Newtonian, multipolar-post-Minowskian (PN-MPM) approach
Blanchet (2014); Buonanno  _et al._ (2013).

###  I.1 Summary of the current work

The current work can be seen as an extension to earlier efforts of Refs. Arun
_et al._ (2009); Buonanno  _et al._ (2013); Mishra  _et al._ (2015). On one
hand Refs. Kidder (1995); Buonanno  _et al._ (2013); Arun  _et al._ (2009)
together present the computation of spin effects in GW modes contributing up
to 2PN for binaries in inspiralling circular orbits. On the other hand, Ref.
Mishra  _et al._ (2015) lists explicitly nonspinning contributions to each
mode for binaries in noncircular orbits. Typically, in PN literature,
generalizations to circular orbit results are outlined under two separate
heads. First, one computes results for general orbits characterized by a set
of position and velocity variables. In the next step, one then specializes to
a specific orbit (an ellipse, a hyperbola or a parabola) with the aid of
relations that connect these general orbit variables to the parameters of the
special orbit one intends to study. This presentation style is followed here
too. Note that, through the 2PN order, only two kinds of spin terms appear in
mode expressions; linear-in-spin terms that arise due to couplings between
individual component spins and the orbital angular momentum and referred to as
spin-orbit (SO) terms, and the second, quadratic-in-spin terms due to
interactions between individual spin vectors referred to as spin-spin (SS)
terms Kidder (1995); Arun  _et al._ (2009); Buonanno  _et al._ (2013).
Additionally, not all modes contributing to the gravitational waveforms to 2PN
have spin contributions. Modes that have spin contributions are ($\ell$,
$|m|$)=($(2,2)$, $(2,1)$, $(3,3)$, $(3,2)$, $(3,1)$, $(4,3)$, $(4,1)$) and we
choose to present only these here. In addition, $(2,0)$ and $(3,0)$ modes are
also included as these are the two $m=0$ modes that contribute to 2PN order.
Nonspinning contributions to modes which do not have spin dependencies through
2PN can be found in Mishra  _et al._ (2015) both for general and elliptical
cases. Further, in the current work we restrict ourselves to nonprecessing (no
spin precession) binaries and choose to present our general orbit results in
terms of two different combinations of spin parameters following the approach
of Refs. Buonanno  _et al._ (2013) and Mishra  _et al._ (2016). We first list
our general orbit results in terms of a set of radial ($r,\dot{r}$) and
angular ($\phi,\dot{\phi}$) position and velocity variables in Sec. III. These
results are applicable to compact binaries in any noncircular orbits although
these results are valid only for binaries with nonprecessing spins. Next, we
specialize to the case of elliptical orbits using the quasi-Keplerian
representation of Klein and Jetzer (2010); Klein  _et al._ (2018) for spinning
compact binaries. While the representation can be used to describe binaries
with even precessing spins, we restrict ourselves to the case of non-
precessing systems. We list our results for the elliptical case in Sec. IV as
well as in Appendix B and also as Supplemental Material kp: . We would like to
highlight here that the results presented in Appendix B are applicable to
elliptical orbits of arbitrary eccentricity and hence are probably the most
useful results of the current paper. However, these have been pushed to the
appendix to avoid any disruption of the flow of the paper as these expressions
run over a couple of pages. Instead in Sec. IV we list these results as
leading eccentric corrections to the circular case by expanding our arbitrary
eccentricity results treating eccentricity as a small parameter. While these
should only be applied to orbits with small eccentricity, these significantly
improve clarity of the presentation as the circular limit can easily be
isolated. Nevertheless, we list these results to $6^{\rm th}$ power in
eccentricity, ${\mathcal{O}(e^{7})}$, as part of the Supplemental Material kp:
.333This means seventh and higher powers of eccentricity ($e$) have been
neglected while writing the results. We also provide for completeness, the
relations connecting the variables of general and elliptical orbits derived
from the results presented in Klein and Jetzer (2010); Klein  _et al._ (2018).
Throughout the paper, we only list the spin contributions to both inputs and
results expressions to avoid unnecessary duplication of nonspinning results
already available in the literature, although, leading nonspinning expressions
are included for ease in PN counting. Finally, complete nonspinning
contributions to 3PN and spinning contributions to 2PN level for binaries in
elliptical orbits to $\mathcal{O}(e^{7})$ are listed when writing the
expressions to the Supplemental Material kp: . Organization of this paper is
as follows. In Sec. II, we recall from literature the necessary PN inputs
including – relations connecting the spherical harmonic modes of gravitational
waveforms to a set of radiative multipole moments, relations connecting
radiative and source multipole moments as well as necessary expressions for
the source moments and equations of motion (EoM). We list our general orbit
results (spin contributions to each contributing mode) in terms of a set of
radial position and velocity variables in Sec. III. Next, we specialize to the
case of elliptical orbit and list our results for this case in Sec. IV as well
as in Appendix B and also as Supplemental Material kp: . Finally, we provide a
summary of results and conclude the paper in Sec. VI.

##  II Inputs from the post-Newtonian theory

###  II.1 Spherical harmonic decomposition of gravitational waveforms

Spherical harmonic modes of the gravitational waveform ($h^{\ell m}$) are
expressed in terms of the standard (non-STF444STF designates symmetric and
trace-free.) radiative mass multipoles ($U^{\ell m}$) and radiative current
multipoles ($V^{\ell m}$) as (Kidder, 2008; Blanchet  _et al._ , 2008)

$h^{\ell m}=-\frac{G}{\sqrt{2}Rc^{\ell+2}}\left[U^{\ell
m}-\frac{{i}}{c}V^{\ell m}\right]\,,$ (1)

where $G$ is the universal gravitational constant, $R$ is the distance to the
source, and $c$ is the speed of light. It was observed in Ref. (Kidder, 2008)
and explicitly shown in Ref. Faye  _et al._ (2012), that, if the binary’s
orbit is restricted to a plane (as would happen in the absence of spin-
precession), then there exists a mode separation. For instance, all “even”
$\ell+m$ modes would depend only on radiative mass multipoles ($U^{\ell m}$)
and all “odd” $\ell+m$ modes on current radiative multipoles ($V^{\ell m}$).
This observation leads to the following simplified expressions for each mode:

$\displaystyle h^{\ell m}$
$\displaystyle=-\frac{G}{\sqrt{2}Rc^{\ell+2}}\times\begin{cases}U^{\ell
m}&\text{ if }\ell+m\text{ is even, }\\\ \\\ -{i\over c}V^{\ell m}&\text{ if
}\ell+m\text{ is odd. }\end{cases}$ (2)

The standard (non-STF) moments ($U^{\ell m}$, $V^{\ell m}$) are related to
corresponding STF moments ($U_{L}$, $V_{L}$) as follows:

$\displaystyle U^{\ell m}$
$\displaystyle=\frac{4}{\ell!}\sqrt{\frac{(\ell+1)(\ell+2)}{2\ell(\ell-1)}}\alpha^{\ell
m}_{L}U_{L}\,,$ (3a) $\displaystyle V^{\ell m}$
$\displaystyle=-\frac{8}{\ell!}\sqrt{\frac{\ell(\ell+2)}{2(\ell+1)(\ell-1)}}\alpha^{\ell
m}_{L}V_{L}\,.$ (3b)

In the above, $L$=$i_{1}\,i_{2}\cdots i_{\ell}$ represents a multi-index
comprising of $\ell$ spatial indices. The object $\alpha^{\ell m}_{L}$ is
again a STF tensor that connects the usual basis of spherical harmonics
($Y^{\ell m}$) to another set of STF tensors, $N_{L}=N_{i_{1}}...N_{i_{\ell}}$
as Blanchet  _et al._ (2008)

$\alpha_{L}^{\ell m}=\int d\Omega N_{\langle L\rangle}\overline{Y}^{\ell
m}\,.$ (4)

Note that $\bm{N}$ is a unit vector pointing toward the detector along the
line joining the source and the detector. For instance, if we fix the binary’s
plane to be the $x$-$y$ plane ${\bm{N}}$ in terms of angles ($\Theta$, $\Phi$)
giving the binary’s location reads

$\displaystyle{\bm{N}}=\sin\Theta\cos\Phi\,\bm{\hat{x}}+\sin\Theta\sin\Phi\,\bm{\hat{y}}+\cos\Theta\,\bm{\hat{z}}\,.$
(5)

Finally, the $Y^{\ell m}$ in terms of location angles ($\Theta$, $\Phi$) are
given by the following relation:

$\displaystyle Y^{\ell m}(\Theta,\Phi)$
$\displaystyle=(-)^{m}\frac{1}{2^{\ell}\ell!}\left[\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}\right]^{1/2}\mathrm{e}^{im\Phi}\,$
$\displaystyle(\sin\Theta)^{m}\frac{\mathrm{d}^{\ell+m}}{\mathrm{\leavevmode\nobreak\
d}(\cos\Theta)^{\ell+m}}\left(\cos^{2}\Theta-1\right)^{\ell}\,.$ (6)

###  II.2 Source moments

Using the inputs of Sec. II.1 each mode can explicitly be expressed in terms
of a mass-type or current-type radiative multipole ($U_{L}$, $V_{L}$). In the
PN-MPM approach Blanchet (2014); Blanchet and Faye (2001); Blanchet  _et al._
(2002); Blanchet and Iyer (2005); Blanchet  _et al._ (2004, 2005), these
radiative mass and current multipole moments ($U_{L}$,$V_{L}$) are related to
a set of source multipole moments ($I_{L}$, $J_{L}$) through a set of mass and
current type canonical moments ($M_{L}$, $S_{L}$). However, as we shall see
below, based on the accuracy of inputs that are required in the current work,
the radiative multipoles ($U_{L}$,$V_{L}$) appearing in mode expressions can
simply be replaced by source moments ($I_{L}$,$J_{L}$). At 2PN the two sets of
moments are related as555In principle, there exist hereditary contributions
starting at 1.5PN order. However, these can be ignored when computing spin
effects to 2PN order.

$\displaystyle U_{L}(U)$
$\displaystyle=I^{(\ell)}_{L}(U)+\mathcal{O}\bigg{(}\frac{1}{c^{5}}\bigg{)}\,,$
(7a) $\displaystyle V_{L}(U)$
$\displaystyle=J^{(\ell)}_{L}(U)+\mathcal{O}\bigg{(}\frac{1}{c^{5}}\bigg{)}\,.$
(7b)

In the above, superscript denotes $\ell$th time-derivatives of source moments
and $\mathcal{O}(1/c^{5})$ indicates that contributions beyond 2PN have been
ignored.666An expression is said to have an accuracy of ${(n/2)}$PN if the
results are expressed as a power series in (1/c) to its $n$th power. Source
multipole moments ($I_{L},J_{L}$) are further expressed as a PN series in
terms of binary parameters. In this work we focus on computing the spin
effects in $h^{\ell m}$ to 2PN order for systems with nonprecessing spins.
Specifically, spin-orbit and spin-spin effects are computed. In order to have
sufficient clarity when listing relevant inputs and desired results, we choose
to split the two spin contributions. Further, to enable easy PN counting we
also provide leading nonspinning (NS) contributions to various quantities.
With this view in mind, we express the source multipoles as777The SS
contributions from current moments ($J_{L}$) appear beyond 2PN order and hence
are not included here.

$\displaystyle I_{L}$
$\displaystyle=I^{{\rm{NS}}}_{L}+I^{{\rm{SO}}}_{L}+I^{{\rm{SS}}}_{L},\,$ (8a)
$\displaystyle J_{L}$ $\displaystyle=J^{{\rm{NS}}}_{L}+J^{{\rm{SO}}}_{L}\,,$
(8b)

where NS, SO and SS in the superscript denote the nonspinning, spin-orbit, and
spin-spin pieces of the moments. It may be important to note at this stage
that, except for the ($\ell=2,|m|=1$) mode, computing all other modes,
($\ell,|m|$)=$((2,2),\,(3,3),\,(3,2),\,(3,1),\,(4,3),\,(4,1))$ through 2PN
order requires only “leading” inputs (NS, SO, or SS). The $(2,\,1)$ mode, on
the other hand, requires 1PN inputs as it would contribute both at 1PN and 2PN
level. Further, computing $(2,0)$ and $(3,0)$ modes also requires only leading
inputs. It should be clear from Eq. (2) [when combined with Eq. (3) and Eq.
(7)] that the $(2,1)$ mode depends on the current quadrupole moment ($J_{ij}$)
and hence spin contributions to this moment are required with 1PN accuracy
[see Eq. (10) below]. Computation of all other modes requires source moments
with only leading SO and SS effect. Explicit expressions for source moments
with each of these contributions have been computed in Refs. Kidder (1995);
Blanchet  _et al._ (2006); Arun  _et al._ (2009); Buonanno  _et al._ (2013).
Below we reproduce these expressions, with the accuracy that is needed in our
calculations. These are expressed in terms of position and velocity vectors,
($x^{i},v^{i}$) or ($\bm{x},\bm{v}$), and spin vectors ($\bm{S},\bm{\Sigma}$).
The spin vectors (${\bm{S}}$, ${\bm{\Sigma}}$) are useful combinations of
individual spin vectors ($\bm{S_{1,2}}$) and component masses ($m_{1,2}$),
where subscript labels represent the two compact objects of the binary. They
read

$\displaystyle\bm{S}$ $\displaystyle\equiv\bm{S}_{1}+\bm{S}_{2}\,,$ (9a)
$\displaystyle\bm{\Sigma}$ $\displaystyle\equiv
M\left[\frac{\bm{S}_{2}}{m_{2}}-\frac{\bm{S}_{1}}{m_{1}}\right]\,,$ (9b)

where $M$ is the total mass of the binary. The only moment needed with
subleading spin corrections is the current quadrupole moment $(J_{ij})$. The
relative 1PN SO and leading NS pieces read

$\displaystyle J^{\rm{NS}}_{ij}$
$\displaystyle=-M\nu\,\delta\,\varepsilon_{ab\langle
i}x^{j\rangle}x^{a}v^{b}+\mathcal{O}\left(\frac{1}{c^{2}}\right)\,,$ (10a)
$\displaystyle J^{\rm{SO}}_{ij}$
$\displaystyle=\frac{\nu}{c}\Bigg{(}-\frac{3}{2}\,x^{\langle
i}\,\Sigma^{j\rangle}\Bigg{)}+\frac{\nu}{c^{3}}\Bigg{\\{}\frac{3}{7}\left(1-\frac{16\nu}{3}\right)r\,\dot{r}\,v^{\langle
i}\,\Sigma^{j\rangle}+\frac{3}{7}\,\delta\,r\,\dot{r}\,v^{\langle
i}\,S^{j\rangle}+\bigg{[}\frac{27}{14}\left(1-\frac{109\nu}{27}\right)(\bm{v}\cdot\bm{\Sigma})+\frac{27}{14}\delta\,(\bm{v}\cdot\bm{S})\bigg{]}\,$
$\displaystyle x^{\langle
i}\,v^{j\rangle}+\left[-\frac{11}{14}\left(1-\frac{47\nu}{11}\right)(\bm{x}\cdot\bm{\Sigma})-\frac{11}{14}\delta\,(\bm{x}\cdot\bm{S})\right]v^{\langle
i}\,v^{j\rangle}+\bigg{[}\frac{19}{28}\left(1+\frac{13\nu}{19}\right)\frac{GM}{r}-\frac{29}{28}\left(1-\frac{143\nu}{29}\right)v^{2}\bigg{]}x^{\langle
i}\,\Sigma^{j\rangle}\,$
$\displaystyle+\left[-\frac{4}{7}\left(1-\frac{31\nu}{8}\right)(\bm{x}\cdot\bm{\Sigma})-\frac{29}{14}\delta\,(\bm{x}\cdot\bm{S})\right]\frac{GM}{r^{3}}\,x^{\langle
i}\,x^{j\rangle}+\left[-\frac{1}{14}\frac{GM}{r}-\frac{2}{7}v^{2}\right]\delta\,x^{\langle
i}\,S^{j\rangle}\Bigg{\\}}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,.$ (10b)

All other moments are needed with only leading spin effects. The NS, SO, and
SS pieces for relevant moments read

$\displaystyle I^{\rm{NS}}_{ij}$ $\displaystyle=M\nu\,x^{\langle
i}x^{j\rangle}+\mathcal{O}\left(\frac{1}{c^{2}}\right)\,,$ (11a)
$\displaystyle I^{\rm{SO}}_{ij}$
$\displaystyle=\frac{8\nu}{3c^{3}}\,x^{\langle
i}\left[\bm{v}\times\left(\bm{S}+\delta\bm{\Sigma}\right)\right]^{j\rangle}$
$\displaystyle-\frac{4\nu}{3c^{3}}\,v^{\langle
i}\left[\bm{x}\times\left(\bm{S}+\delta\bm{\Sigma}\right)\right]^{j\rangle}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(11b) $\displaystyle I^{\rm{SS}}_{ij}$
$\displaystyle=-\frac{\kappa_{1}}{m_{1}c^{4}}S_{1}^{\langle
i}S_{1}^{j\rangle}-\frac{\kappa_{2}}{m_{2}c^{4}}S_{2}^{\langle
i}S_{2}^{j\rangle}+\mathcal{O}\left(\frac{1}{c^{6}}\right)\,,$ (11c)
$\displaystyle I^{\rm{NS}}_{ijk}$ $\displaystyle=-M\nu\,\delta\,x^{\langle
i}x^{j}x^{k\rangle}+\mathcal{O}\left(\frac{1}{c^{2}}\right)\,,$ (11d)
$\displaystyle I^{\rm{SO}}_{ijk}$
$\displaystyle=\frac{\nu}{c^{3}}\,\Bigg{[}-\frac{9}{2}\,\delta\,x^{\langle
i}x^{j}\left(\bm{v}\times\bm{S}\right)^{k\rangle}$
$\displaystyle-\frac{9}{2}\,\left(1-\frac{11\nu}{3}\right)\,x^{\langle
i}x^{j}\left(\bm{v}\times\bm{\Sigma}\right)^{k\rangle}$
$\displaystyle+3\,\delta\,x^{\langle
i}v^{j}\left(\bm{x}\times\bm{S}\right)^{k\rangle}+3\,\left(1-3\nu\right)x^{\langle
i}v^{j}\left(\bm{x}\times\bm{\Sigma}\right)^{k\rangle}\Bigg{]}\,$
$\displaystyle+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$ (11e)
$\displaystyle J^{\rm{SO}}_{ijk}$
$\displaystyle=\frac{\nu}{c}\Bigg{(}2x^{\langle
i}x^{j}S^{k\rangle}+2\delta\,x^{\langle
i}x^{j}\Sigma^{k\rangle}\Bigg{)}+\mathcal{O}\left(\frac{1}{c^{3}}\right)\,,$
(11f) $\displaystyle J^{\rm{SO}}_{ijkl}$
$\displaystyle=-\frac{5\nu}{2c}\,\Bigg{[}\delta\,x^{\langle
i}x^{j}x^{k}S^{l\rangle}+\left(1-3\nu\right)x^{\langle
i}x^{j}x^{k}\Sigma^{l\rangle}\Bigg{]}$
$\displaystyle+\mathcal{O}\left(\frac{1}{c^{3}}\right)\,.$ (11g)

In the above expressions, $\nu$=$(m_{1}m_{2}/M^{2})$ is the symmetric mass
ratio parameter and $\delta$=$(m_{1}-m_{2})/M$, with $m_{1}$, $m_{2}$ being
component masses. (Note that the above definitions lead to
$\delta=\pm\sqrt{1-4\nu}$ depending upon the sign of the difference
$m_{1}-m_{2}$.) Note also, the angular brackets ($\langle\rangle$) around
indices indicate that expressions are symmetric and trace-free in indices they
enclose. Additionally, $r=|\bm{x}|$, is the binary’s relative separation and
$\dot{r}=({\bm{x}\cdot\bm{v})/r}$ the radial component of the relative
velocity vector. Finally, the parameters $\kappa_{1,2}$ characterize the spin-
induced mass quadrupole moment of the two components, defined such that for
black holes $\kappa_{1,2}=1$.

###  II.3 Equations of motion

Equations of motion, which help writing mode expressions completely in terms
of position and velocity variables, have been computed with the accuracy
desired in (Buonanno  _et al._ , 2013). Similar to the source moments these
have contributions from SO and SS effects. These are expressed in terms of
vectors along the relative separation ($\bm{x}$) and relative velocity
($\bm{v}$) and involve suitable combinations of spin vectors introduced in
(Kidder, 1995; Buonanno  _et al._ , 2013). We reproduce these here for the
convenience of the reader. Again, we split the related expression in terms of
NS, SO, and SS pieces for clarity in presentation. The structure for EoM (or
simply acceleration, ${\bm{a}}$) is as follows:

$\displaystyle\bm{a}$ $\displaystyle=$
$\displaystyle\bm{a}^{{\rm{NS}}}+\bm{a}^{{\rm{SO}}}+\bm{a}^{{\rm{SS}}}\,,$
(12)

with,

$\displaystyle\bm{a}^{{\rm{NS}}}$
$\displaystyle=-\frac{GM}{r^{3}}\bm{x}-\frac{1}{c^{2}}\frac{GM}{r^{3}}\Bigg{\\{}\bigg{[}(1+3\nu)v^{2}-\frac{3}{2}\nu\dot{r}^{2}$
$\displaystyle-\frac{4GM}{r}\left(1+\frac{\nu}{2}\right)\bigg{]}\bm{x}\,-\,4r\dot{r}\Big{(}1-\frac{\nu}{2}\Big{)}\bm{v}\Bigg{\\}}+\mathcal{O}\left(\frac{1}{c^{4}}\right),$
(13a) $\displaystyle\bm{a}^{{\rm{SO}}}$
$\displaystyle=\frac{1}{c^{3}}\frac{GM}{r^{3}}\left(\frac{GM}{r}\right)\Bigg{\\{}\left[\left(\frac{\bm{x}\times\bm{v}}{r}\right)\cdot\left(\frac{12\bm{S}}{GM^{2}}\right.\right.$
$\displaystyle\left.\left.+\frac{6\delta\bm{\Sigma}}{GM^{2}}\right)\right]\bm{x}-r\bigg{[}\bm{v}\times\bigg{(}\frac{7\bm{S}}{GM^{2}}+\frac{3\delta\bm{\Sigma}}{GM^{2}}\bigg{)}\bigg{]}$
$\displaystyle+3\dot{r}\bigg{[}\bm{x}\times\bigg{(}\frac{3\bm{S}}{GM^{2}}+\frac{\delta\bm{\Sigma}}{GM^{2}}\bigg{)}\bigg{]}\Bigg{\\}}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(13b) $\displaystyle\bm{a}^{{\rm{SS}}}$ $\displaystyle=-{3\over
2}\frac{1}{c^{4}}\frac{GM}{r^{3}}\left(\frac{GM}{r}\right)^{2}\Bigg{[}\bm{x}\,\bigg{(}\frac{{\bm{S}_{0}^{+}}}{GM^{2}}\cdot\frac{{\bm{S}_{0}^{-}}}{GM^{2}}\bigg{)}\,$
$\displaystyle+\bigg{(}\bm{x}\cdot\frac{{\bm{S}_{0}^{+}}}{GM^{2}}\bigg{)}\frac{{\bm{S}_{0}^{-}}}{GM^{2}}+\bigg{(}\bm{x}\cdot\frac{{\bm{S}_{0}^{-}}}{GM^{2}}\bigg{)}\frac{{\bm{S}_{0}^{+}}}{GM^{2}}\,$
$\displaystyle-\frac{5\bm{x}}{r^{2}}\,\bigg{(}\bm{x}\cdot\frac{{\bm{S}_{0}^{+}}}{GM^{2}}\bigg{)}\bigg{(}\bm{x}\cdot\frac{{\bm{S}_{0}^{-}}}{GM^{2}}\bigg{)}\Bigg{]}+\mathcal{O}\left(\frac{1}{c^{6}}\right)\,.$
(13c)

Note again, the NS part above is 1PN accurate. This again is keeping in mind
the computation of the ($\ell$=2, $m$=1) mode which has leading (1PN) and
subleading (2PN) contributions. For all other modes, contributing to the 2PN
waveform Newtonian acceleration in the NS piece together with SO and SS pieces
should suffice. Below we summarize notation of Buonanno  _et al._ (2013) used
in expressing the EoM. The set of spin vectors ($\bm{S_{0}^{+}}$,
$\bm{S_{0}^{-}}$) that appear in the SS part of the EoM are combinations of
individual spin vectors (${\bm{S}_{1}}$, ${\bm{S}_{2}}$), component masses
($m_{1,2}$), and parameters characterizing spin-induced effects
($\kappa_{1,2}$). These read

$\displaystyle\bm{S}^{+}_{0}=\frac{M}{m_{1}}\bigg{(}\frac{\kappa_{1}}{\kappa_{2}}\bigg{)}^{1/4}\left(1+\sqrt{1-\kappa_{1}\kappa_{2}}\right)^{1/2}{\bm{S_{1}}}$
$\displaystyle\
+\frac{M}{m_{2}}\bigg{(}\frac{\kappa_{2}}{\kappa_{1}}\bigg{)}^{1/4}\left(1-\sqrt{1-\kappa_{1}\kappa_{2}}\right)^{1/2}{\bm{S_{2}}}\,,$
(14a)
$\displaystyle\bm{S}^{-}_{0}=\frac{M}{m_{2}}\bigg{(}\frac{\kappa_{2}}{\kappa_{1}}\bigg{)}^{1/4}\left(1+\sqrt{1-\kappa_{1}\kappa_{2}}\right)^{1/2}{\bm{S_{2}}}$
$\displaystyle\
+\frac{M}{m_{1}}\bigg{(}\frac{\kappa_{1}}{\kappa_{2}}\bigg{)}^{1/4}\left(1-\sqrt{1-\kappa_{1}\kappa_{2}}\right)^{1/2}{\bm{S_{1}}}\,.$
(14b)

Additionally, expressions for $\ddot{r}$ and $\dot{v}$ will also be required
and can be computed using the following relations,

$\displaystyle\dot{v}$ $\displaystyle=\frac{\bm{a}\cdot\bm{v}}{v}\,,$ (15a)
$\displaystyle\ddot{r}$
$\displaystyle=\frac{1}{r}\left[\left(v^{2}-\dot{r}^{2}\right)+{\bm{a}}\cdot\bm{x}\right]\,.$
(15b)

with $v=|\bm{v}|$.

##  III Spherical Harmonic modes with spin effects to 2PN : General orbits,
Nonprecessing case

We now have all the inputs that are needed to compute spin contributions up to
2PN in spherical harmonic mode expressions. While the computations are algebra
intensive they are straightforward.888All computations are performed using
Mathematica and the tensor algebra package, xACT (Martin-Garcia  _et al._ ,
2019). It should be clear that the computation of radiative multipoles
($U_{L},V_{L}$) and hence the computation of modes ($h^{\ell m}$) involves
computing time derivatives of source multipoles [see for instance, Eq. (7)].
Further, as pointed out in Sec. II.3, EoM help writing mode expressions solely
in terms of position and velocity variables. [With source moments written in
terms of position and velocity variables, time derivatives of source moment
will induce acceleration terms ($\bm{a},\ddot{r},\dot{v}$) and EoM can be used
to reexpress the time derivatives again in terms of position and velocity
variables.] Results expressed in terms of position and velocity, valid for all
possible orbits, are referred to as _general orbit_ results. In this section,
we list spin contributions through 2PN for each relevant mode in terms of
position and velocity variables, namely, the radial separation ($r$), radial
velocity ($\dot{r}$), orbital phase ($\phi$), and the angular velocity
($\dot{\phi}$). Note also that, from here onward, we specialize to the case of
nonprecessing systems. Note however, that inputs of Sec. II together with
evolution equations for spin vectors ($\bm{S}_{1,2}$) listed in Buonanno  _et
al._ (2013) can be used to compute waveforms including the effect of
precession. For us, the choice to work with a nonprecessing system drastically
simplifies the resulting expressions and helps us provide a completely
analytical model, which otherwise would be a semianalytical prescription in
the absence of analytical solutions to spin-evolution equations for binaries
in noncircular orbits. The first obvious simplification occurs due to the use
of spin vectors of constant magnitude ($\bm{S}^{c}_{1,2}$ with
$\bm{S}^{c}_{i}\cdot\bm{\dot{S}}^{c}_{i}=0$) as opposed to the usual spin
vectors ($\bm{S}_{1,2}$) as in Blanchet  _et al._ (2006); Buonanno  _et al._
(2013). Further, unit vectors ($\bm{\hat{n}}$, $\bm{\hat{\lambda}}$) along the
relative separation and velocity vector ($\bm{x}$, $\bm{v}$), together with
the unit vector along the orbital angular momentum, $\bm{\hat{\ell}}$, form an
orthonormal triad.999Note that we use the same letter, $\ell$, for
representing the mode number in $h^{\ell m}$ and the orbital angular momentum
unit vector, however, throughout the paper we express the angular momentum
unit vector as $\bm{\hat{\ell}}$. This further helps in expressing results
solely in terms of $(\bm{\hat{\ell}}\cdot\bm{S}_{\rm{c}})$ and
$(\bm{\hat{\ell}}\cdot\bm{\Sigma}_{\rm{c}})$, where $\bm{S_{c}}$ and
$\bm{\Sigma_{c}}$ in terms of spin vectors ($\bm{S}$, $\bm{\Sigma}$) are
defined in Buonanno  _et al._ (2013) and we simply reproduce them here for the
ease of the reader,101010See Sec. IV E of Buonanno  _et al._ (2013) for a
detailed discussion.

$\displaystyle\bm{S}_{\rm c}$
$\displaystyle=\bm{S}+\frac{GM\nu}{rc^{2}}\Big{[}2\bm{S}+\delta\,\bm{\Sigma}\Big{]}+\mathcal{O}\left(\frac{1}{c^{3}}\right)\,,$
(16a) $\displaystyle\bm{\Sigma}_{\rm c}$
$\displaystyle=\bm{\Sigma}+\frac{GM}{rc^{2}}\Big{[}\delta\,\bm{S}+(1-2\nu)\bm{\Sigma}\Big{]}+\mathcal{O}\left(\frac{1}{c^{3}}\right).$
(16b)

We are now ready to write final expressions for spin effects in each mode
contributing to gravitational waveforms through 2PN order. Following Mishra
_et al._ (2016) we work with a source frame similar to the one chosen for
spinning systems in Refs. Arun  _et al._ (2009); Buonanno  _et al._ (2013). As
was pointed out in Mishra  _et al._ (2016), this choice of the source frame is
different from the one in Blanchet  _et al._ (2008) which does not discuss
spin contributions. Nevertheless, mode expressions in conventions used in
Blanchet  _et al._ (2008) can be obtained by simply dividing by a factor of
$i^{m}$ to our results below. [This is also the reason we have introduced a
multiplicative factor of $i^{2m}$ when presenting the results; see for
instance, Eq. (17) below.) In terms of generic position and velocity variables
($r,\dot{r},\phi,\dot{\phi}$), mode expressions take the following form,

$h^{\ell
m}=-4\,{i^{2m}}\sqrt{\frac{\pi}{5}}\,\left(\frac{GM\nu}{c^{4}R}\right)\mathrm{e}^{-im\phi}H^{\ell
m}$ (17)

with

$\displaystyle H^{\ell m}=H^{\ell m}_{\rm NS}+H^{\ell m}_{\rm SO}+H^{\ell
m}_{\rm SS}\,,$ (18)

where NS, SO, and SS contributions for all relevant modes read

$\displaystyle H^{22}_{\rm{NS}}$
$\displaystyle=\frac{GM}{r}+\left(i\dot{r}+r\dot{\phi}\right)^{2}+\mathcal{O}\left(\frac{1}{c^{2}}\right)\,,$
(19a) $\displaystyle H^{22}_{\rm{SO}}$
$\displaystyle=-\frac{1}{3}\frac{1}{c^{3}}\left(\frac{GM}{r}\right)^{2}\bigg{[}\left(4i\dot{r}+7r\dot{\phi}\right)\left(\bm{\hat{\ell}}\cdot\bm{s}_{c}\right)+\delta\left(-2i\dot{r}+r\dot{\phi}\right)\left(\bm{\hat{\ell}}\cdot\bm{\sigma}_{c}\right)\bigg{]}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(19b) $\displaystyle H^{22}_{\rm{SS}}$
$\displaystyle=\frac{3}{2}\frac{1}{c^{4}}\left(\frac{GM}{r}\right)^{3}\bigg{[}\left(\bm{s}_{0}^{+}\cdot\bm{s}_{0}^{-}\right)\bigg{]}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(19c) $\displaystyle H^{21}_{\rm{NS}}$
$\displaystyle=-\frac{2}{3}\frac{1}{c}\left(\frac{GM}{\,r}\right)\delta\left(r\dot{\phi}\right)+\mathcal{O}\left(\frac{1}{c^{3}}\right)\,,$
(19d) $\displaystyle H^{21}_{\rm{SO}}$
$\displaystyle=-\frac{1}{c^{2}}\left(\frac{GM}{r}\right)^{2}\left(\bm{\hat{\ell}}\cdot\bm{\sigma}_{c}\right)+\frac{1}{42}\frac{1}{c^{4}}\left(\frac{GM}{r}\right)^{2}\Bigg{\\{}2\delta\bigg{[}\frac{24GM}{r}+2\dot{r}^{2}+48i\dot{r}\left(r\dot{\phi}\right)+62\left(r\dot{\phi}\right)^{2}\bigg{]}\left(\bm{\hat{\ell}}\cdot\bm{s}_{c}\right)-\bigg{[}-\frac{202GM}{r}\,$
$\displaystyle\times\left(1-\frac{37\nu}{101}\right)-109\left(1-\frac{68\nu}{109}\right)\dot{r}^{2}+198i\dot{r}\left(r\dot{\phi}\right)\left(1+\frac{109\nu}{99}\right)-40\left(r\dot{\phi}\right)^{2}\left(1-\frac{29\nu}{5}\right)\bigg{]}\left(\bm{\hat{\ell}}\cdot\bm{\sigma}_{c}\right)\Bigg{\\}}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(19e) $\displaystyle H^{20}_{\rm{NS}}$
$\displaystyle=\sqrt{\frac{2}{3}}\Bigg{\\{}\frac{GM}{r}-\left(r\dot{\phi}\right)^{2}-\dot{r}^{2}\Bigg{\\}}+\mathcal{O}\left(\frac{1}{c^{2}}\right)\,,$
(19f) $\displaystyle H^{20}_{\rm{SO}}$
$\displaystyle=-\sqrt{\frac{2}{3}}\frac{1}{c^{3}}\left(\frac{GM}{r}\right)^{2}\left(r\dot{\phi}\right)\Bigg{\\{}5\left(\bm{\hat{\ell}}\cdot\bm{s}_{c}\right)+3\delta\left(\bm{\hat{\ell}}\cdot\bm{\sigma}_{c}\right)\Bigg{\\}}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(19g) $\displaystyle H^{20}_{\rm{SS}}$
$\displaystyle=\sqrt{\frac{3}{2}}\frac{1}{c^{4}}\left(\frac{GM}{r}\right)^{3}\bigg{[}\left(\bm{s}_{0}^{+}\cdot\bm{s}_{0}^{-}\right)\bigg{]}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(19h) $\displaystyle H^{33}_{\rm{NS}}$
$\displaystyle=-\frac{\sqrt{5}}{2\sqrt{42}}\frac{1}{c}\delta\,\Bigg{\\{}2\left(i\dot{r}+r\dot{\phi}\right)^{3}+\frac{GM}{r}\left[4i\dot{r}+7r\dot{\phi}\right]\Bigg{\\}}+\mathcal{O}\left(\frac{1}{c^{3}}\right)\,,$
(19i) $\displaystyle H^{33}_{\rm{SO}}$
$\displaystyle=\frac{\sqrt{5}}{4\sqrt{42}}\frac{1}{c^{4}}\left(\frac{GM}{\,r}\right)^{2}\Bigg{\\{}\delta\,\left[-10\dot{r}^{2}+38i\dot{r}\left(r\dot{\phi}\right)+43\left(r\dot{\phi}\right)^{2}\right]\left(\bm{\hat{\ell}}\cdot\bm{s}_{c}\right)+\bigg{[}-\frac{4GM}{r}\left(1-5\nu\right)+2\dot{r}^{2}(1-5\nu)+2i\dot{r}\,$
$\displaystyle\times\left(r\dot{\phi}\right)\left(1+\nu\right)+19\left(r\dot{\phi}\right)^{2}\left(1-\frac{53\nu}{19}\right)\bigg{]}\left(\bm{\hat{\ell}}\cdot\bm{\sigma}_{c}\right)\Bigg{\\}}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(19j) $\displaystyle H^{32}_{\rm{NS}}$
$\displaystyle=\frac{\sqrt{5}}{6\sqrt{7}}\frac{1}{c^{2}}\left(\frac{GM}{r}\right)\left(1-3\nu\right)\left(r\dot{\phi}\right)\bigg{[}i\dot{r}+4r\dot{\phi}\bigg{]}+\mathcal{O}\left(\frac{1}{c^{4}}\right)\,,$
(19k) $\displaystyle H^{32}_{\rm{SO}}$
$\displaystyle=\frac{\sqrt{5}}{3\sqrt{7}}\frac{1}{c^{3}}\left(\frac{GM}{r}\right)^{2}\bigg{[}i\dot{r}+4r\dot{\phi}\bigg{]}\bigg{[}\left(\bm{\hat{\ell}}\cdot\bm{s}_{c}\right)+\delta\left(\bm{\hat{\ell}}\cdot\bm{\sigma}_{c}\right)\bigg{]}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(19l) $\displaystyle H^{31}_{\rm{NS}}$
$\displaystyle=\frac{1}{6\sqrt{14}}\frac{1}{c}\delta\Bigg{\\{}6\left(-i\dot{r}+r\dot{\phi}\right)\left(i\dot{r}+r\dot{\phi}\right)^{2}-\frac{GM}{r}\bigg{[}12i\dot{r}+7r\dot{\phi}\bigg{]}\Bigg{\\}}+\mathcal{O}\left(\frac{1}{c^{3}}\right)\,,$
(19m) $\displaystyle H^{31}_{\rm{SO}}$
$\displaystyle=\frac{1}{12\sqrt{14}}\frac{1}{c^{4}}\left(\frac{GM}{r}\right)^{2}\Bigg{\\{}\delta\left[-10\dot{r}^{2}+66i\dot{r}\left(r\dot{\phi}\right)+59\left(r\dot{\phi}\right)^{2}\right]\left(\bm{\hat{\ell}}\cdot\bm{s}_{c}\right)+\bigg{[}-\frac{4GM}{r}\left(1-5\nu\right)+2\dot{r}^{2}\left(1-5\nu\right)+54i\dot{r}\,$
$\displaystyle\times\left(r\dot{\phi}\right)\left(1-\frac{101\nu}{27}\right)+35\left(r\dot{\phi}\right)^{2}\left(1-\frac{149\nu}{35}\right)\bigg{]}\left(\bm{\hat{\ell}}\cdot\bm{\sigma}_{c}\right)\Bigg{\\}}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(19n) $\displaystyle H^{30}_{\rm{NS}}$
$\displaystyle=\frac{i}{\sqrt{42}}\frac{1}{c^{2}}\left(1-3\nu\right)\frac{GM}{r}\dot{r}\left(r\dot{\phi}\right)+\mathcal{O}\left(\frac{1}{c^{4}}\right)\,,$
(19o) $\displaystyle H^{30}_{\rm{SO}}$
$\displaystyle=i\sqrt{\frac{2}{21}}\frac{1}{c^{3}}\left(\frac{GM}{r}\right)^{2}\dot{r}\bigg{[}\left(\bm{\hat{\ell}}\cdot\bm{s}_{c}\right)+\delta\left(\bm{\hat{\ell}}\cdot\bm{\sigma}_{c}\right)\bigg{]}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(19p) $\displaystyle H^{43}_{\rm{NS}}$
$\displaystyle=-\frac{2}{3\sqrt{70}}\frac{1}{c^{3}}\left(\frac{GM}{\,r}\right)\delta\left(1-2\nu\right)\left(r\dot{\phi}\right)\bigg{[}\frac{GM}{r}-\frac{1}{2}\dot{r}^{2}+\frac{5}{2}i\dot{r}\left(r\dot{\phi}\right)+\frac{23}{4}\left(r\dot{\phi}\right)^{2}\bigg{]}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(19q) $\displaystyle H^{43}_{\rm{SO}}$
$\displaystyle=-\frac{\sqrt{5}}{3\sqrt{14}}\frac{1}{c^{4}}\left(\frac{GM}{r}\right)^{2}\left[\frac{GM}{r}-\frac{1}{2}\dot{r}^{2}+\frac{5}{2}i\dot{r}\left(r\dot{\phi}\right)+\frac{23}{4}\left(r\dot{\phi}\right)^{2}\right]\bigg{[}\delta\left(\bm{\hat{\ell}}\cdot\bm{s}_{c}\right)+\left(1-3\nu\right)\left(\bm{\hat{\ell}}\cdot\bm{\sigma}_{c}\right)\bigg{]}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(19r) $\displaystyle H^{41}_{\rm{NS}}$
$\displaystyle=-\frac{2}{7\sqrt{70}}\frac{1}{c^{3}}\left(\frac{GM}{r}\right)\delta\left(1-2\nu\right)\left(r\dot{\phi}\right)\left[\frac{GM}{r}-\frac{1}{2}\dot{r}^{2}+\frac{5}{6}i\dot{r}\left(r\dot{\phi}\right)-\frac{11}{12}\left(r\dot{\phi}\right)^{2}\right]+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(19s) $\displaystyle H^{41}_{\rm{SO}}$
$\displaystyle=-\frac{\sqrt{5}}{7\sqrt{2}}\frac{1}{c^{4}}\left(\frac{GM}{r}\right)^{2}\left[\frac{GM}{r}-{1\over
2}\dot{r}^{2}+{5\over 6}i\dot{r}\left(r\dot{\phi}\right)-{11\over
12}\left(r\dot{\phi}\right)^{2}\right]\bigg{[}\delta\left(\bm{\hat{\ell}}\cdot\bm{s}_{c}\right)+\left(1-3\nu\right)\left(\bm{\hat{\ell}}\cdot\bm{\sigma}_{c}\right)\bigg{]}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(19t)

where we have used $v^{2}=\dot{r}^{2}+r^{2}\dot{\phi}^{2}$. Note also in the
above, we have redefined the spin variables
($\bm{S}_{c},\bm{\Sigma}_{c},\bm{S}^{+}_{0},\bm{S}^{-}_{0}$) as

$\displaystyle\bm{s}_{c}$
$\displaystyle=\frac{\bm{S}_{c}}{GM^{2}},\,\bm{\sigma}_{c}=\frac{\bm{\Sigma}_{c}}{GM^{2}}\,,\bm{s}^{+}_{0}=\frac{\bm{S}^{+}_{0}}{GM^{2}},\,\bm{s}^{-}_{0}=\frac{\bm{S}^{-}_{0}}{GM^{2}}\,.$
(20)

Following Mishra  _et al._ (2016), we also express below the spin
contributions in terms of symmetric and antisymmetric combinations of
dimensionless component spin parameters,
$\bm{\chi}^{c}_{i}={\bm{S}^{c}_{i}}/G\,m^{2}_{i}$, useful in data-analysis
applications. [Note that this change of spin variables does not affect the
leading nonspinning part included in Eq .(19) and hence is not included in Eq.
(21) to avoid duplication.] With
$\bm{\chi}^{c}_{s}=\frac{1}{2}(\bm{\chi}^{c}_{1}+\bm{\chi}^{c}_{2})$ and
$\bm{\chi}^{c}_{a}=\frac{1}{2}(\bm{\chi}^{c}_{1}-\bm{\chi}^{c}_{2})$, spin
contributions to each mode expressed in terms of ($\bm{\chi}^{c}_{s}$,
$\bm{\chi}^{c}_{a}$) read

$\displaystyle H^{22}_{\rm{SO}}$
$\displaystyle=-\frac{2}{3}\frac{1}{c^{3}}\left(\frac{GM}{r}\right)^{2}\Bigg{\\{}3\delta\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)\left(r\dot{\phi}+i\dot{r}\right)+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\left[3\left(1-\frac{5\nu}{3}\right)r\dot{\phi}+3i\dot{r}\left(1-\frac{8\nu}{3}\right)\right]\Bigg{\\}}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(21a) $\displaystyle H^{22}_{\rm{SS}}$
$\displaystyle=-\frac{3}{2}\frac{1}{c^{4}}\left(\frac{GM}{r}\right)^{3}\Bigg{\\{}2\,\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\bigg{[}-(1-2\nu)\kappa_{a}-\delta\kappa_{s}\bigg{]}+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right){}^{2}\bigg{[}-2\nu-\delta\kappa_{a}-(1-2\nu)\kappa_{s}\bigg{]}\,$
$\displaystyle+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right){}^{2}\bigg{[}2\nu-\delta\kappa_{a}-(1-2\nu)\kappa_{s}\bigg{]}\Bigg{\\}}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(21b) $\displaystyle H^{21}_{\rm{SO}}$
$\displaystyle=\frac{1}{c^{2}}\left(\frac{GM}{r}\right)^{2}\bigg{[}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)+\delta\,\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\bigg{]}-\frac{1}{42}\frac{1}{c^{4}}\left(\frac{GM}{r}\right)^{2}\Bigg{\\{}\delta\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\Bigg{[}\frac{154GM}{r}\left(1+\frac{\nu}{7}\right)-84\left(1-\frac{4\nu}{21}\right)\left(r\dot{\phi}\right)^{2}\,$
$\displaystyle-294i\left(1+\frac{13\nu}{147}\right)\dot{r}\left(r\dot{\phi}\right)+105\dot{r}^{2}\left(1-\frac{4\nu}{7}\right)\Bigg{]}+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)\Bigg{[}\frac{154GM}{r}\left(1+\frac{59\nu}{77}\right)-84\left(1-\frac{22\nu}{7}\right)\left(r\dot{\phi}\right)^{2}\,$
$\displaystyle-294i\left(1-\frac{83\nu}{147}\right)\dot{r}\left(r\dot{\phi}\right)+105\dot{r}^{2}\left(1-\frac{52\nu}{105}\right)\Bigg{]}\Bigg{\\}}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(21c) $\displaystyle H^{20}_{\rm{SO}}$
$\displaystyle=-2\sqrt{\frac{2}{3}}\frac{1}{c^{3}}\left(\frac{GM}{r}\right)^{2}\left(r\dot{\phi}\right)\bigg{[}\delta\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)+(1+\nu)\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\bigg{]}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(21d) $\displaystyle H^{20}_{\rm{SS}}$
$\displaystyle=\sqrt{\frac{3}{2}}\frac{1}{c^{4}}\left(\frac{GM}{r}\right)^{3}\Bigg{\\{}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)^{2}\bigg{[}-2\nu+\delta\kappa_{a}+\kappa_{s}\left(1-2\nu\right)\bigg{]}+2\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\bigg{[}\delta\kappa_{s}+\kappa_{a}\left(1-2\nu\right)\bigg{]}\,$
$\displaystyle+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)^{2}\bigg{[}2\nu+\delta\kappa_{a}+\kappa_{s}\left(1-2\nu\right)\bigg{]}\Bigg{\\}}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(21e) $\displaystyle H^{33}_{\rm{SO}}$
$\displaystyle=-\frac{\sqrt{5}}{4\sqrt{42}}\frac{1}{c^{4}}\left(\frac{GM}{r}\right)^{2}\Bigg{\\{}\delta\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\bigg{[}-\frac{4GM}{r}\left(1-5\nu\right)-24\left(1-\frac{11\nu}{8}\right)\left(r\dot{\phi}\right)^{2}-36i\left(1-\frac{13\nu}{6}\right)\dot{r}\left(r\dot{\phi}\right)+12\dot{r}^{2}\,$
$\displaystyle\times\left(1-\frac{5\nu}{2}\right)\bigg{]}+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)\bigg{[}-\frac{4GM}{r}\left(1-5\nu\right)-24\left(1-\frac{119\nu}{24}\right)\left(r\dot{\phi}\right)^{2}-36i\left(1-\frac{77\nu}{18}\right)\dot{r}\left(r\dot{\phi}\right)+12\dot{r}^{2}\,$
$\displaystyle\times\left(1-\frac{25\nu}{6}\right)\bigg{]}\Bigg{\\}}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(21f) $\displaystyle H^{32}_{\rm{SO}}$
$\displaystyle=\frac{2\sqrt{5}}{3\sqrt{7}}\frac{1}{c^{3}}\left(\frac{GM}{r}\right)^{2}\nu\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\bigg{[}4r\dot{\phi}+i\dot{r}\bigg{]}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(21g) $\displaystyle H^{31}_{\rm{SO}}$
$\displaystyle=\frac{1}{12\sqrt{14}}\frac{1}{c^{4}}\left(\frac{GM}{r}\right)^{2}\Bigg{\\{}\delta\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\bigg{[}\frac{4GM}{r}\left(1-5\nu\right)+24\left(1+\frac{31\nu}{24}\right)\left(r\dot{\phi}\right)^{2}+12i\left(1+\frac{35\nu}{6}\right)\dot{r}\left(r\dot{\phi}\right)-12\dot{r}^{2}\,$
$\displaystyle\times\left(1-\frac{5\nu}{2}\right)\bigg{]}+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)\left[\frac{4GM}{r}\left(1-5\nu\right)+24\left(1-\frac{29\nu}{8}\right)\left(r\dot{\phi}\right)^{2}+12i\left(1-\frac{31\nu}{6}\right)\dot{r}\left(r\dot{\phi}\right)-12\dot{r}^{2}\left(1-\frac{25\nu}{6}\right)\right]\Bigg{\\}}\,$
$\displaystyle+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$ (21h)
$\displaystyle H^{30}_{\rm{SO}}$
$\displaystyle=2i\sqrt{\frac{2}{21}}\frac{1}{c^{3}}\left(\frac{GM}{r}\right)^{2}\nu\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\dot{r}+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(21i) $\displaystyle H^{43}_{\rm{SO}}$
$\displaystyle=\frac{\sqrt{5}}{3\sqrt{14}}\frac{1}{c^{4}}\left(\frac{GM}{r}\right)^{2}\nu\bigg{[}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)-\delta\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\bigg{]}\left[\frac{GM}{r}+\frac{23}{4}\left(r\dot{\phi}\right)^{2}+\frac{5}{2}i\dot{r}\left(r\dot{\phi}\right)-\frac{1}{2}\dot{r}^{2}\right]+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(21j) $\displaystyle H^{41}_{\rm{SO}}$
$\displaystyle=\frac{\sqrt{5}}{7\sqrt{2}}\frac{1}{c^{4}}\left(\frac{GM}{r}\right)^{2}\nu\bigg{[}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)-\delta\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\bigg{]}\left[\frac{GM}{r}-\frac{11}{12}\left(r\dot{\phi}\right)^{2}+\frac{5}{6}i\dot{r}\left(r\dot{\phi}\right)-\frac{1}{2}\dot{r}^{2}\right]+\mathcal{O}\left(\frac{1}{c^{5}}\right)\,,$
(21k)

where we have used $\kappa_{s}=(\kappa_{1}+\kappa_{2})/2$ and
$\kappa_{a}=(\kappa_{1}-\kappa_{2})/2$.

Recently we learned about a related work (Khalil  _et al._ , 2021) which
explicitly writes general orbital expressions for various modes with spins up
to 2PN order in harmonic coordinates and in terms of dimensionless spin
variables (${\bm{\chi}_{1}},{\bm{\chi}_{2}}$) associated with _nonconstant_
spins ($\bm{S}_{1},\bm{S}_{2}$) as $\bm{\chi}_{i}={\bm{S}_{i}}/G\,m^{2}_{i}$.
[See Eq. (C1) there for complete expression.] The connection between
(${\bm{\chi}_{1}},{\bm{\chi}_{2}}$) and the _constant_ spin variables
(${\bm{\chi}^{c}_{1}},{\bm{\chi}^{c}_{2}}$) used above can be obtained by
employing Eq. (16) and they read

$\displaystyle\bm{\chi}^{c}_{1}=\bm{\chi}_{1}+\frac{1}{2}\left(\frac{GM}{c^{2}r}\right)\left(1-\delta\right)\bm{\chi}_{1},$
(22a)
$\displaystyle\bm{\chi}^{c}_{2}=\bm{\chi}_{2}+\frac{1}{2}\left(\frac{GM}{c^{2}r}\right)\left(1+\delta\right)\bm{\chi}_{2}.$
(22b)

Using the above connection one can show the correspondence between the results
of Ref. (Khalil  _et al._ , 2021) and those presented here. We confirm that
our results are in complete agreement with those of Ref. (Khalil  _et al._ ,
2021).111111Note also that the spin quadrupole constants
($C_{1ES^{2}},C_{2ES^{2}}$) in our notation are $\kappa_{1,2}$.

##  IV quasi-keplerian representation

The general orbit expressions with spin contributions to 2PN in all the
relevant spherical harmonic modes were listed in the previous section. Leading
expressions for the nonspinning part was also included for ease in PN
counting. These results can be applied to compact binaries with nonprecessing
spins and in orbits of arbitrary shapes. In this section, we specialize to the
case of elliptical orbits by making use of generalized quasi-Keplerian (QK)
representation for the conservative dynamics of the spinning compact binaries
in elliptical orbits to 2PN (Klein and Jetzer, 2010; Klein  _et al._ , 2018).
(Reference Klein  _et al._ (2018) also includes the corresponding 3PN QK
representation of Damour and Deruelle (1985); Damour and Schaefer (1988);
Schäfer and Wex (1993); Wex (1995); Memmesheimer  _et al._ (2004) for
nonspinning systems in harmonic coordinates.) It may be worth noting that the
QK representation presented in Klein  _et al._ (2018) is in harmonic
coordinates or more specifically in modified harmonic (MH) coordinates that
were introduced to remove the presence of logarithmic terms in expressions for
mass quadrupole moment at 3PN order Memmesheimer  _et al._ (2004); Arun  _et
al._ (2008) and have been extensively used in numerous efforts focusing on
modeling compact binary inspirals in elliptical orbits Mishra  _et al._
(2015); Boetzel  _et al._ (2019); Moore  _et al._ (2016); Boetzel  _et al._
(2017); Klein and Jetzer (2010); Klein  _et al._ (2018). The QK representation
involves connecting variables describing compact binaries in general orbits to
those characterizing the orbit one wishes to specialize to. Once these
relations are established, they can be used to convert the general orbit
results to corresponding ones for an orbit of interest. The QK representation
of Klein  _et al._ (2018) explicitly writes these relations for spinning
systems in an elliptical orbit. Moreover, as far as we know this is the only
prescription available that describes the binary’s conservative dynamics and
evolution of orbital elements to 2PN order for spinning systems and we choose
to use these results here. We also keep the symbols and notations as used in
Klein  _et al._ (2018) to avoid any confusion unless it conflicts with our
presentation; in that case, we inform the reader about the differences in
notations adopted in the current work. However, in order to avoid duplication,
we only wish to reproduce representation of Klein  _et al._ (2018) partly,
although the results included here should be sufficient to convert the general
orbit results of Sec. III to the elliptical case. Here, we wish to list
expressions for each mode included in Sec. III specialized to the case of
elliptical orbits. Since the general orbit results are explicitly written in
terms of a set of radial ($r$, $\dot{r}$) and angular variables ($\phi$, and
$\dot{\phi}$), converting the general orbit results simply requires using
expressions relating these general orbit variables to those of elliptical
orbit. Below we reproduce the relations that connect $r$, $\dot{r}$ $\phi$,
and $\dot{\phi}$ to a set of variables characterizing an elliptical orbit.
They read

$\displaystyle r$ $\displaystyle=a(1-e_{r}\cos u)+f_{r}(V)\,,$ (23a)
$\displaystyle\dot{r}$ $\displaystyle=\frac{a\,e_{r}\,n\,\sin u}{1-e_{t}\cos
u}\,,$ (23b) $\displaystyle\phi$ $\displaystyle=(1+k)V+f_{\phi}(V)\,,$ (23c)
$\displaystyle\dot{\phi}$
$\displaystyle=\frac{\sqrt{1-e_{\phi}^{2}}\,(1+k)\,n}{(1-e_{t}\cos
u)(1-e_{\phi}\cos u)}\,.$ (23d)

Here, $a$ and $e$ denote the semimajor axis and orbital eccentricity. Note
that parameter subscripts ($r,\phi,t$) indicate that the concerned parameter
is associated with a specific coordinate. For instance, all three types of
eccentricities ($e_{r},\,e_{\phi},\,e_{t}$) are featured in the above
relations. (The subscript is dropped in case there is no need to distinguish
between them.) Further, the symbols ($n,\,k,\,u$ and $V$), represent mean
motion, advance of periastron, eccentric anomaly, and true
anomaly,121212Reference Klein  _et al._ (2018) uses the symbol $v$ for true
anomaly; although, here we already use $v$ for the magnitude of the velocity
vector and thus alternatively use symbol $V$ following Arun  _et al._ (2008)
to represent true anomaly. respectively. Finally, $f_{t},\,f_{r}$ and
$f_{\phi}$ are 2PN corrections to these relations; however, they contribute to
spin effects beyond 2PN for nonprecessing systems and hence are ignored here.
Explicit expressions for QK parameters characterizing the binary’s motion in
an elliptical orbit in terms of a PN parameter ($y$), time eccentricity
($e_{t}$) and two spin related parameters ($\beta$, $\gamma_{1}$), with the
accuracy desired in our work, have been listed in Appendix B of Ref. Klein
_et al._ (2018) and we do not wish to reproduce them here. Nevertheless, we
provide explicit relations for radial and angular position and velocity
variables in terms of the parameters chosen in Klein  _et al._ (2018) in
Appendix A. These could directly be used in general orbital expressions for
modes presented in Sec. III in order to reexpress them for the elliptical
case. Leading NS and SO, SS contributions up to 2PN to all relevant modes
valid for compact binaries in orbits of arbitrary eccentricity have been
listed in Appendix B. These are probably the most useful results of the
current paper, although, they have been pushed to the appendix to avoid
disruption of the flow of the paper. Instead, in this section we list these
contributions as leading eccentric corrections to the circular case by
expanding our arbitrary eccentricity results by treating eccentricity as a
small parameter. Only leading corrections are included so as to have clarity
in presentation and again to maintain the flow of the paper. However, the
arbitrary eccentricity results of Appendix B can be expanded to an arbitrary
power of eccentricity and thus become suitable for small to moderate
eccentricities. In fact, we do provide these results to $\mathcal{O}(e^{7})$
as Supplemental Material kp: . These complement the nonspinning results
available to $\mathcal{O}(e^{7})$ in eccentricity up to 3PN order and were
computed in Boetzel  _et al._ (2019); Ebersold  _et al._ (2019). Expressions
for each relevant GW mode with leading NS and SO, SS terms to 2PN and with
leading correction of eccentricity to the circular case read

$h^{\ell
m}=-i^{2m}\,\frac{8\,G\,M\,\nu\,x}{c^{2}\,R}\,\sqrt{\frac{\pi}{5}}\,\mathrm{e}^{-im\phi}\,H^{\ell
m}$ (24)

with

$\displaystyle H^{\ell m}=H^{\ell m}_{\rm NS}+H^{\ell m}_{\rm SO}+H^{\ell
m}_{\rm SS}\,,$ (25)

where NS, SO, and SS pieces for all relevant modes read

$\displaystyle H^{22}_{{\rm{NS}}}$
$\displaystyle=\Bigg{[}1+e\left(\frac{1}{4}\mathrm{e}^{-il}+\frac{5}{4}\mathrm{e}^{il}\right)\Bigg{]}+\mathcal{O}\left(x\right)\,,$
(26a) $\displaystyle H^{22}_{\rm{SO}}$
$\displaystyle=x^{3/2}\Bigg{\\{}-\frac{4}{3}\delta\,\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)-\frac{4}{3}(1-\nu)\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)+e\Bigg{[}\left(-\frac{7}{3}\mathrm{e}^{-il}-\frac{8}{3}\mathrm{e}^{il}\right)\delta\,\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)+\bigg{[}-\frac{7}{3}\mathrm{e}^{-il}\left(1-\frac{5\nu}{14}\right)\,$
$\displaystyle-\frac{8}{3}\mathrm{e}^{il}\left(1-\frac{13\nu}{16}\right)\bigg{]}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\Bigg{]}\Bigg{\\}}+\mathcal{O}\left(x^{5/2}\right)\,,$
(26b) $\displaystyle H^{22}_{\rm{SS}}$
$\displaystyle=\frac{x^{2}}{8}\Bigg{\\{}\bigg{[}8+15\,e\left(\mathrm{e}^{-il}+\mathrm{e}^{il}\right)\bigg{]}\bigg{[}2\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)\,\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\,\Big{[}(1-2\nu)\kappa_{a}+\delta\,\kappa_{s}\Big{]}+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right){}^{2}\Big{[}-2\nu+\delta\,\kappa_{a}+(1-2\nu)\,$
$\displaystyle\times\kappa_{s}\Big{]}+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right){}^{2}\Big{[}2\nu+\delta\,\kappa_{a}+(1-2\nu)\kappa_{s}\Big{]}\bigg{]}\Bigg{\\}}+\mathcal{O}\left(x^{5/2}\right)\,,$
(26c) $\displaystyle H^{21}_{{\rm{NS}}}$
$\displaystyle=-\frac{\delta}{3}\,\sqrt{x}\Bigg{[}1+e\,\left(\mathrm{e}^{il}+\mathrm{e}^{-il}\right)\Bigg{]}+\mathcal{O}\left(x^{3/2}\right)\,,$
(26d) $\displaystyle H^{21}_{\rm{SO,1PN}}$
$\displaystyle=\frac{x}{2}\,\Bigg{\\{}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)+\delta\,\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)+e\bigg{[}\left[\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)+\delta\,\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\right]\,\left(\mathrm{e}^{il}+\mathrm{e}^{-il}\right)\bigg{]}\Bigg{\\}}+\mathcal{O}\left(x^{2}\right)\,,$
(26e) $\displaystyle H^{21}_{\rm{SO,2PN}}$
$\displaystyle=x^{2}\Bigg{\\{}\frac{1}{6}\left(1-\frac{205\nu}{7}\right)\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)+\frac{\delta}{6}\left(1-\frac{33\nu}{7}\right)\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)+e\Bigg{[}\bigg{[}\frac{5}{6}\mathrm{e}^{-il}\left(1-\frac{165\nu}{14}\right)+\frac{13}{3}\mathrm{e}^{il}\left(1-\frac{991\nu}{364}\right)\bigg{]}$
$\displaystyle\times\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)+\delta\left[\frac{5}{6}\mathrm{e}^{-il}\left(1-\frac{169\nu}{70}\right)+\frac{13}{3}\mathrm{e}^{il}\left(1-\frac{11\nu}{28}\right)\right]\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\Bigg{]}\Bigg{\\}}+\mathcal{O}\left(x^{5/2}\right)\,,$
(26f) $\displaystyle H^{20}_{\rm{NS}}$
$\displaystyle=-\frac{1}{2\sqrt{6}}e\left(\mathrm{e}^{il}+\mathrm{e}^{-il}\right)+\mathcal{O}\left(x\right)\,,$
(26g) $\displaystyle H^{20}_{\rm{SO}}$
$\displaystyle=-\frac{1}{3\sqrt{6}}x^{3/2}\bigg{[}e\left(\mathrm{e}^{il}+\mathrm{e}^{-il}\right)\bigg{]}\bigg{[}7\delta\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)+7\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\left(1-\frac{5\nu}{7}\right)\bigg{]}+\mathcal{O}\left(x^{5/2}\right)\,,$
(26h) $\displaystyle H^{20}_{\rm{SS}}$
$\displaystyle=\frac{1}{4}\sqrt{\frac{3}{2}}x^{2}e\left(\mathrm{e}^{il}+\mathrm{e}^{-il}\right)\Bigg{\\{}2\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\bigg{[}\delta\kappa_{s}+\kappa_{a}\left(1-2\nu\right)\bigg{]}+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)^{2}\bigg{[}-2\nu+\delta\kappa_{a}+\kappa_{s}\left(1-2\nu\right)\bigg{]}\,$
$\displaystyle+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)^{2}\bigg{[}2\nu+\delta\kappa_{a}+\kappa_{s}\left(1-2\nu\right)\bigg{]}\Bigg{\\}}+\mathcal{O}\left(x^{5/2}\right)\,,$
(26i) $\displaystyle H^{33}_{{\rm{NS}}}$
$\displaystyle=-\frac{3\sqrt{15}}{4\sqrt{14}}\delta\,\sqrt{x}\,\Bigg{[}1+e\left(\frac{5\,\mathrm{e}^{-il}}{9}+\frac{5\,\mathrm{e}^{il}}{3}\right)\Bigg{]}+\mathcal{O}\left(x^{3/2}\right)\,,$
(26j) $\displaystyle H^{33}_{\rm{SO}}$
$\displaystyle=x^{2}\Bigg{\\{}\frac{3}{2}\sqrt{\frac{15}{14}}\left(1-\frac{19\nu}{4}\right)\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)+\frac{3}{2}\sqrt{\frac{15}{14}}\delta\left(1-\frac{5\nu}{4}\right)\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)+e\Bigg{[}\bigg{[}\frac{15}{4}\sqrt{\frac{15}{14}}\mathrm{e}^{il}\left(1-\frac{139\nu}{30}\right)+\frac{5}{4}\sqrt{\frac{35}{6}}\mathrm{e}^{-il}\,$
$\displaystyle\times\bigg{(}1-\frac{327\nu}{70}\bigg{)}\bigg{]}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)+\delta\bigg{[}\frac{5}{4}\sqrt{\frac{35}{6}}\mathrm{e}^{-il}\left(1-\frac{9\nu}{14}\right)+\frac{15}{4}\sqrt{\frac{15}{14}}\mathrm{e}^{il}\left(1-\frac{11\nu}{10}\right)\bigg{]}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\Bigg{]}\Bigg{\\}}+\mathcal{O}\left(x^{5/2}\right)\,,$
(26k) $\displaystyle H^{32}_{{\rm{NS}}}$
$\displaystyle=\frac{\sqrt{5}}{3\sqrt{7}}\left(1-3\nu\right)\,x\Bigg{[}1+\frac{1}{8}e\left(11\,\mathrm{e}^{-il}+13\,\mathrm{e}^{il}\right)\Bigg{]}+\mathcal{O}\left(x^{2}\right)\,,$
(26l) $\displaystyle H^{32}_{\rm{SO}}$
$\displaystyle=\frac{\sqrt{5}}{6\sqrt{7}}\,\nu\,x^{3/2}\Bigg{[}8+e\left(11\,\mathrm{e}^{-il}+13\,\mathrm{e}^{il}\right)\Bigg{]}\,\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)+\mathcal{O}\left(x^{5/2}\right)\,,$
(26m) $\displaystyle H^{31}_{{\rm{NS}}}$
$\displaystyle=-\frac{1}{12\sqrt{14}}\,\delta\,\sqrt{x}\Bigg{[}1-e\left(5\,\mathrm{e}^{-il}-\mathrm{e}^{il}\right)\Bigg{]}+\mathcal{O}\left(x^{3/2}\right)\,,$
(26n) $\displaystyle H^{31}_{\rm{SO}}$
$\displaystyle=x^{2}\Bigg{\\{}\frac{1}{6\sqrt{14}}\left(1-\frac{11\nu}{4}\right)\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)+\frac{\delta}{6\sqrt{14}}\left(1-\frac{13\nu}{4}\right)\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)+e\Bigg{[}\bigg{[}\frac{1}{12}\sqrt{\frac{7}{2}}\mathrm{e}^{il}\left(1-\frac{51\nu}{14}\right)+\frac{25}{12\sqrt{14}}\mathrm{e}^{-il}\,$
$\displaystyle\times\left(1-\frac{181\nu}{50}\right)\bigg{]}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)+\delta\bigg{[}\frac{1}{12}\sqrt{\frac{7}{2}}\mathrm{e}^{il}\left(1-\frac{17\nu}{14}\right)+\frac{25}{12\sqrt{14}}\mathrm{e}^{-il}\left(1-\frac{63\nu}{50}\right)\bigg{]}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\Bigg{]}\Bigg{\\}}+\mathcal{O}\left(x^{5/2}\right)\,,$
(26o) $\displaystyle H^{30}_{\rm{NS}}$
$\displaystyle=\frac{1}{4\sqrt{42}}x\,e\left(\mathrm{e}^{il}-\mathrm{e}^{-il}\right)\left(1-3\nu\right)+\mathcal{O}\left(x^{2}\right)\,,$
(26p) $\displaystyle H^{30}_{\rm{SO}}$ $\displaystyle=\frac{1}{\sqrt{42}}\nu
x^{3/2}e\left(\mathrm{e}^{il}-\mathrm{e}^{-il}\right)\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)+\mathcal{O}\left(x^{5/2}\right)\,,$
(26q) $\displaystyle H^{43}_{{\rm{NS}}}$
$\displaystyle=-\frac{9}{4\sqrt{70}}\,\delta\left(1-2\nu\right)\,x^{3/2}\Bigg{[}1+\frac{1}{27}e\left(47\mathrm{e}^{-il}+57\mathrm{e}^{il}\right)\Bigg{]}+\mathcal{O}\left(x^{5/2}\right)\,,$
(26r) $\displaystyle H^{43}_{\rm{SO}}$
$\displaystyle=\frac{\sqrt{5}}{8\sqrt{14}}\,\nu\,x^{2}\,\Bigg{\\{}9\bigg{[}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)-\delta\,\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\bigg{]}+e\left(19\,\mathrm{e}^{il}+\frac{47}{3}\mathrm{e}^{-il}\right)\bigg{[}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)-\delta\,\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\bigg{]}\Bigg{\\}}+\mathcal{O}\left(x^{5/2}\right)\,,$
(26s) $\displaystyle H^{41}_{{\rm{NS}}}$
$\displaystyle=-\frac{1}{84\sqrt{10}}\delta\left(1-2\nu\right)x^{3/2}\Bigg{[}1-e\left(9\,\mathrm{e}^{-il}-\mathrm{e}^{il}\right)\Bigg{]}+\mathcal{O}\left(x^{5/2}\right)\,,$
(26t) $\displaystyle H^{41}_{\rm{SO}}$
$\displaystyle=\frac{\sqrt{5}}{168\sqrt{2}}\nu\,x^{2}\Bigg{\\{}\bigg{[}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)-\delta\,\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\bigg{]}-e\left(9\,\mathrm{e}^{-il}-\mathrm{e}^{il}\right)\bigg{[}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)-\delta\,\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\bigg{]}\Bigg{\\}}+\mathcal{O}\left(x^{5/2}\right)\,.$
(26u)

Note that the PN parameter ($x$) used above is related to the PN parameter
($y$) as $x=y^{2}(1-e^{2})$. Note that we continue to use the parameter $y$
for the arbitrary eccentricity results following the suggestion of Klein  _et
al._ (2018) as can be seen in Appendix B. When writing eccentricity expanded
results, however, we change to the parameter $x$ which is defined as

$\displaystyle x$
$\displaystyle=\Bigg{[}\frac{G\,M\,(1+k)\,n}{c^{3}}\Bigg{]}^{2/3}\,,$ (27)

where different symbols have been defined above. This choice is to provide
results in a notation consistent with the 3PN nonspinning counterparts listed
in Boetzel  _et al._ (2019); Ebersold  _et al._ (2019) without running into
problems related to its use in arbitrary eccentricity results as we treat
eccentricity as a small parameter. Note also, while eccentric anomaly ($u$)
has been used while writing arbitrary eccentricity results, we use mean
anomaly ($l$) again to be consistent with the choices of Boetzel  _et al._
(2019); Ebersold  _et al._ (2019). Note that this requires one to make use of
$u$-$l$ relation obtained by solving the Kepler equation that connects the
two. A General solution to the Kepler equation for nonspinning systems has
been outlined in Boetzel  _et al._ (2019) and can be used to write the desired
relation to arbitrary powers of eccentricity. For our current purposes (SO and
SS contributions through 2PN), we only require 1PN relation with
leading131313For the results in the Supplemental Material kp:  the $u$-$l$
relation to $\mathcal{O}(e^{7})$ is used. eccentricity correction. It reads

$\displaystyle u$ $\displaystyle=\Big{[}l+e\,\sin
l+\mathcal{O}\left(e^{2}\right)\Big{]}+\mathcal{O}\left(x^{2}\right)\,.$ (28)

##  V quasicircular limit

Both QK and general orbit expressions for various modes reduce to circular
versions available in literature Arun  _et al._ (2009); Buonanno  _et al._
(2013). While the circular limit of elliptical orbit results is listed in Sec.
IV and Appendix B by setting $e\rightarrow 0$, general orbit results can be
reduced to the circular case by setting $\dot{r}$=0 and
$\dot{\phi}=\omega_{\phi}=n$.141414Leading corrections to $\dot{r}$ appear
only at 2.5PN order. For the circular case, Eq. (27) becomes

$\displaystyle x$ $\displaystyle=\left(\frac{GMn}{c^{3}}\right)^{2/3}\,.$ (29)

The dependence on the radial separation $r$ can then be expressed in terms of
$n$ [and hence in terms of $x$ using Eq. (29)]. For binaries in circular
orbits, the radial separation $r$ in terms of $x$ with spin corrections to 2PN
is given in Refs. Blanchet  _et al._ (2006); Bohé  _et al._ (2015). We only
reproduce NS, SO, and SS pieces that are needed to reproduce the limits. They
read

$\displaystyle\gamma=x\Big{[}\gamma^{\rm NS}+\gamma^{\rm SO}+\gamma^{\rm
SS}\Big{]}\,,$ (30)

where individual pieces read

$\displaystyle\gamma^{\rm NS}$
$\displaystyle=1+x\,\left(1-\frac{\nu}{3}\right)+\mathcal{O}\left(x^{2}\right)\,,$
(31a) $\displaystyle\gamma^{\rm SO}$
$\displaystyle=\frac{1}{3}\,x^{3/2}\Bigg{[}5\,\left(\bm{\hat{\ell}}\cdot\bm{s}_{c}\right)+3\,\delta\,\left(\bm{\hat{\ell}}\cdot\bm{\sigma}_{c}\right)\Bigg{]}+\mathcal{O}\left(x^{5/2}\right)\,,$
(31b) $\displaystyle\gamma^{\rm SS}$
$\displaystyle=-\frac{1}{2}x^{2}\left(\bm{s}^{+}_{0}\cdot\bm{s}^{-}_{0}\right)+\mathcal{O}\left(x^{3}\right)\,.$
(31c)

Here, $\gamma$ is a another PN parameter and is related to the binary’s radial
separation ($r$) as

$\displaystyle\gamma=\left(\frac{GM}{c^{2}r}\right)\,.$ (32)

With $\dot{r}=0$ and both ($r$, $\dot{\phi}$) expressed in terms of $x$
through Eqs. (29)-(31), the circular limit of general orbit results can be
written completely in terms of the PN parameter $x$. We have verified that our
general orbit results presented in Sec. III reduce to the circular versions
presented in Arun  _et al._ (2009); Buonanno  _et al._ (2013). It may be worth
noting that the $(2,0)$ mode at the leading (nonspinning) order also has
nonoscillatory memory contributions originating from memory integrals
appearing in the expression for the mass quadrupole moment Ebersold  _et al._
(2019). Since we do not discuss hereditary contributions in this work, we
ignore these terms here. A Null circular limit for the expression of the
$(2,0)$ mode [see Eq. (26g) and Eq. (41g)] is indicative of the same. Note
also, the circular orbit limit in terms of (anti)symmetric combinations of
dimensionless spin parameters
($\bm{\chi}^{c}_{s},\bm{\chi}^{c}_{a},\kappa_{s},\kappa_{a}$) is automatically
extracted from Eq. (26).

##  VI Summary and Discussion

In this paper, we presented computations of spin-orbit and spin-spin effects
in spherical harmonic modes of gravitational waveforms through 2PN order
within the generalized framework of Kidder (2008); Buonanno  _et al._ (2013).
Section II lists the inputs required for these computations. The first set of
results include SO and SS contributions to each mode contributing to the
waveform through 2PN order for compact binaries in general orbits with
nonprecessing spins and have been listed in Sec. III in terms of radial
($r,\dot{r}$) and angular ($\phi,\dot{\phi}$) variables for position and
velocity as well as in terms of suitable combinations of individual spins. We
choose to express these results in terms of two sets of spin variables.
Equation (19) presents results in terms of spin variables defined in Buonanno
_et al._ (2013) so as to reproduce exact expressions of Arun  _et al._ (2009);
Buonanno  _et al._ (2013) when the circular limit is taken.151515In fact, we
use a dimensionless version of these variables defined in Eq. (20).
Additionally, following Mishra  _et al._ (2016) we also list our results as
Eq. (21) in terms of symmetric ($\bm{\chi}^{c}_{s}$) and antisymmetric
($\bm{\chi}^{c}_{a}$) combinations of dimensionless spin variables
$\bm{\chi}^{c}_{1,2}={\bm{S^{c}}_{1,2}}/G\,m^{2}_{1,2}$. In this presentation
of results, the dependence on spin-induced parameters for individual binary
components ($\kappa_{1,2}$) appearing in the SS part for the dominant
($\ell$=$2$, $m$=$2$) mode also becomes explicit.161616Note that this is also
true for the ($\ell$=$2$, $m$=0) mode. Again, we choose to write the final
expressions in terms of a symmetric and antisymmetric combination of
($\kappa_{1,2}$) defined above. Note that through the 2PN order not all modes
have spin corrections and only ($\ell$,
$|m|$)=($(2,2),(2,1),(2,0),(3,3),(3,2),(3,1),(3,0),(4,3),\,\\\ (4,1)$) modes
contribute. We do not list the modes that do not have spin corrections through
2PN when presenting the results in this paper. Note also that through 2PN only
the ($\ell$=$2$, $|m|$=$2$) mode has SS contributions and hence a dependence
on spin-induced parameters.171717 Note again that the leading $m=0$ mode
$(\ell=2,m=0)$ also has SS contribution as can be seen in Eq. (19h), (21),
(26). Additionally, up to 2PN order only the (2, 1) mode has leading (at 1PN)
and subleading (at 2PN) SO contribution, for other modes sub-leading terms
appear beyond 2PN. Finally, the circular orbit limit of our general orbit
results can be obtained using the transformations discussed in Sec. V. We have
verified that our results reduce to circular results of Arun  _et al._ (2009);
Buonanno  _et al._ (2013) up to 2PN. Next, we specialize to the case of
elliptical orbits using the QK representation of Klein and Jetzer (2010);
Klein  _et al._ (2018) for spinning compact binaries. These results are listed
in Sec. IV and in Appendix B. Here, we have used the relations connecting
general orbit variables to those characterizing elliptical orbits (listed in
Appendix A) in the expressions for each mode written in terms of $r,\,\dot{r}$
and $\dot{\phi}$ and given by Eq. (19). While the results presented in Sec.
III are useful for orbits with small eccentricities, arbitrary eccentricity
results have been pushed to Appendix B to avoid any disruption of the flow of
the paper as these expressions run over a couple of pages. Similar to the
general orbit results, we only list modes which have spin contributions
through 2PN with SO, SS, and leading NS pieces presented separately. Also, we
choose to present arbitrary eccentricity results in terms of spin combinations
($\beta,\gamma_{1}$) defined in Klein  _et al._ (2018), although, in our
presentation we scale them with appropriate factors of $G$ and $M$ (omitted in
Klein  _et al._ (2018)) to make them dimensionless181818We redefine
$(\beta,\gamma_{1})$ of Klein  _et al._ (2018) as
$\frac{{\beta(a,b)}}{G\,M}\rightarrow\beta(a,b)$ and
$\frac{\gamma_{1}}{G^{2}\,M^{2}\,}\rightarrow\gamma_{1}$. and reexpressed as
Eq. (39) in terms of dimensionless spin variables
($\bm{\chi}_{s},\bm{\chi}_{a}$) for compactness. Further, these results have
been expressed as explicit functions of orbital time eccentricity ($e$),
eccentric anomaly ($u$), and the PN parameter $(y)$ introduced in Klein  _et
al._ (2018). The small eccentricity results presented in Sec. IV include
leading-order eccentric corrections to circular results, expressed explicitly
in terms of another PN parameter ($x$) related to $y$ and $e$ (see a
discussion in Sec. IV). Other parameters that appear include orbital time
eccentricity $e$, mean anomaly ($l$), and spin parameters $\bm{\chi}^{c}_{s}$
and $\bm{\chi}^{c}_{a}$. Higher order corrections to eccentricity [to
$\mathcal{O}(e^{7})$] ofthese results together with nonspinning contributions
to 3PN are included in the Supplemental Material kp: . This paper lists
explicitly the spin contributions to different spherical harmonic modes
through 2PN order for binaries in general orbits. These together with the
nonspinning results of Mishra  _et al._ (2015) provide state-of-the-art
results for compact binaries in orbits of arbitrary shape and nature.
Moreover, elliptical orbit results presented in Appendix B together with the
related results of Mishra  _et al._ (2015) are again the most up-to-date
versions of PN eccentric waveforms valid for arbitrary eccentricities. We
recall here that the nonspinning results of Mishra  _et al._ (2015) do not
include the contributions from hereditary terms and were computed later in
follow-up papers Boetzel  _et al._ (2017); Ebersold  _et al._ (2019) in the
small eccentricity approximation. In fact, as was argued in Boetzel  _et al._
(2017), closed-form expressions for hereditary terms (for any noncircular
orbit) can only be obtained under the small eccentricity approximation. While
a hereditary contribution associated with spins appears in waveforms beyond
the 2PN order and is not our immediate concern, future works involving these
contributions shall also involve an expansion of eccentricity parameter. (See
however a very recent work Henry  _et al._ (2022) where these contributions
were computed for binaries in circular orbits.) References Boetzel  _et al._
(2017); Ebersold  _et al._ (2019) provide complete 3PN contributions to GW
modes for eccentric binaries with nonspinning components to
$\mathcal{O}(e^{7})$. Our current computations add to these results the spin
contributions through 2PN order for the nonprecessing case again to
$\mathcal{O}(e^{7})$. Complete results with a 3PN accurate nonspinning part
and a 2PN spinning part valid for nonprecessing systems are being provided as
Supplemental Material kp:  and should be useful for systems with small to
moderate eccentricities. Results presented here together with the nonspinning
results presented in earlier works Mishra  _et al._ (2015); Boetzel  _et al._
(2017); Ebersold  _et al._ (2019) should be useful for accurate comparisons
with state-of-the art eccentric numerical relativity simulations for spinning
BBHs Joshi  _et al._ (2022) and should set the platform for construction of
complete inspiral-merger-ringdown models for eccentric, nonprecessing BBH
models including the effect of higher order modes. Note, however, that in this
work we only discuss the computation of spin effects in the amplitude of each
relevant mode. The spinning corrections to the secular phase can be computed
using the energy and flux expressions given in Ref. (Klein  _et al._ , 2018)
following an approach similar to Moore  _et al._ (2016) which explicitly
computes the GW secular phase to 3PN order assuming eccentricity as a small
parameter Paul  _et al._ . Finally, these spherical harmonic mode expressions
can be combined using the usual basis of spherical harmonics to obtain
polarization waveforms that are typically used in data-analysis applications.
The combined effect of spins, eccentricity, and higher modes computed here
should help reduce systematic biases in various parameter estimation analyses
for inspiral dominated signals. Moreover, these also could be used to obtain
an analytical frequency domain model (say based on stationary phase
approximation) that can be used for performing detailed parameter estimation
with eccentricity as a parameter.

## ACKNOWLEDGEMENTS

We thank Guillaume Faye and K. G. Arun for useful discussions. We thank the
referee and Mohammed Khalil for pointing out a possible error in our
computations of the $(2,1)$ mode. We sincerely thank Quentin Henry, Guillaume
Faye, and Mohammed Khalil for the discussions that helped us resolve the
discrepancy. We also thank members of the Gravitation & Cosmology group at IIT
Madras for numerous insightful discussions. All computations were performed
using Mathematica and the tensor algebra package, xACT (Martin-Garcia  _et
al._ , 2019).

##  Appendix A ORBITAL DYNAMICS

General orbit variables $(r,\dot{r},\phi,\dot{\phi})$ are written explicitly
in terms of parameters characterizing the binary’s orbital dynamics in
elliptical orbits $(y,e,u)$. Only relevant pieces (desired for investigations
performed here) are computed starting with the QK representation of Ref. Klein
_et al._ (2018). These read as

$\displaystyle r$ $\displaystyle=r^{{\rm{NS}}}+r^{{\rm{SO}}}+r^{{\rm{SS}}}\,,$
(33a) $\displaystyle\dot{r}$
$\displaystyle=\dot{r}^{{\rm{NS}}}+\dot{r}^{{\rm{SO}}}+\dot{r}^{{\rm{SS}}}\,,$
(33b) $\displaystyle\phi$
$\displaystyle=\phi^{{\rm{NS}}}+\phi^{{\rm{SO}}}+\phi^{{\rm{SS}}}\,,$ (33c)
$\displaystyle\dot{\phi}$
$\displaystyle=\dot{\phi}^{{\rm{NS}}}+\dot{\phi}^{{\rm{SO}}}+\dot{\phi}^{{\rm{SS}}}\,,$
(33d)

where

$\displaystyle r_{{\rm{Newt}}}$ $\displaystyle=\frac{GM(1-e\cos
u)}{c^{2}\left(1-e^{2}\right)y^{2}}\,,$ (34a) $\displaystyle r^{{\rm{NS}}}$
$\displaystyle=r_{{\rm{Newt}}}\Bigg{\\{}1+y^{2}\bigg{[}\frac{1}{3}\left[-3+\nu+9e^{2}\left(1-\frac{\nu}{9}\right)\right]\,$
$\displaystyle-\frac{4e\left(1-e^{2}\right)\cos u}{1-e\cos
u}\,\left(1-\frac{3\nu}{8}\right)\bigg{]}\Bigg{\\}}\,,$ (34b) $\displaystyle
r^{{\rm{SO}}}$
$\displaystyle=r_{{\rm{Newt}}}\,y^{3}\Bigg{[}-\frac{e\,\left(1-e^{2}\right)\cos
u}{2\left(1-e\cos u\right)}\beta(4,2)\,$
$\displaystyle+\beta\left(\frac{2}{3}+2e^{2},1+e^{2}\right)\Bigg{]}\,,$ (34c)
$\displaystyle r^{{\rm{SS}}}$
$\displaystyle=r_{{\rm{Newt}}}\,y^{4}\Bigg{\\{}\gamma_{1}\bigg{[}\frac{1+e^{2}}{2}-\frac{e\,\left(1-e^{2}\right)\cos
u}{2\left(1-e\cos u\right)}\bigg{]}\Bigg{\\}}\,,$ (34d)
$\displaystyle\dot{r}_{{\rm{Newt}}}$
$\displaystyle=\frac{c\,e\,\sqrt{1-e^{2}}\,\sin u\,y}{(1-e\cos u)}\,,$ (35a)
$\displaystyle\dot{r}^{{\rm{NS}}}$
$\displaystyle=\dot{r}_{{\rm{Newt}}}\Bigg{\\{}1+y^{2}\left[-e^{2}+\left(-\frac{7}{6}+\frac{7e^{2}}{6}\right)\nu\right]\Bigg{\\}}\,,$
(35b) $\displaystyle\dot{r}^{{\rm{SO}}}$
$\displaystyle=\dot{r}_{{\rm{Newt}}}\,y^{3}\Bigg{[}\left(\frac{1}{2}-\frac{e^{2}}{2}\right)\beta\left(4,2\right)-\beta\left(4,3\right)\,$
$\displaystyle+\beta\left(\frac{2}{3}+2e^{2},1+e^{2}\right)\Bigg{]}\,,$ (35c)
$\displaystyle\dot{r}^{{\rm{SS}}}$
$\displaystyle=-\dot{r}_{{\rm{Newt}}}\,\frac{\gamma_{1}}{2}y^{4}\,,$ (35d)
$\displaystyle\phi^{{\rm{NS}}}$
$\displaystyle=\Bigg{\\{}V\left(e,u\right)+y^{2}\bigg{[}3V\left(e,u\right)+\frac{4e\sqrt{1-e^{2}}\sin
u}{\left(1-e\cos u\right)}\,$
$\displaystyle\left(1-\frac{\nu}{4}\right)\bigg{]}\Bigg{\\}}\,,$ (36a)
$\displaystyle\phi^{{\rm{SO}}}$
$\displaystyle=y^{3}\Bigg{[}V\left(e,u\right)\beta\left(4,3\right)+\frac{e\sqrt{1-e^{2}}\sin
u}{2\left(1-e\cos u\right)}\beta\left(4,4\right)\Bigg{]}\,,$ (36b)
$\displaystyle\phi^{{\rm{SS}}}$
$\displaystyle=y^{4}\Bigg{\\{}\gamma_{1}\bigg{[}\frac{3}{2}V\left(e,u\right)+\frac{e\sqrt{1-e^{2}}\sin
u}{\left(1-e\cos u\right)}\bigg{]}\Bigg{\\}}\,,$ (36c)

with

$\displaystyle V(e,u)$
$\displaystyle=2\,\mathrm{arctan}\left[\sqrt{\frac{\left(1+e\right)}{\left(1-e\right)}}\mathrm{tan}\frac{u}{2}\right]\,,$
(37) $\displaystyle\dot{\phi}_{{\rm{Newt}}}$
$\displaystyle=\frac{c^{3}\left(1-e^{2}\right)^{2}y^{3}}{GM\left(1-e\cos
u\right)^{2}}\,,$ (38a) $\displaystyle\dot{\phi}^{{\rm{NS}}}$
$\displaystyle=\dot{\phi}_{{\rm{Newt}}}\Bigg{\\{}1+y^{2}\,\bigg{[}\frac{4e\left(\cos
u-e\right)}{\left(1-e\cos
u\right)}\left(1-\frac{\nu}{4}\right)\bigg{]}\Bigg{\\}}\,,$ (38b)
$\displaystyle\dot{\phi}^{{\rm{SO}}}$
$\displaystyle=\dot{\phi}_{{\rm{Newt}}}\,y^{3}\Bigg{\\{}\bigg{[}-\frac{1}{2}+\frac{1-e^{2}}{2\left(1-e\cos
u\right)}\bigg{]}\beta\left(4,4\right)\Bigg{\\}}\,,$ (38c)
$\displaystyle\dot{\phi}^{{\rm{SS}}}$
$\displaystyle=\dot{\phi}_{{\rm{Newt}}}\,y^{4}\Bigg{\\{}\gamma_{1}\bigg{[}-1+\frac{1-e^{2}}{\left(1-e\cos
u\right)}\bigg{]}\Bigg{\\}}\,.$ (38d)

In the above,

$\displaystyle\beta\left(a,b\right)$
$\displaystyle=-\,\Bigg{\\{}a\Big{[}\left(1-2\nu\right)\,\left(\bm{\hat{\ell}}\cdot\bm{\chi}_{s}\right)+\delta\,\left(\bm{\hat{\ell}}\cdot\bm{\chi}_{a}\right)\Big{]}\,$
$\displaystyle+2b\nu\,\left(\bm{\hat{\ell}}\cdot\bm{\chi}_{s}\right)\Bigg{\\}},$
(39a) $\displaystyle\gamma_{1}$
$\displaystyle=\Bigg{\\{}\Big{[}\delta\,\kappa_{a}+\kappa_{s}(1-2\nu)-2\nu\Big{]}\left(\bm{\hat{\ell}}\cdot\bm{\chi}_{a}\right)^{2}+2\Big{[}\delta\,\kappa_{s}$
$\displaystyle+\kappa_{a}\left(1-2\nu\right)\Big{]}\left(\bm{\hat{\ell}}\cdot\bm{\chi}_{a}\right)\left(\bm{\hat{\ell}}\cdot\bm{\chi}_{s}\right)+\Big{[}\delta\,\kappa_{a}+\kappa_{s}\,$
$\displaystyle\times\left(1-2\nu\right)+2\nu\Big{]}\left(\bm{\hat{\ell}}\cdot\bm{\chi}_{s}\right)^{2}\Bigg{\\}}.$
(39b)

##  Appendix B SPHERICAL HARMONIC MODES WITH SPIN EFFECTS TO 2PN: ARBITRARY
ECCENTRICITY CASE

Spherical harmonic modes with spin effects to 2PN, valid for elliptical orbits
of arbitrary eccentricity, in terms of parameters $(e,u,y)$ read

$h^{\ell
m}=-i^{2m}\frac{8\,G\,M\,\nu\,\left(1-e^{2}\right)\,y^{2}}{c^{2}\,R}\,\sqrt{\frac{\pi}{5}}\,\mathrm{e}^{-im\phi}\,H^{\ell
m}$ (40)

where

$\displaystyle H^{22}_{\rm{NS}}$ $\displaystyle=\frac{1}{4\left(1-e\cos
u\right)^{2}}\Bigg{(}4-3e^{2}-2e\cos u+e^{2}\cos 2u+4i\,e\sqrt{1-e^{2}}\sin
u\Bigg{)}+\mathcal{O}\left(y^{2}\right)\,,$ (41a) $\displaystyle
H^{22}_{{\rm{SO}}}$ $\displaystyle=-\frac{y^{3}}{6\left(1-e\cos
u\right)^{3}}\Bigg{\\{}2\delta\,\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)\Bigg{[}4-11e^{2}+3e^{4}+e\left(3+4e^{2}\right)\cos
u-4e^{2}\cos 2u+e^{3}\cos 3u+\sqrt{1-e^{2}}\bigg{[}\Big{(}i\,e\,$
$\displaystyle-3i\,e^{3}\Big{)}\sin u+i\,e^{2}\sin
2u\bigg{]}\Bigg{]}+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\Bigg{[}8-22e^{2}+6e^{4}-8\nu+10e^{2}\nu+2e^{4}\nu+\Big{(}2e^{3}-e^{3}\nu\Big{)}\cos
3u-8e^{2}\,$ $\displaystyle\times\cos 2u\left(1-\frac{\nu}{2}\right)+\cos
u\left[6e(1+\nu)+8e^{3}\left(1-\frac{13\nu}{8}\right)\right]+\sqrt{1-e^{2}}\bigg{[}\Big{(}2i\,e-6i\,e^{3}-8i\,e\nu+4i\,e^{3}\nu\Big{)}\sin
u\,$ $\displaystyle+\Big{(}2i\,e^{2}+2i\,e^{2}\nu\Big{)}\sin
2u\bigg{]}\Bigg{]}\Bigg{\\}}+\mathcal{O}\left(y^{5}\right)\,,$ (41b)
$\displaystyle H^{22}_{{\rm{SS}}}$ $\displaystyle=\frac{y^{4}}{8\left(1-e\cos
u\right)^{3}}\Bigg{\\{}\Bigg{[}8-22e^{2}+10e^{4}+e\left(6+e^{2}\right)\cos
u-4e^{2}\cos 2u+e^{3}\cos 3u-4i\,e^{3}\sqrt{1-e^{2}}\sin u\,$
$\displaystyle+2i\,e^{2}\sqrt{1-e^{2}}\sin
2u\Bigg{]}\Bigg{[}2\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\bigg{[}\left(1-2\nu\right)\kappa_{a}+\delta\,\kappa_{s}\bigg{]}+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right){}^{2}\bigg{[}-2\nu+\delta\,\kappa_{a}+\left(1-2\nu\right)\kappa_{s}\bigg{]}\,$
$\displaystyle+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right){}^{2}\bigg{[}2\nu+\delta\,\kappa_{a}+\left(1-2\nu\right)\kappa_{s}\bigg{]}\Bigg{]}\Bigg{\\}}+\mathcal{O}\left(y^{5}\right)\,,$
(41c) $\displaystyle H^{21}_{\rm{NS}}$
$\displaystyle=-\frac{\delta\,\left(1-e^{2}\right)y}{3\left(1-e\cos
u\right)^{2}}+\mathcal{O}\left(y^{3}\right)\,,$ (41d) $\displaystyle
H^{21}_{\rm{SO,1PN}}$
$\displaystyle=\frac{\left(1-e^{2}\right)y^{2}}{2\left(1-e\cos
u\right)^{2}}\Bigg{[}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)+\delta\,\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\Bigg{]}+\mathcal{O}\left(y^{4}\right)\,,$
(41e) $\displaystyle H^{21}_{\rm{SO,2PN}}$
$\displaystyle=-\frac{\left(1-e^{2}\right)y^{4}}{168(1-e\cos(u))^{4}}\Bigg{\\{}\delta\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\Bigg{[}-28+1057e^{2}-357e^{4}+132\nu-322e^{2}\nu+190e^{4}\nu+e^{2}\bigg{[}203-38\nu+21e^{2}\,$
$\displaystyle\times\left(1+\frac{38\nu}{21}\right)\bigg{]}\cos 2u-4e\cos
u\bigg{[}189-24\nu+35e^{2}\left(1+\frac{24\nu}{35}\right)\bigg{]}+\sqrt{1-e^{2}}\bigg{[}-588ie+588ie^{3}-52ie\nu\,$
$\displaystyle+52ie^{3}\nu\bigg{]}\sin
u\Bigg{]}+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)\Bigg{[}-28+1057e^{2}-357e^{4}+820\nu-2170e^{2}\nu+678e^{4}\nu+4e\bigg{[}-189-35e^{2}\,$
$\displaystyle\times\left(1-\frac{136\nu}{35}\right)+88\nu\bigg{]}\cos
u+e^{2}\bigg{[}203-270\nu+21e^{2}\left(1+\frac{46\nu}{21}\right)\bigg{]}\cos
2u+\sqrt{1-e^{2}}\bigg{[}-588ie+588ie^{3}\,$
$\displaystyle+332ie\nu-332ie^{3}\nu\bigg{]}\sin(u)\Bigg{]}\Bigg{\\}}+\mathcal{O}\left(y^{5}\right)\,,$
(41f) $\displaystyle H^{20}_{\rm{NS}}$
$\displaystyle=-\frac{1}{\sqrt{6}}\frac{e\cos u}{(1-e\cos
u)}+\mathcal{O}\left(y^{2}\right)\,,$ (41g) $\displaystyle H^{20}_{\rm{SO}}$
$\displaystyle=-\frac{ey^{3}}{3\sqrt{6}(1-e\cos
u)^{3}}\Bigg{\\{}\delta\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)\bigg{[}\left(6e^{3}-14e\right)+2e^{2}\cos
3u-8e\cos 2u+14\cos
u\bigg{]}+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\,$
$\displaystyle\times\bigg{[}6e^{3}(1-\nu)+2e^{2}\left(1-\frac{\nu}{2}\right)\cos
3u+\left(3e^{2}\nu-10\nu+14\right)\cos
u+10e\nu-8e\left(1-\frac{\nu}{2}\right)\cos
2u-14e\bigg{]}\Bigg{\\}}+\mathcal{O}\left(y^{5}\right)\,,$ (41h)
$\displaystyle H^{20}_{\rm{SS}}$
$\displaystyle=\frac{ey^{4}}{4\sqrt{6}(1-e\cos
u)^{3}}\Bigg{\\{}\Bigg{[}\left(e^{2}+6\right)\cos u+e\left(2e^{2}+e\cos
3u-4\cos
2u-6\right)\Bigg{]}\Bigg{[}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)^{2}\bigg{[}-2\nu+\delta\kappa_{a}+\kappa_{s}\,$
$\displaystyle\times\left(1-2\nu\right)\bigg{]}+2\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\bigg{[}\delta\kappa_{s}+\kappa_{a}\left(1-2\nu\right)\bigg{]}+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)^{2}\bigg{[}2\nu+\delta\kappa_{a}+\kappa_{s}\left(1-2\nu\right)\bigg{]}\Bigg{]}\Bigg{\\}}+\mathcal{O}\left(y^{5}\right)\,,$
(41i) $\displaystyle H^{33}_{\rm{NS}}$
$\displaystyle=-\frac{\sqrt{5}\delta\,y}{8\sqrt{42}\left(1-e\cos
u\right)^{3}}\Bigg{\\{}18-28e^{2}+10e^{4}+\left(1-e^{2}\right)\bigg{[}-14e\cos
u+6e^{2}\cos
2u\bigg{]}+\sqrt{1-e^{2}}\bigg{[}\Big{(}20i\,e-15i\,e^{3}\Big{)}\,$
$\displaystyle\times\sin u-4i\,e^{2}\sin 2u+i\,e^{3}\sin
3u\bigg{]}\Bigg{\\}}+\mathcal{O}\left(y^{3}\right)\,,$ (41j) $\displaystyle
H^{33}_{\rm{SO}}$ $\displaystyle=\frac{\sqrt{5}y^{4}}{8\sqrt{42}\left(1-e\cos
u\right)^{4}}\Bigg{\\{}\delta\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\Bigg{[}36-132e^{2}+126e^{4}-30e^{6}-45\nu+102e^{2}\nu-69e^{4}\nu+12e^{6}\nu+\left(6e^{3}-6e^{5}\right)\,$
$\displaystyle\times\cos 3u+e^{2}\left(1-e^{2}\right)\cos
2u\bigg{[}-20-3\nu-6e^{2}\left(1-\frac{\nu}{2}\right)\bigg{]}-2e\left(1-e^{2}\right)\cos
u\bigg{[}-8\left(1+\frac{9\nu}{4}\right)-17e^{2}\,$
$\displaystyle\times\left(1-\frac{18\nu}{17}\right)\bigg{]}+\sqrt{1-e^{2}}\bigg{[}\Big{(}20i\,e-84i\,e^{3}+36i\,e^{5}-54i\,e\nu+98i\,e^{3}\nu-30i\,e^{5}\nu\Big{)}\sin
u+\Big{(}24i\,e^{2}+4i\,e^{4}\,$
$\displaystyle+4i\,e^{2}\nu-18i\,e^{4}\nu\Big{)}\sin
2u+\Big{(}-12i\,e^{3}+6i\,e^{3}\nu\Big{)}\sin
3u+\Big{(}2i\,e^{4}-i\,e^{4}\nu\Big{)}\sin
4u\bigg{]}\Bigg{]}+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)\Bigg{[}36-132e^{2}\,$
$\displaystyle+126e^{4}-30e^{6}-171\nu+606e^{2}\nu-579e^{4}\nu+144e^{6}\nu+\Big{(}6e^{3}-6e^{5}-24e^{3}\nu+24e^{5}\nu\Big{)}\cos
3u+e^{2}\left(1-e^{2}\right)\,$ $\displaystyle\times\cos
2u\bigg{[}-20+79\nu-6e^{2}\left(1-\frac{25\nu}{6}\right)\bigg{]}-2e\left(1-e^{2}\right)\cos
u\bigg{[}-8+30\nu-17e^{2}\left(1-\frac{70\nu}{17}\right)\bigg{]}+\sqrt{1-e^{2}}\,$
$\displaystyle\times\bigg{[}\Big{(}20i\,e-84i\,e^{3}+36i\,e^{5}-90i\,e\nu+356i\,e^{3}\nu-154i\,e^{5}\nu\Big{)}\sin
u+\Big{(}24i\,e^{2}+4i\,e^{4}-96i\,e^{2}\nu-16i\,e^{4}\nu\Big{)}\sin 2u\,$
$\displaystyle+\Big{(}-12i\,e^{3}+48i\,e^{3}\nu\Big{)}\sin
3u+\Big{(}2i\,e^{4}-8i\,e^{4}\nu\Big{)}\sin
4u\bigg{]}\Bigg{]}\Bigg{\\}}+\mathcal{O}\left(y^{5}\right)\,,$ (41k)
$\displaystyle H^{32}_{\rm{NS}}$
$\displaystyle=\frac{\sqrt{5}\left(1-e^{2}\right)\left(1-3\nu\right)y^{2}}{12\sqrt{7}\left(1-e\cos
u\right)^{3}}\,\Bigg{(}4-4e^{2}+i\,e\sqrt{1-e^{2}}\sin
u\Bigg{)}+\mathcal{O}\left(y^{4}\right)\,,$ (41l) $\displaystyle
H^{32}_{\rm{SO}}$
$\displaystyle=\frac{\sqrt{5}\left(1-e^{2}\right)y^{3}\nu}{3\sqrt{7}\left(1-e\cos
u\right)^{3}}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\Bigg{(}4-4e^{2}+i\,e\sqrt{1-e^{2}}\sin
u\Bigg{)}+\mathcal{O}\left(y^{5}\right)\,,$ (41m) $\displaystyle
H^{31}_{\rm{NS}}$ $\displaystyle=\frac{\delta\,y}{12\sqrt{14}\left(1-e\cos
u\right)^{2}}\Bigg{\\{}-1+e^{2}+6e\left(1-e^{2}\right)\cos
u+\sqrt{1-e^{2}}\bigg{[}-6i\,e\sin u+3i\,e^{2}\sin
2u\bigg{]}\Bigg{\\}}+\mathcal{O}\left(y^{3}\right)\,,$ (41n) $\displaystyle
H^{31}_{\rm{SO}}$ $\displaystyle=\frac{y^{4}}{24\sqrt{14}\left(1-e\cos
u\right)^{4}}\Bigg{\\{}\delta\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\Bigg{[}4-68e^{2}+94e^{4}-30e^{6}-13\nu+70e^{2}\nu-101e^{4}\nu+44e^{6}\nu+2e\left(1-e^{2}\right)\,$
$\displaystyle\times\bigg{[}24-14\nu+e^{2}\left(1+14\nu\right)\bigg{]}\cos
u+\left(6e^{3}-6e^{5}\right)\cos 3u+e^{2}\left(1-e^{2}\right)\cos
2u\bigg{[}-20-3\nu-6e^{2}\left(1-\frac{\nu}{2}\right)\bigg{]}\,$
$\displaystyle+\sqrt{1-e^{2}}\bigg{[}\Big{(}-36i\,e-60i\,e^{3}+12i\,e^{5}+46i\,e\nu-26i\,e^{3}\nu+22i\,e^{5}\nu\Big{)}\sin
u+\Big{(}72i\,e^{2}+12i\,e^{4}-36i\,e^{2}\nu-6i\,e^{4}\nu\Big{)}\,$
$\displaystyle\times\sin 2u+\Big{(}-36i\,e^{3}+18i\,e^{3}\nu\Big{)}\sin
3u+\Big{(}6i\,e^{4}-3i\,e^{4}\nu\Big{)}\sin
4u\bigg{]}\Bigg{]}+\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)\Bigg{[}4-68e^{2}+94e^{4}-30e^{6}-11\nu\,$
$\displaystyle+254e^{2}\nu-355e^{4}\nu+112e^{6}\nu+\Big{(}6e^{3}-6e^{5}-24e^{3}\nu+24e^{5}\nu\Big{)}\cos
3u-2e\left(1-e^{2}\right)\cos u\bigg{[}-24+94\nu-e^{2}\,$
$\displaystyle\times\left(1-6\nu\right)\bigg{]}+e^{2}\left(1-e^{2}\right)\cos
2u\bigg{[}-20+79\nu-6e^{2}\left(1-\frac{25\nu}{6}\right)\bigg{]}+\sqrt{1-e^{2}}\bigg{[}\Big{(}-36i\,e-60i\,e^{3}+12i\,e^{5}\,$
$\displaystyle+130i\,e\nu+268i\,e^{3}\nu-62i\,e^{5}\nu\Big{)}\sin
u+\Big{(}72i\,e^{2}+12i\,e^{4}-288i\,e^{2}\nu-48i\,e^{4}\nu\Big{)}\sin
2u+\Big{(}-36i\,e^{3}+144i\,e^{3}\nu\Big{)}\,$ $\displaystyle\times\sin
3u+\Big{(}6i\,e^{4}-24i\,e^{4}\nu\Big{)}\sin
4u\bigg{]}\Bigg{]}\Bigg{\\}}+\mathcal{O}\left(y^{5}\right)\,,$ (41o)
$\displaystyle H^{30}_{\rm{NS}}$
$\displaystyle=\frac{i}{2\sqrt{42}}\frac{e\left(1-e^{2}\right)^{3/2}(1-3\nu)y^{2}\sin
u}{(1-e\cos u)^{3}}+\mathcal{O}\left(y^{4}\right)\,,$ (41p) $\displaystyle
H^{30}_{\rm{SO}}$
$\displaystyle=i\sqrt{\frac{2}{21}}\frac{e\left(1-e^{2}\right)^{3/2}\nu\,y^{3}\sin
u}{(1-e\cos
u)^{3}}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)+\mathcal{O}\left(y^{5}\right)\,,$
(41q) $\displaystyle H^{43}_{\rm{NS}}$
$\displaystyle=-\frac{\left(1-e^{2}\right)^{2}\delta\,\left(1-2\nu\right)\,y^{3}}{12\sqrt{70}\left(1-e\cos
u\right)^{4}}\Bigg{(}27-24e^{2}-4e\cos u+e^{2}\cos 2u+10i\,e\sqrt{1-e^{2}}\sin
u\Bigg{)}+\mathcal{O}\left(y^{5}\right)\,,$ (41r) $\displaystyle
H^{43}_{\rm{SO}}$
$\displaystyle=\frac{\sqrt{5}\left(1-e^{2}\right)^{2}\nu\,y^{4}}{24\sqrt{14}\left(1-e\cos
u\right)^{4}}\Bigg{[}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)-\delta\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\Bigg{]}\Bigg{(}27-24e^{2}-4e\cos
u+e^{2}\cos 2u+10i\,e\sqrt{1-e^{2}}\sin u\Bigg{)}\,$
$\displaystyle+\mathcal{O}\left(y^{5}\right)\,,$ (41s) $\displaystyle
H^{41}_{\rm{NS}}$
$\displaystyle=-\frac{\left(1-e^{2}\right)^{2}\delta\,\left(1-2\nu\right)\,y^{3}}{84\sqrt{10}\left(1-e\cos
u\right)^{4}}\Bigg{(}1+8e^{2}-12e\cos u+3e^{2}\cos 2u+10i\,e\sqrt{1-e^{2}}\sin
u\Bigg{)}+\mathcal{O}\left(y^{5}\right)\,,$ (41t) $\displaystyle
H^{41}_{\rm{SO}}$
$\displaystyle=\frac{\sqrt{5}\left(1-e^{2}\right)^{2}\nu\,y^{4}}{168\sqrt{2}\left(1-e\cos
u\right)^{4}}\Bigg{[}\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{a}\right)-\delta\left(\bm{\hat{\ell}}\cdot\bm{\chi}^{c}_{s}\right)\Bigg{]}\Bigg{(}1+8e^{2}-12e\cos
u+3e^{2}\cos 2u+10i\,e\sqrt{1-e^{2}}\sin u\Bigg{)}\,$
$\displaystyle+\mathcal{O}\left(y^{5}\right)\,.$ (41u)

## References

  * Aasi  _et al._ (2015) J. Aasi _et al._ (LIGO Scientific), Class. Quant. Grav. 32, 074001 (2015), arXiv:1411.4547 [gr-qc] .
  * Acernese  _et al._ (2015) F. Acernese _et al._ (VIRGO), Class. Quant. Grav. 32, 024001 (2015), arXiv:1408.3978 [gr-qc] .
  * Abbott  _et al._ (2019a) B. P. Abbott _et al._ (LIGO Scientific, Virgo), Phys. Rev. X 9, 031040 (2019a), arXiv:1811.12907 [astro-ph.HE] .
  * Abbott  _et al._ (2021a) R. Abbott _et al._ (LIGO Scientific, Virgo), Phys. Rev. X 11, 021053 (2021a), arXiv:2010.14527 [gr-qc] .
  * Abbott  _et al._ (2021b) R. Abbott _et al._ (LIGO Scientific, VIRGO), (2021b), arXiv:2108.01045 [gr-qc] .
  * Abbott  _et al._ (2021c) R. Abbott _et al._ (LIGO Scientific, VIRGO, KAGRA), arXiv preprint (2021c), arXiv:2111.03606 [gr-qc] .
  * Abbott  _et al._ (2017) B. P. Abbott _et al._ (LIGO Scientific, Virgo), Phys. Rev. Lett. 119, 161101 (2017), arXiv:1710.05832 [gr-qc] .
  * Abbott  _et al._ (2020a) B. P. Abbott _et al._ (LIGO Scientific, Virgo), Astrophys. J. Lett. 892, L3 (2020a), arXiv:2001.01761 [astro-ph.HE] .
  * Abbott  _et al._ (2021d) R. Abbott _et al._ (LIGO Scientific, KAGRA, VIRGO), Astrophys. J. Lett. 915, L5 (2021d), arXiv:2106.15163 [astro-ph.HE] .
  * Shoemaker (2019) D. Shoemaker (LIGO Scientific), (2019), arXiv:1904.03187 [gr-qc] .
  * McClelland  _et al._ (2016) D. McClelland, M. Evans, R. Schnabel, B. Lantz, I. Martin, and V. Quetschke,  (2016).
  * Dwyer  _et al._ (2015) S. Dwyer, D. Sigg, S. W. Ballmer, L. Barsotti, N. Mavalvala, and M. Evans, Phys. Rev. D 91, 082001 (2015), arXiv:arXiv:1410.0612 [astro-ph.IM] .
  * Punturo  _et al._ (2010) M. Punturo _et al._ , Class. Quant. Grav. 27, 194002 (2010).
  * Sato  _et al._ (2017) S. Sato _et al._ , Journal of Physics: Conference Series 840, 012010 (2017).
  * Amaro-Seoane  _et al._ (2017) P. Amaro-Seoane, H. Audley, S. Babak, J. Baker, E. Barausse, P. Bender, E. Berti, P. Binetruy, M. Born, D. Bortoluzzi, _et al._ , arXiv preprint arXiv:1702.00786 (2017).
  * Cutler and Flanagan (1994) C. Cutler and E. E. Flanagan, Phys. Rev. D 49, 2658 (1994), arXiv:gr-qc/9402014 .
  * Poisson and Will (1995) E. Poisson and C. M. Will, Phys. Rev. D 52, 848 (1995), arXiv:gr-qc/9502040 .
  * Krolak  _et al._ (1995) A. Krolak, K. D. Kokkotas, and G. Schaefer, Phys. Rev. D 52, 2089 (1995), arXiv:gr-qc/9503013 .
  * Abbott  _et al._ (2016) B. P. Abbott _et al._ (LIGO Scientific, Virgo), Phys. Rev. D 93, 122003 (2016), arXiv:1602.03839 [gr-qc] .
  * Peters (1964) P. C. Peters, Phys. Rev. 136, B1224 (1964).
  * Peters and Mathews (1963) P. C. Peters and J. Mathews, Phys. Rev. 131, 435 (1963).
  * Kozai (1962) Y. Kozai, Astron. J. 67, 591 (1962).
  * Lidov (1962) M. Lidov, Planetary and Space Science 9, 719 (1962).
  * Abbott  _et al._ (2019b) B. P. Abbott _et al._ (LIGO Scientific, Virgo), Astrophys. J. 883, 149 (2019b), arXiv:1907.09384 [astro-ph.HE] .
  * Abbott  _et al._ (2020b) R. Abbott _et al._ (LIGO Scientific, Virgo), Astrophys. J. Lett. 900, L13 (2020b), arXiv:2009.01190 [astro-ph.HE] .
  * Kimball  _et al._ (2021) C. Kimball _et al._ , Astrophys. J. Lett. 915, L35 (2021), arXiv:2011.05332 [astro-ph.HE] .
  * Romero-Shaw  _et al._ (2021) I. M. Romero-Shaw, P. D. Lasky, and E. Thrane, Astrophys. J. Lett. 921, L31 (2021), arXiv:2108.01284 [astro-ph.HE] .
  * Romero-Shaw  _et al._ (2022) I. M. Romero-Shaw, P. D. Lasky, and E. Thrane, (2022), arXiv:2206.14695 [astro-ph.HE] .
  * O’Shea and Kumar (2021) E. O’Shea and P. Kumar, (2021), arXiv:2107.07981 [astro-ph.HE] .
  * Brown and Zimmerman (2010) D. A. Brown and P. J. Zimmerman, Phys. Rev. D 81, 024007 (2010), arXiv:0909.0066 [gr-qc] .
  * Huerta and Brown (2013) E. A. Huerta and D. A. Brown, Phys. Rev. D 87, 127501 (2013), arXiv:1301.1895 [gr-qc] .
  * Favata  _et al._ (2022) M. Favata, C. Kim, K. G. Arun, J. Kim, and H. W. Lee, Phys. Rev. D 105, 023003 (2022), arXiv:2108.05861 [gr-qc] .
  * Lower  _et al._ (2018) M. E. Lower, E. Thrane, P. D. Lasky, and R. Smith, Phys. Rev. D 98, 083028 (2018), arXiv:1806.05350 [astro-ph.HE] .
  * (34) S. Kumar, S. Tibrewal, Divyajyoti, and C. K. Mishra, In preparation (2022).
  * Chattaraj  _et al._ (2022) A. Chattaraj, T. RoyChowdhury, Divyajyoti, C. K. Mishra, and A. Gupta, (2022), arXiv:2204.02377 [gr-qc] .
  * Hinder  _et al._ (2018) I. Hinder, L. E. Kidder, and H. P. Pfeiffer, Phys. Rev. D 98, 044015 (2018), arXiv:1709.02007 [gr-qc] .
  * Rebei  _et al._ (2019) A. Rebei, E. A. Huerta, S. Wang, S. Habib, R. Haas, D. Johnson, and D. George, Phys. Rev. D 100, 044025 (2019), arXiv:1807.09787 [gr-qc] .
  * Mishra  _et al._ (2015) C. K. Mishra, K. G. Arun, and B. R. Iyer, Phys. Rev. D 91, 084040 (2015), arXiv:1501.07096 [gr-qc] .
  * Moore  _et al._ (2016) B. Moore, M. Favata, K. G. Arun, and C. K. Mishra, Phys. Rev. D 93, 124061 (2016), arXiv:1605.00304 [gr-qc] .
  * Tanay  _et al._ (2016) S. Tanay, M. Haney, and A. Gopakumar, Phys. Rev. D 93, 064031 (2016), arXiv:1602.03081 [gr-qc] .
  * Boetzel  _et al._ (2019) Y. Boetzel, C. K. Mishra, G. Faye, A. Gopakumar, and B. R. Iyer, Phys. Rev. D 100, 044018 (2019), arXiv:1904.11814 [gr-qc] .
  * Ebersold  _et al._ (2019) M. Ebersold, Y. Boetzel, G. Faye, C. K. Mishra, B. R. Iyer, and P. Jetzer, Phys. Rev. D 100, 084043 (2019), arXiv:1906.06263 [gr-qc] .
  * Konigsdorffer and Gopakumar (2006) C. Konigsdorffer and A. Gopakumar, Phys. Rev. D 73, 124012 (2006), arXiv:gr-qc/0603056 .
  * Moore and Yunes (2019) B. Moore and N. Yunes, Class. Quant. Grav. 36, 185003 (2019), arXiv:1903.05203 [gr-qc] .
  * Arun  _et al._ (2009) K. G. Arun, A. Buonanno, G. Faye, and E. Ochsner, Phys. Rev. D 79, 104023 (2009), [Erratum: Phys.Rev.D 84, 049901 (2011)], arXiv:0810.5336 [gr-qc] .
  * Hartung and Steinhoff (2011) J. Hartung and J. Steinhoff, Annalen Phys. 523, 783 (2011), arXiv:1104.3079 [gr-qc] .
  * Marsat  _et al._ (2013) S. Marsat, A. Bohe, G. Faye, and L. Blanchet, Class. Quant. Grav. 30, 055007 (2013), arXiv:1210.4143 [gr-qc] .
  * Levi and Steinhoff (2016) M. Levi and J. Steinhoff, JCAP 01, 011 (2016), arXiv:1506.05056 [gr-qc] .
  * Bohe  _et al._ (2013) A. Bohe, S. Marsat, G. Faye, and L. Blanchet, Class. Quant. Grav. 30, 075017 (2013), arXiv:1212.5520 [gr-qc] .
  * Bohé  _et al._ (2013) A. Bohé, S. Marsat, and L. Blanchet, Class. Quant. Grav. 30, 135009 (2013), arXiv:1303.7412 [gr-qc] .
  * Marsat  _et al._ (2014) S. Marsat, A. Bohé, L. Blanchet, and A. Buonanno, Class. Quant. Grav. 31, 025023 (2014), arXiv:1307.6793 [gr-qc] .
  * Bohé  _et al._ (2015) A. Bohé, G. Faye, S. Marsat, and E. K. Porter, Class. Quant. Grav. 32, 195010 (2015), arXiv:1501.01529 [gr-qc] .
  * Blanchet  _et al._ (2011) L. Blanchet, A. Buonanno, and G. Faye, Phys. Rev. D 84, 064041 (2011), arXiv:1104.5659 [gr-qc] .
  * Klein  _et al._ (2013) A. Klein, N. Cornish, and N. Yunes, Phys. Rev. D 88, 124015 (2013), arXiv:1305.1932 [gr-qc] .
  * Chatziioannou  _et al._ (2013) K. Chatziioannou, A. Klein, N. Yunes, and N. Cornish, Phys. Rev. D 88, 063011 (2013), arXiv:1307.4418 [gr-qc] .
  * Henry  _et al._ (2022) Q. Henry, S. Marsat, and M. Khalil, (2022), arXiv:2209.00374 [gr-qc] .
  * Kidder (1995) L. E. Kidder, Phys. Rev. D 52, 821 (1995), arXiv:gr-qc/9506022 .
  * Buonanno  _et al._ (2013) A. Buonanno, G. Faye, and T. Hinderer, Phys. Rev. D 87, 044009 (2013), arXiv:1209.6349 [gr-qc] .
  * Majar and Vasuth (2008) J. Majar and M. Vasuth, Phys. Rev. D 77, 104005 (2008), arXiv:0806.2273 [gr-qc] .
  * Klein and Jetzer (2010) A. Klein and P. Jetzer, Phys. Rev. D 81, 124001 (2010), arXiv:1005.2046 [gr-qc] .
  * Klein  _et al._ (2018) A. Klein, Y. Boetzel, A. Gopakumar, P. Jetzer, and L. de Vittori, Phys. Rev. D 98, 104043 (2018), arXiv:1801.08542 [gr-qc] .
  * Vasuth and Majar (2007) M. Vasuth and J. Majar, Int. J. Mod. Phys. A 22, 2405 (2007), arXiv:0705.3481 [gr-qc] .
  * Klein (2021) A. Klein, (2021), arXiv:2106.10291 [gr-qc] .
  * Blanchet  _et al._ (2008) L. Blanchet, G. Faye, B. R. Iyer, and S. Sinha, Class. Quant. Grav. 25, 165003 (2008), [Erratum: Class.Quant.Grav. 29, 239501 (2012)], arXiv:0802.1249 [gr-qc] .
  * Kidder (2008) L. E. Kidder, Phys. Rev. D 77, 044016 (2008), arXiv:0710.0614 [gr-qc] .
  * Blanchet (2014) L. Blanchet, Living Rev. Rel. 17, 2 (2014), arXiv:1310.1528 [gr-qc] .
  * Mishra  _et al._ (2016) C. K. Mishra, A. Kela, K. G. Arun, and G. Faye, Phys. Rev. D 93, 084054 (2016), arXiv:1601.05588 [gr-qc] .
  * (68) See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevD.108.024023 for spin effects in spherical harmonic modes valid for elliptical orbits are expanded up to 6th power in eccentricity, are given in the supplemental material.
  * Faye  _et al._ (2012) G. Faye, S. Marsat, L. Blanchet, and B. R. Iyer, Class. Quant. Grav. 29, 175004 (2012), arXiv:1204.1043 [gr-qc] .
  * Blanchet and Faye (2001) L. Blanchet and G. Faye, Phys. Rev. D 63, 062005 (2001), arXiv:gr-qc/0007051 .
  * Blanchet  _et al._ (2002) L. Blanchet, B. R. Iyer, and B. Joguet, Phys. Rev. D 65, 064005 (2002), [Erratum: Phys.Rev.D 71, 129903 (2005)], arXiv:gr-qc/0105098 .
  * Blanchet and Iyer (2005) L. Blanchet and B. R. Iyer, Phys. Rev. D 71, 024004 (2005), arXiv:gr-qc/0409094 .
  * Blanchet  _et al._ (2004) L. Blanchet, T. Damour, and G. Esposito-Farese, Phys. Rev. D 69, 124007 (2004), arXiv:gr-qc/0311052 .
  * Blanchet  _et al._ (2005) L. Blanchet, T. Damour, G. Esposito-Farese, and B. R. Iyer, Phys. Rev. D 71, 124004 (2005), arXiv:gr-qc/0503044 .
  * Blanchet  _et al._ (2006) L. Blanchet, A. Buonanno, and G. Faye, Phys. Rev. D 74, 104034 (2006), [Erratum: Phys.Rev.D 75, 049903 (2007), Erratum: Phys.Rev.D 81, 089901 (2010)], arXiv:gr-qc/0605140 .
  * Martin-Garcia  _et al._ (2019) J. M. Martin-Garcia _et al._ (xAct:Efficient tensor computer algebra for the wolfram language), http://www.xact.es/ (2019).
  * Khalil  _et al._ (2021) M. Khalil, A. Buonanno, J. Steinhoff, and J. Vines, Phys. Rev. D 104, 024046 (2021), arXiv:2104.11705 [gr-qc] .
  * Damour and Deruelle (1985) T. Damour and N. Deruelle, Annales de l’I.H.P. Physique théorique 43, 107 (1985).
  * Damour and Schaefer (1988) T. Damour and G. Schaefer, Nuovo Cim. B 101, 127 (1988).
  * Schäfer and Wex (1993) G. Schäfer and N. Wex, Physics Letters A 174, 196 (1993).
  * Wex (1995) N. Wex, Classical and Quantum Gravity 12, 983 (1995).
  * Memmesheimer  _et al._ (2004) R.-M. Memmesheimer, A. Gopakumar, and G. Schaefer, Phys. Rev. D 70, 104011 (2004), arXiv:gr-qc/0407049 .
  * Arun  _et al._ (2008) K. G. Arun, L. Blanchet, B. R. Iyer, and M. S. S. Qusailah, Phys. Rev. D 77, 064035 (2008), arXiv:0711.0302 [gr-qc] .
  * Boetzel  _et al._ (2017) Y. Boetzel, A. Susobhanan, A. Gopakumar, A. Klein, and P. Jetzer, Phys. Rev. D 96, 044011 (2017), arXiv:1707.02088 [gr-qc] .
  * Joshi  _et al._ (2022) A. V. Joshi, S. G. Rosofsky, R. Haas, and E. A. Huerta, (2022), arXiv:2210.01852 [gr-qc] .
  * (86) K. Paul, K. Srinivasan, A. Klein, and C. K. Mishra, In preparation (2022).