# Post-Keplerian waveform model for binary compact object as sources of space-
based gravitational wave detector and its implications

Li-Fang Li  Institute of Mechanics, Chinese Academy of Sciences, Beijing
100190, China  Zhoujian Cao 111corresponding author<EMAIL_ADDRESS>Institute
of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese
Academy of Sciences, Beijing 100190, China  School of Fundamental Physics and
Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou
310024, China

###### Abstract

Binary compact objects will be among the important sources for the future
space-based gravitational wave detectors. Such binary compact objects include
stellar massive binary black hole, binary neutron star, binary white dwarf and
mixture of these compact objects. Regarding to the relatively low frequency,
the gravitational interaction between the two objects of the binary is weak.
Post-Newtonian approximation of general relativity is valid. Previous works
about the waveform model for such binaries in the literature consider the
dynamics for specific situations which involve detailed complicated matter
dynamics between the two objects. We here take a different idea. We adopt the
trick used in pulsar timing detection. For any gravity theories and any
detailed complicated matter dynamics, the motion of the binary can always be
described as a post-Keplerian expansion. And a post-Keplerian gravitational
waveform model will be reduced. Instead of object masses, spins, matter’s
equation of state parameters and dynamical parameters beyond general
relativity, the involved parameters in our post-Keplerian waveform model are
the Keplerian orbit elements and their adiabatic variations. Respect to
current planning space-based gravitational wave detectors including LISA,
Taiji and Tianqin, we find that the involved waveform model parameters can be
well determined. And consequently the detail matter dynamics of the binary can
be studied then. For binary with purely gravitational interactions, gravity
theory can be constrained well.

##  I Introduction

Binary stars are unique astrophysical laboratories for the study of
fundamental physics and cosmology. Such laboratories have been realized
through pulsar timing detection of binary pulsar Kramer et al. (2021); Lorimer
and Kramer (2004) and gravitational wave detection of binary coalescence The
LIGO Scientific Collaboration et al. (2021). But pulsar timing detection and
current gravitational wave detection concern two different life stages of the
binary system. Pulsar timing detection is about the early inspiral stage.
While current gravitational wave detection by ground based detectors is about
the stage near merger. In the coming future the space-based gravitational wave
detectors will change this situation. Then gravitational wave detection can
also provides information about early inspiral stage Willems et al. (2008);
Huang et al. (2020). For example, LISA may resolve $\sim 10^{4}$ double white
dwarfs (DWDs) Nelemans et al. (2001); Seto (2022).

Waveform model is important for gravitational wave data analysis and for
extracting the information of the sources. For the merger stage of a binary,
extremal general relativity (GR) condition makes numerical relativity
calculation necessary to construct the waveform model Cao and Han (2017); Liu
et al. (2020, 2022). For the early inspiral stage, the gravitational
interaction is much weaker. And Newtonian approximation is valid. The binary
compact objects including stellar massive binary black hole, binary neutron
star, binary white dwarf and mixture of these compact objects belong to such
early inspiral binaries for the future space-based gravitational wave
detectors. The simplest waveform model for such sources is based on quadrupole
formula for monochromatic approximation Huang et al. (2020). When the GR
leading order of inspiral behavior is taken into consideration, the chirp
waveform model has been widely used Takahashi and Seto (2002). Besides the GR
effect, the mass transfer between the two compact objects Kremer et al. (2017)
and the tidal deformation of the two objects Benacquista (2011); Wolz et al.
(2020) can also contribute to the chirp signal. Recently, the Ref. Willems et
al. (2008) used Newtonian approximation to construct a waveform model and
focused on the effect of white dwarf interior on the periastron precession.
These different waveform models focus on different physical mechanism and
introduce different waveform parameters consequently. Such diversity of
waveform models make data analysis and source property study hard. In order to
catch weak GW signals, ones have to analyze the same data many many times with
different specific waveform models. Even worse, some signals may be lost due
to the uncomplete analysis. As promising laboratory to test gravity theories,
it is interesting and important to construct waveform model for different
gravity theories other than GR. Unfortunately such waveform models have not
been studied yet.

In contrast to the aforementioned specific waveform models for different
physical situations, an unified waveform model will be useful. Such an unified
waveform model will definitely facilitate the data analysis for the space-
based detectors. With such an unified waveform model, ones can not only catch
the signal much easier but also distinguish different physical situations and
even constraint different gravity theories. We will propose such an unified
waveform model in the current paper.

Since the component objects of a binary in early inspiral stage are well
separated, the orbital dynamics can be well described by a perturbation to a
Kepler orbit. Such perturbation may come from gravity theory, environment,
tidal interaction and mass transfer between the two components. All kinds of
effects on the binary dynamics can be included. Consequently, the
parameterized post-Keplerian formalism Damour and Taylor (1992); Moore et al.
(2016) is a good framework to describe the system. This unified waveform model
uses post-Keplerian parameters instead of component masses and other physical
parameters. This unified waveform model is consequently valid for any kinds of
gravity theories and any detailed complicated matter dynamics. If ones can
measure the post-Keplerian parameters accurately, gravity theory and the
theory of stellar structure and evolution can be tested and investigated
Kramer et al. (2021); Willems et al. (2008). We will show later in the current
paper that this is exactly true for the current planning space-based
gravitational wave detectors including LISA, Taiji and Tianqin.

The arrangement of the rest of this paper is as following. We will present the
post-Keplerian waveform model in the next section. Then we relate the involved
post-Keplerian parameters to general relativity (GR) effect, gravity theory
effect beyond GR, tidal interaction and the mass transfer effect between the
two components of the binary. When we construct such relation we also give
guide lines to use post-Keplerian parameters to distinguish these different
effects. After that we use Fisher matrix technique to estimate the measurement
accuracy of the post-Keplerian parameters based on LISA, Taiji and Tianqin
respectively. We find that the involved post-Keplerian parameters can be well
determined. And consequently the detail matter dynamics of the binary can be
studied then. For binary with purely gravitational interactions, gravity
theory can be constrained well. That is to say, equipped with our post-
Keplerian waveform model, the future space-based gravitational wave detectors
can do good science for compact object binary observations. Throughout this
paper we will use units $c=G=1$.

##  II Post-Keplerian waveform model

The general bound Keplerian orbit is an eccentric orbit. The binary moves
along an eccentric orbit and the corresponding gravitational wave form respect
to the detector can be written as Takahashi and Seto (2002); Barack and Cutler
(2004); Willems et al. (2008)

$\displaystyle
h=\frac{\sqrt{3}}{2}\sum_{n=1}^{\infty}[F_{+}h_{n}^{+}+F_{\times}h_{n}^{\times}],$
(1) $\displaystyle
h_{n}^{+}(t)=A\\{(1+\cos^{2}\iota)a_{n}(e)\cos[n\phi(t)]\cos(2\gamma)$
$\displaystyle-(1+\cos^{2}\iota)b_{n}(e)\sin[n\phi(t)]\sin(2\gamma)$
$\displaystyle+(\sin^{2}\iota)c_{n}(e)\cos[n\phi(t)]\\},$ (2) $\displaystyle
h_{n}^{\times}(t)=-2A\cos\iota\\{b_{n}(e)\sin[n\phi(t)]\cos(2\gamma)$
$\displaystyle+a_{n}(e)\cos[n\phi(t)]\sin(2\gamma)\\},$ (3) $\displaystyle
a_{n}(e)=n[J_{n-2}(ne)-2eJ_{n-1}(ne)+(2/n)J_{n}(ne)$
$\displaystyle+2eJ_{n+1}(ne)-J_{n+2}(ne)],$ (4) $\displaystyle
b_{n}(e)=n\sqrt{1-e^{2}}[J_{n-2}(ne)-2J_{n}(ne)+J_{n+2}(ne)],$ (5)
$\displaystyle c_{n}(e)=J_{n}(ne),$ (6)

where $J_{i}(x)$ is the $i$-th order Bessel function of the first kind. The
amplitude $A$ depends on the gravity theory and the specific dynamics of the
binary. $n$ corresponds to higher harmonics excited by the eccentric orbit. In
Liu et al. (2022), we call it tones to distinguish with the spherical
harmonics. $\phi(t)$ is the Doppler shifted orbital phase

$\displaystyle\phi(t)=\phi_{0}+2\pi ft$ (7)

with initial phase $\phi_{0}$.

Perturbations to the above Kepler orbit may result in changing of orbital
frequency $f$, inclination angle $\iota$, orbit eccentricity $e$, periastron
direction $\gamma$. In all we have adiabatic changing

$\displaystyle f=f_{0}+\dot{f}_{0}t,$ (8)
$\displaystyle\iota=\iota_{0}+\dot{\iota}_{0}t,$ (9) $\displaystyle
e=e_{0}+\dot{e}_{0}t,$ (10)
$\displaystyle\gamma=\gamma_{0}+\dot{\gamma}_{0}t.$ (11)

Suitable coordinate choice can eliminate $\gamma_{0}=0$. The investigation in
Willems et al. (2008) neglected parameters $\dot{\iota}_{0}$ and
$\dot{e}_{0}$. The authors in Wolz et al. (2020) only considered the changing
of $f$ but to second order.

On the first glance, our waveform model has nothing to do with the masses of
the two components of the binary. That is because the information of mass and
spin are coded in post-Keplerian parameters like $\dot{f}_{0}$. And just
because of this fact, our post-Keplerian waveform model is quite generic for
stellar binary sources of space detector.

Besides the above intrinsic parameters, the pattern function $F_{+,\times}$
will introduce three more extrinsic parameters Cutler (1998)

$\displaystyle F^{+}(\alpha,\delta,\psi)$
$\displaystyle\equiv\frac{1}{2}(1+\sin^{2}\delta)\cos 2\alpha\cos 2\psi$
$\displaystyle\,\,\,\,-\sin\delta\sin 2\alpha\sin 2\psi,$ (12) $\displaystyle
F^{\times}(\alpha,\delta,\psi)$
$\displaystyle\equiv\frac{1}{2}(1+\sin^{2}\delta)\cos 2\alpha\sin 2\psi$
$\displaystyle\,\,\,\,+\sin\delta\sin 2\alpha\cos 2\psi.$ (13)

The above pattern functions depends on the source right ascension $\alpha$ and
declination $\delta$, and the wave polarization angle $\psi$.

Including both the intrinsic and extrinsic parameters we have 12 parameters
$(A,f_{0},\dot{f}_{0},\iota_{0},\dot{\iota}_{0},e_{0},\dot{e}_{0},\dot{\gamma}_{0},\phi_{0},\alpha,\delta,\psi)$
in all. Source localization can be analyzed as Kocsis et al. (2007); Baibhav
et al. (2020); Wang et al. (2020); Ruan et al. (2021); Hu et al. (2021); Zhang
et al. (2021). Since we do not care about the source localization in the
current work and the source location can not be determined by a single
detector, we instead fix $\alpha=\psi=0$ and $\delta=\pi/2$ in the current
work. So we need only consider 9 parameters
$(A,f_{0},\dot{f}_{0},\iota_{0},\dot{\iota}_{0},e_{0},\dot{e}_{0},\dot{\gamma}_{0},\phi_{0})$
and the pattern functions become

$\displaystyle F^{+}=1,F^{\times}=0.$ (14)

Eqs. (1)-(11) correspond to the post-Keplerian waveform model. The
approximated pattern function shown above is only for simplified discussion
involved in the next sections. Such simplification does not affect the general
conclusion. And the post-Keplerian waveform model can be straight forwardly
applied to any realistic situations.

##  III Theoretical predictions on the post-Keplerian parameters

Assuming the masses of the two components of the binary are $M_{1,2}$, the
chirp mass, symmetric mass ratio and total mass are given by

$\displaystyle\mathcal{M}=\eta^{3/5}M^{2/5},$ (15)
$\displaystyle\eta=\frac{M_{1}M_{2}}{M},$ (16) $\displaystyle M=M_{1}+M_{2}.$
(17)

In order to make the mass transfer between the two compact objects stable till
merger, the mass ratio should be limited in $(0,0.24)$ Verbunt and Rappaport
(1988); Han and Webbink (1999); Yu and Jeffery (2010).

###  III.1 Leading order gravitational radiation reaction effect in GR

The general relativity contribution to the post-Keplerian parameters can be
calculated through post-Newtonian approximation as Seto (2001); Willems et al.
(2007); Maggiore (2008); Yunes et al. (2009); Tanay et al. (2016); Boetzel et
al. (2017)

$\displaystyle
A=\frac{(2\pi\mathcal{M}f_{0})^{2/3}}{1-e^{2}_{0}}\frac{\mathcal{M}}{D_{L}},$
(18)
$\displaystyle\dot{f}_{0}=\frac{\pi^{8/3}\mathcal{M}^{5/3}f_{0}^{11/3}}{5(1-e_{0}^{2})^{7/2}}[96+292e_{0}^{2}+37e_{0}^{4}],$
(19) $\displaystyle\dot{\iota}_{0}=0,$ (20)
$\displaystyle\dot{e}_{0}=-\frac{2}{3}\frac{\sigma}{\sigma^{\prime}}\frac{\dot{f}_{0}}{f_{0}},$
(21)
$\displaystyle\sigma=\frac{e_{0}^{12/19}}{1-e_{0}^{2}}\left[1+\frac{121}{304}e_{0}^{2}\right]^{870/2299},$
(22) $\displaystyle\sigma^{\prime}=\frac{1}{2^{1181/2299}19^{870/2299}}\times$
$\displaystyle\frac{96+292e_{0}^{2}+37e_{0}^{4}}{e_{0}^{7/19}(1-e_{0}^{2})^{2}(304+121e_{0}^{2})^{1429/2299}},$
(23)

where $D_{L}$ is the luminosity distance between the source and the detector.
$f_{0}$, $\iota_{0}$, $e_{0}$ and $\dot{\gamma}_{0}$ correspond to the orbital
status of the binary. Approximately Seto (2001)

$\displaystyle f_{0}$
$\displaystyle\approx\frac{1}{2\pi}\sqrt{\frac{M}{a^{3}}},$ (24)
$\displaystyle\dot{\gamma}_{0}$ $\displaystyle=6\pi
f_{0}\frac{M}{a(1-e_{0}^{2})}$ (25) $\displaystyle\approx
24\pi^{3}\frac{a^{2}f_{0}^{3}}{1-e_{0}^{2}}$ (26) $\displaystyle\approx
3(2\pi)^{7/3}\frac{M^{2/3}f_{0}^{5/3}}{1-e_{0}^{2}}$ (27)

where $a$ is the separation of the two components of the binary, approximately
the semi-major axis of the binary orbit. In principle $\iota_{0}\in(0,\pi)$,
$e_{0}\in(0,1)$. Since $\dot{\iota}_{0}=0$ and
$(\dot{e}_{0},\dot{\gamma}_{0})$ can be determined by other parameters, there
are only six independent parameters involved for pure GR situation.

From the above relations we can find that for a binary with given components
mass and source localization

$\displaystyle A$ $\displaystyle\propto f_{0}^{2/3},$ (28)
$\displaystyle\dot{f}_{0}$ $\displaystyle\propto f_{0}^{11/3},$ (29)
$\displaystyle\dot{e}_{0}$ $\displaystyle\propto e_{0}f_{0}^{8/3},$ (30)
$\displaystyle\dot{\gamma}_{0}$ $\displaystyle\propto f_{0}^{5/3}.$ (31)

This is to say when the system frequency $f_{0}$ increases, all the above
parameters increase correspondingly. If just the eccentricity increases, only
$\dot{e}_{0}$ is affected. But if we only consider $e_{0}$ takes order $0.1$
which means $e_{0}\in(0.1,1)$, the difference of $\dot{e}_{0}$ due to $e_{0}$
is small compared to the difference due to $f_{0}$.

If we use the above gravitational radiation reaction to estimate the post-
Keplerian parameters, we have typically Stroeer et al. (2005); Stroeer and
Vecchio (2006); Luo et al. (2016)

$\displaystyle M\approx 1M_{\odot}\approx 10^{-6}{\rm s},$ (32) $\displaystyle
A\approx 10^{-20},$ (33) $\displaystyle f_{0}\approx 10^{-3}{\rm Hz},$ (34)
$\displaystyle a\approx 1{\rm s},$ (35) $\displaystyle\dot{f}_{0}\approx
10^{-18}{\rm Hz}^{2},$ (36) $\displaystyle\dot{e}_{0}\approx 10^{-16}{\rm
Hz},$ (37) $\displaystyle\dot{\gamma}_{0}\approx 10^{-6}{\rm Hz}.$ (38)

Here Eq. (32) is because we care about compact stars including neutron star
and white dwarf. Eq. (33) corresponds to the verification binaries for space-
based detectors. Eq. (34) is determined by the sensitive frequency band of
space-based detectors. Eqs. (35)-(38) are determined by the relations shown in
(18)-(27). The variation time scale is much smaller than the orbit period
indicate. This fact means that our post-Keplerian waveform model introduced in
the above section is valid quite well for gravitational radiation reaction.

We have two clues to show whether the detected binary is dominated by GR
effect or not. The first one is the detected $\dot{\iota}_{0}$ strongly
deviates from 0, and/or $\dot{f}_{0}$, $\dot{e}_{0}$ and $\dot{\gamma}_{0}$
strongly deviate from the predicted values (36)-(38). The GR effect on the
orbit plan precession comes from the compact object spin precession
Apostolatos et al. (1994)

$\displaystyle\dot{\vec{L}}=$
$\displaystyle\frac{1}{a^{3}}\left[\frac{4M^{2}_{1}+3M_{1}M_{2}}{2}\vec{\chi}_{1}+\frac{4M^{2}_{2}+3M_{1}M_{2}}{2}\vec{\chi}_{2}\right]\times\vec{L}$
$\displaystyle-\frac{3}{2}\frac{M_{1}^{2}M_{2}^{2}}{a^{3}}\left[(\vec{\chi}_{2}\cdot\hat{L})\vec{\chi}_{1}+(\vec{\chi}_{1}\cdot\hat{L})\vec{\chi}_{2}\right]\times\hat{L}$
$\displaystyle-\frac{32}{5}\frac{\mu^{2}}{a}\left(\frac{M}{a}\right)^{5/2}\hat{L},$
(39)

where $\vec{L}$ is the total angular momentum of the binary system,
$\vec{\chi}_{1,2}$ are the dimensionless spin vectors of two individual
components. For black holes, the dimensionless spin parameter $\chi$ is less
than 1 due to the cosmic censorship. For other objects like neutron star and
white dwarf, it will be even much smaller, typically $\chi<0.1$. So we can
estimate the GR induced

$\displaystyle\dot{\iota}_{\rm
GR0}<\chi\left[2\frac{M^{2}}{a^{3}}+\frac{3}{2}\frac{M_{1}^{2}M_{2}^{2}}{a^{3}M\sqrt{aM}}\right]\sim\chi\frac{M^{2}}{a^{3}}\sim
10^{-13}$ (40)

which can be safely neglected compared to the measurement accuracy (check
Table. 1).

The second clue is the relation between $\dot{f}_{0}$ and $\dot{\gamma}_{0}$.
Using the analysis technique similar to the one in pulsar timing Manchester
(2015), we can draw lines relating $M_{1}$ and $M_{2}$ based on the
measurement of post-Keplerian parameters $\dot{f}_{0}$ and $\dot{\gamma}_{0}$.
As an example, we plot such a diagram in Fig. 1 which is similar to Fig. 6 of
Manchester (2015) in pulsar timing.

---
Figure 1: Plot of companion mass ($M_{2}$) versus compact object mass
($M_{1}$) for an example case $M_{1}=M_{2}=1M_{\odot}$. Constraints derived
from the measured post-Keplerian parameters assuming GR are shown pairs of
lines with separation indicating the uncertainty range. The measurement
accuracy is taken from Table. 1. The red lines correspond to $\dot{f}_{0}$ and
the green lines correspond to $\dot{\gamma}_{0}$. The blue point corresponds
to the expected value from general relativity.

###  III.2 Gravitational effect beyond GR

Using the Brans-Dicke theory as an example Ma and Yunes (2019); Barausse et
al. (2013), the gravity theory beyond GR will introduce extra driving terms
beside the ones predicted by GR in (19) and (21)

$\displaystyle\dot{f}_{\rm
0BD}=\frac{96\pi^{2}}{5}b\mathcal{M}\eta^{2/5}f_{0}^{3}\frac{1+\frac{1}{2}e_{0}^{2}}{(1-e_{0}^{2})^{5/2}},$
(41) $\displaystyle\dot{e}_{\rm
0BD}=-\frac{48\pi^{2}}{5}b\mathcal{M}\eta^{2/5}f_{0}^{2}\frac{e_{0}}{(1-e_{0}^{2})^{3/2}}.$
(42)

Here $b$ is a parameter determined by the gravity theory Will and Zaglauer
(1989). When $b=0$ GR is recovered. Periastron precession becomes

$\displaystyle\dot{\gamma}_{0}$ $\displaystyle=6\pi
f_{0}\frac{M}{a(1-e_{0}^{2})}\frac{\mathcal{P}}{\mathcal{G}}$ (43)

where parameters $\mathcal{P}$ and $\mathcal{G}$ are determined by the gravity
theory. For black holes we have

$\displaystyle\mathcal{G}=1-\frac{\xi}{2},$ (44)
$\displaystyle\mathcal{P}=\mathcal{G}^{2},$ (45)
$\displaystyle\xi=\frac{1}{2+\omega_{\rm BD}}.$ (46)

For neutron star and white dwarf, $b$, $\mathcal{P}$ and $\mathcal{G}$ also
depend on the sensitivity of the stars.

If we use parametrized post-newtonian (PPN) formalism to describe the
alternative gravity theory other than GR, the factor
$\frac{\mathcal{P}}{\mathcal{G}}$ becomes Soffel et al. (1987); Zhao and Xie
(2013)

$\displaystyle\frac{\mathcal{P}}{\mathcal{G}}=\frac{2\gamma+2-\beta}{3}.$ (47)

When the PPN parameters $\gamma=\beta=1$, GR is recovered. In addition, PPN
will correct the relation (24) to

$\displaystyle f_{0}\approx\frac{1}{2\pi}\sqrt{\frac{M}{d^{3}}}\times$
$\displaystyle\left[1-(3\beta+6\gamma-9\eta+6)\frac{M}{d\sqrt{1-e_{0}^{2}}}+(2\gamma-7\eta+4)\frac{M}{d}\right].$
(48)

If the gravitational constant $G$ is varying, additional driving term beside
the one predicted by GR in (19) comes in Damour et al. (1988)

$\displaystyle\dot{f}_{\rm 0VG}=-2\frac{\dot{G}_{0}}{G_{0}}f_{0},$ (49)

where $G_{0}$ is the current value of the gravitational constant.

If dipole radiation exists which results from the equivalence principle
violation, the post-Keplerian parameter $\dot{f}_{0}$ will change to Barausse
et al. (2016); Zhao et al. (2021)

$\displaystyle\dot{f}_{0}=\left[1+B\frac{d}{M}\right]\dot{f}_{\rm 0GR},$ (50)
$\displaystyle B=\frac{5}{96}|\Delta\alpha|^{2},$ (51)

where $\Delta\alpha$ describes the difference between the effective scalar
couplings of two objects in the binary. Here we have used $\dot{f}_{\rm 0GR}$
to denote the expression (19).

If monopole radiation exists, the post-Keplerian parameter $\dot{f}_{0}$ will
change to Damour and Esposito-Farese (1992); Kramer et al. (2021); Schön and
Doneva (2022)

$\displaystyle\dot{f}_{0}=\left[1+Ce_{0}^{2}(1+\frac{e_{0}^{2}}{4})\right]\dot{f}_{\rm
0GR},$ (52)

where $C$ depends on specific gravity theory.

Here we just calculated typical gravity theories beyond GR. General gravity
theory can be similar related to the post-Keplerian parameters. If only we
find a GR effect dominated binary with space-based gravitational wave
detector, such binary will be a very good laboratory for gravity theory study.

###  III.3 Tidal interactions

Tidal interactions may affect the properties of the binary compact objects and
their evolutions Shakura (1985); Yu and Jeffery (2010); Kremer et al. (2015,
2017). Since the gravitational wave amplitude just depends on the collective
motion of the two components, Eq. (18) still holds even tidal interaction
appears. But the dynamical variables $\dot{f}_{0}$, $\dot{\iota}_{0}$,
$\dot{e}_{0}$ and $\dot{\gamma}_{0}$ will evolve following different rule.
Theoretical models of tidal interaction include equilibrium type Iben et al.
(1998); Willems et al. (2010); Piro (2011) and dynamical type Fuller and Lai
(2012, 2013, 2014). All of these models predict

$\displaystyle\dot{f}_{0}=C_{1}(f_{1}-f_{0}),$ (53)
$\displaystyle\dot{\iota}_{0}=-\frac{2\pi f_{0}}{(1-e_{0}^{2})^{2}}\times$
$\displaystyle\sum_{i=1}^{2}k_{i}\left(\frac{R_{i}}{a}\right)^{5}\left(\frac{f_{i}}{f_{0}}\right)^{2}\left(1+\frac{M_{3-i}}{M_{i}}\right)\cos\alpha_{i}\cos\gamma_{i},$
(54) $\displaystyle\dot{e}_{0}=-C_{2}e_{0},$ (55)
$\displaystyle\dot{\gamma}_{0}=\frac{2\pi f_{0}}{(1-e_{0}^{2})^{2}}\times$
$\displaystyle\sum_{i=1}^{2}\frac{k_{i}}{M_{i}}\left(\frac{R_{i}}{a}\right)^{5}\left[15M_{3-i}\frac{1+\frac{3}{2}e_{0}^{2}+\frac{1}{8}e_{0}^{4}}{(1-e_{0}^{2})^{3}}+M\left(\frac{f_{i}}{f_{0}}\right)^{2}\right]$
(56)

where $R_{i}$ is the radius of the $i$-th component star, $f_{i}$ is its self-
rotation frequency, and $k_{i}\in(0,0.75)$ is its quadrupolar apsidal-motion
constant. $\alpha_{i}$ denotes the angle between the $i$-th component’s
rotation axis and the normal to the orbit plane. $\gamma_{i}$ denote the angle
between the rotation axis and the binary’s line of ascending node. The factor
$C_{1}$ in (53) depends on the orbital separation, the internal structure of
the two components. The factor $C_{2}$ in (55) depends on the orbital
separation, the internal structure of the two components and their rotation.

When $f_{0}$ is relatively small the contribution of tidal interaction to
$\dot{\gamma}_{0}$ will be larger than the contribution of GR. So if the
detected $\dot{\gamma}_{0}$ is explicitly bigger than the prediction of GR, we
can use relation (56) to analyze the matter property of the binary Willems et
al. (2008). In contrast, if the detected $\dot{\gamma}_{0}$ is consistent to
the one predicted by GR we are sure the binary is dominated by GR effect.

###  III.4 Mass transfer effect

When the two compact objects approach to each other, the material of one
component may be accreted by another one. And the mass transfer happens
consequently Thomas (1977); Wong and Bildsten (2021). The mass transfer effect
will affect the post-Keplerian parameters strongly Marsh et al. (2004);
Gokhale et al. (2007).

The mass transfer between a close binary can be dived into three types Thomas
(1977). Case A: If the orbital separation is small enough (less than a few
days), the star can fill its Roche lobe during its slow expansion through the
main-sequence phase while still burning hydrogen in its core. Case B: If the
orbital period is less than about 100 days, but longer than a few days, the
star will fill its Roche lobe during the rapid expansion to a red giant with a
helium core. If the helium core ignites during this phase and the transfer is
interrupted, the mass transfer is case B. Case C: If the orbital period is
above 100 days, the star can evolve to the red supergiant phase before it
fills its Roche lobe. In this case, the star may have a CO or ONe core. Since
we care about the related gravitational wave in mini-Hz band, we only concern
Case A mass transfer in the current work.

When no ejected matter leaves a binary system, the mass transfer is said to be
conservative. Consequently we have $\dot{J}=\dot{M}=0$ where $J$ is the total
orbital angular momentum

$\displaystyle J=M_{1}M_{2}\sqrt{\frac{a(1-e^{2})}{M}}.$ (57)

Differentiating the above equation we get

$\displaystyle\frac{\dot{a}}{a}$
$\displaystyle=2\frac{\dot{J}}{J}-2\frac{\dot{M}_{1}}{M_{1}}-2\frac{\dot{M}_{2}}{M_{2}}+\frac{\dot{M}}{M},$
(58) $\displaystyle=2\dot{M}_{1}\frac{M_{2}-M_{1}}{M_{1}M_{2}}.$ (59)

Together with the Kepler’s third law
$\frac{1}{a^{3}}=\frac{4\pi^{2}f_{0}^{2}}{M}$, we get the additional
contribution of post-Keplerian parameters due to mass transfer

$\displaystyle\dot{f}_{\rm
0MTC}=3\frac{\dot{M}_{1}(M_{1}-M_{2})}{M_{1}M_{2}}f_{0}.$ (60)

In case of non-conservative mass transfer both mass and angular momentum can
be removed from the system. We can use the moment of inertia of the binary

$\displaystyle I=\frac{M_{1}M_{2}}{M}a^{2}$ (61)

to express the orbital angular momentum

$\displaystyle J=2\pi If_{0}=2\pi\frac{q}{(1+q)^{2}}Ma^{2}f_{0}.$ (62)

For the non-conservative mass transfer part we can approximately assume the
mass ratio $q\equiv\frac{M_{1}}{M_{2}}$ is a constant. Consequently we have

$\displaystyle\frac{\dot{J}}{J}=\frac{\dot{M}}{M}+2\frac{\dot{a}}{a}+\frac{\dot{f}_{0}}{f_{0}}.$
(63)

Based on the Kepler’s third law we have additional relation

$\displaystyle\frac{\dot{M}}{M}=3\frac{\dot{a}}{a}+2\frac{\dot{f}_{0}}{f_{0}}.$
(64)

Combine the above two equations we get

$\displaystyle\frac{\dot{J}}{J}=\frac{5}{3}\frac{\dot{M}}{M}-\frac{1}{3}\frac{\dot{f}_{0}}{f_{0}}.$
(65)

If we additionally assume isotropic mass loss from the surface of the
components we have

$\displaystyle\dot{J}=2\pi\frac{q}{(1+q)^{2}}\dot{M}a^{2}f_{0}.$ (66)

Combine Eqs. (62)-(66) we have additional contribution of post-Keplerian
parameters due to mass transfer

$\displaystyle\dot{f}_{\rm 0MTN}=-2\frac{\dot{M}}{M}f_{0}.$ (67)

Put the conservative and non-conservative mass transfer together we have

$\displaystyle\dot{f}_{\rm
0MT}=\left[3\frac{(\dot{M}_{1}-\dot{M})(M_{1}-M_{2})}{M_{1}M_{2}}-2\frac{\dot{M}}{M}\right]f_{0}.$
(68)

Mass transfer and tidal interaction also affect the stellar oscillations Guo
(2021). If both gravitational wave and stellar oscillations are observed, the
combined information can be used to infer the property of the binary.

Typically $\dot{f}_{\rm 0MT}$ is comparable to $\dot{f}_{\rm 0GR}$. So if the
mass transfer does happen we expect the detected $\dot{f}_{\rm 0}$ is
different to the one predicted by GR. For such case we can subtract
$\dot{f}_{\rm 0GR}$ from the detected $\dot{f}_{\rm 0}$ to get the effects
coming from mass transfer and tidal interaction. Following that the
interesting analysis related to matter property can be done then.

##  IV Measurement accuracy estimation of the post-Keplerian parameters

In this section we will use Fisher information matrix to estimate the
measurement accuracy of the post-Keplerian parameters

$\displaystyle\Gamma_{ij}=\left\langle\frac{\partial h}{\partial
p_{i}}|\frac{\partial h}{\partial p_{j}}\right\rangle,$ (69)

where $p_{i}$ represents the $i$-th parameter among the 9 parameters
$(A,f_{0},\dot{f}_{0},\iota_{0},\dot{\iota}_{0},e_{0},\dot{e}_{0},\dot{\gamma}_{0},\phi_{0})$.
Inversing the Fisher matrix we can estimate the measurement error as

$\displaystyle\Delta p_{i}=\sqrt{\Gamma^{-1}_{ii}}.$ (70)

The inner product in the above equation is defined as

$\displaystyle\left\langle
g|h\right\rangle=\frac{2}{S_{n}(f_{0})}\int_{0}^{T}g(t)h(t)dt,$ (71)

where $f_{0}$ corresponds to the post-Keplerian parameter and the adiabatic
approximation has been used Wolz et al. (2020).

Regarding to the sensitivity of a space-based detector, we use the following
approximation (Eq. (13) of Robson et al. (2019))

$\displaystyle S_{n}(f)$ $\displaystyle=\frac{10}{3L^{2}}\left(P_{\rm
OMS}+2(1+\cos^{2}(f/f_{*}))\frac{P_{\rm acc}}{(2\pi f)^{4}}\right)\times$
$\displaystyle\left(1+\frac{6}{10}\left(\frac{f}{f_{*}}\right)^{2}\right),$
(72) $\displaystyle f_{*}$ $\displaystyle=c/(2\pi L).$ (73)

For LISA Robson et al. (2019) we have

$\displaystyle P_{\rm OMS}$ $\displaystyle=(1.5\times 10^{-11}{\rm m})^{2}{\rm
Hz}^{-1},$ (74) $\displaystyle P_{\rm acc}$ $\displaystyle=(3\times
10^{-15}{\rm ms}^{-2})^{2}\left(1+\left(\frac{4\times 10^{-4}{\rm
Hz}}{f}\right)^{2}\right){\rm Hz}^{-1},$ (75) $\displaystyle L$
$\displaystyle=2.5\times 10^{9}{\rm m}.$ (76)

For Taiji Wang et al. (2020) we have

$\displaystyle P_{\rm OMS}$ $\displaystyle=(8\times 10^{-12}{\rm m})^{2}{\rm
Hz}^{-1},$ (77) $\displaystyle P_{\rm acc}$ $\displaystyle=(3\times
10^{-15}{\rm ms}^{-2})^{2}\left(1+\left(\frac{4\times 10^{-4}{\rm
Hz}}{f}\right)^{2}\right){\rm Hz}^{-1},$ (78) $\displaystyle L$
$\displaystyle=3\times 10^{9}{\rm m}.$ (79)

For Tianqin we have Luo et al. (2016)

$\displaystyle P_{\rm OMS}$ $\displaystyle=(1\times 10^{-12}{\rm m})^{2}{\rm
Hz}^{-1},$ (80) $\displaystyle P_{\rm acc}$ $\displaystyle=(1\times
10^{-15}{\rm ms}^{-2})^{2}\left(1+\left(\frac{1\times 10^{-4}{\rm
Hz}}{f}\right)^{2}\right){\rm Hz}^{-1},$ (81) $\displaystyle L$
$\displaystyle=\sqrt{3}\times 10^{8}{\rm m}.$ (82)

---
Figure 2: Approximated sensitivity curves respectively for LISA Robson et al.
(2019), Taiji Wang et al. (2020) and Tianqin Luo et al. (2016) used in the
current work. Note that the designed sensitivity of these three detectors is
still changing because the final decision has not been reached yet. But such
detail does not affect our analysis done in the current paper.

Due to the relation (71), the measurement accuracy for different detector is
similar only upto the factor $1/S_{n}(f_{0})$. Since this fact we calculate
the measurement accuracy for LISA first and deduce the accuracy for other
detectors accordingly in the following. As Huang et al. (2020), we take the
observation time as $T=5$ years.

Table 1:  Typical measurement accuracy of post-Keplerian parameters. The second column represents the assumed values taken by the parameters themselves. The rest three columns correspond to the measurement accuracy detected by LISA, Taiji and Tianqin respectively. Here four sets of parameters are investigated. Each set corresponds to one group listed below. | value | LISA | Taiji | Tianqin
---|---|---|---|---
$A$ | $10^{-20}$ | $9.5\times 10^{-23}$ | $7.9\times 10^{-23}$ | $4.3\times 10^{-22}$
$f_{0}$ | $10^{-3}$ | $3.1\times 10^{-12}$ | $2.6\times 10^{-12}$ | $1.4\times 10^{-11}$
$\dot{f}_{0}$ | $10^{-21}$ | $1.3\times 10^{-20}$ | $1.1\times 10^{-20}$ | $5.9\times 10^{-20}$
$\iota_{0}$ | $\frac{\pi}{4}$ | $1.4\times 10^{-2}$ | $1.2\times 10^{-2}$ | $6.4\times 10^{-2}$
$\dot{\iota}_{0}$ | $0$ | $1.0\times 10^{-11}$ | $8.7\times 10^{-12}$ | $4.7\times 10^{-11}$
$e_{0}$ | $0.1$ | $2.6\times 10^{-4}$ | $2.2\times 10^{-4}$ | $1.2\times 10^{-3}$
$\dot{e}_{0}$ | $10^{-18}$ | $2.9\times 10^{-12}$ | $2.4\times 10^{-12}$ | $1.3\times 10^{-11}$
$\dot{\gamma}_{0}$ | $10^{-7}$ | $1.5\times 10^{-11}$ | $1.2\times 10^{-11}$ | $6.7\times 10^{-11}$
$\phi_{0}$ | $0$ | $4.6\times 10^{-4}$ | $3.8\times 10^{-4}$ | $2.1\times 10^{-3}$
$A$ | $10^{-20}$ | $9.4\times 10^{-24}$ | $4.4\times 10^{-24}$ | $9.4\times 10^{-24}$
$f_{0}$ | $10^{-2}$ | $3.1\times 10^{-13}$ | $1.4\times 10^{-13}$ | $3.1\times 10^{-13}$
$\dot{f}_{0}$ | $10^{-18}$ | $1.3\times 10^{-21}$ | $6.1\times 10^{-22}$ | $1.3\times 10^{-21}$
$\iota_{0}$ | $\frac{\pi}{4}$ | $1.4\times 10^{-3}$ | $6.6\times 10^{-4}$ | $1.4\times 10^{-3}$
$\dot{\iota}_{0}$ | $0$ | $1.0\times 10^{-12}$ | $4.8\times 10^{-13}$ | $1.0\times 10^{-12}$
$e_{0}$ | $0.1$ | $2.6\times 10^{-4}$ | $1.2\times 10^{-5}$ | $2.6\times 10^{-4}$
$\dot{e}_{0}$ | $10^{-16}$ | $2.9\times 10^{-13}$ | $1.3\times 10^{-13}$ | $2.9\times 10^{-13}$
$\dot{\gamma}_{0}$ | $10^{-6}$ | $1.5\times 10^{-12}$ | $6.8\times 10^{-13}$ | $1.5\times 10^{-12}$
$\phi_{0}$ | $0$ | $4.6\times 10^{-5}$ | $2.1\times 10^{-5}$ | $4.6\times 10^{-5}$
$A$ | $10^{-20}$ | $1.4\times 10^{-23}$ | $1.1\times 10^{-23}$ | $6.1\times 10^{-23}$
$f_{0}$ | $10^{-3}$ | $8.0\times 10^{-13}$ | $6.6\times 10^{-13}$ | $3.6\times 10^{-12}$
$\dot{f}_{0}$ | $10^{-21}$ | $4.7\times 10^{-21}$ | $3.9\times 10^{-21}$ | $2.1\times 10^{-20}$
$\iota_{0}$ | $\frac{\pi}{4}$ | $2.2\times 10^{-3}$ | $1.8\times 10^{-3}$ | $9.7\times 10^{-3}$
$\dot{\iota}_{0}$ | $0$ | $9.2\times 10^{-12}$ | $7.7\times 10^{-12}$ | $4.1\times 10^{-11}$
$e_{0}$ | $0.6$ | $1.8\times 10^{-4}$ | $1.5\times 10^{-4}$ | $8.1\times 10^{-4}$
$\dot{e}_{0}$ | $10^{-18}$ | $2.0\times 10^{-12}$ | $1.7\times 10^{-12}$ | $8.9\times 10^{-12}$
$\dot{\gamma}_{0}$ | $10^{-7}$ | $3.5\times 10^{-12}$ | $2.9\times 10^{-12}$ | $1.6\times 10^{-11}$
$\phi_{0}$ | $0$ | $1.7\times 10^{-4}$ | $1.4\times 10^{-4}$ | $7.4\times 10^{-4}$
$A$ | $10^{-20}$ | $9.5\times 10^{-23}$ | $7.9\times 10^{-23}$ | $4.3\times 10^{-22}$
$f_{0}$ | $10^{-3}$ | $3.1\times 10^{-12}$ | $2.6\times 10^{-12}$ | $1.4\times 10^{-11}$
$\dot{f}_{0}$ | $10^{-18}$ | $1.3\times 10^{-20}$ | $1.1\times 10^{-20}$ | $5.9\times 10^{-20}$
$\iota_{0}$ | $\frac{\pi}{4}$ | $1.4\times 10^{-2}$ | $1.2\times 10^{-2}$ | $6.4\times 10^{-2}$
$\dot{\iota}_{0}$ | $0$ | $1.0\times 10^{-11}$ | $8.7\times 10^{-12}$ | $4.7\times 10^{-11}$
$e_{0}$ | $0.1$ | $2.7\times 10^{-4}$ | $2.2\times 10^{-4}$ | $1.2\times 10^{-3}$
$\dot{e}_{0}$ | $10^{-16}$ | $2.9\times 10^{-12}$ | $2.4\times 10^{-12}$ | $1.3\times 10^{-11}$
$\dot{\gamma}_{0}$ | $10^{-6}$ | $1.5\times 10^{-11}$ | $1.2\times 10^{-11}$ | $6.7\times 10^{-11}$
$\phi_{0}$ | $0$ | $4.6\times 10^{-4}$ | $3.8\times 10^{-4}$ | $2.1\times 10^{-3}$

We list the typical measurement accuracies of post-Keplerian parameters in
Table. 1. Four typical post-Keplerian parameters are investigated. Each set
corresponds to a typical binary system. The corresponding waveforms of these
four groups parameters are shown in Fig. 3.

---
Figure 3: Typical post-Keplerian waveforms correspond to the parameters shown
in Table. 1. Group 2 admits higher frequency than others. Group 3 admits
higher eccentricity than others. Consequently the waveform of group 3 shows
clear higher tone behavior. Compared to group 1, group 4 admits higher
$\dot{e}_{0}$ and $\dot{\gamma}_{0}$. But these parameters difference only
affect long term behavior. The short term waveform shown here can not indicate
such difference.

The first group corresponds to a typical binary white dwarf with pure GR
effect. We can see all three detectors including LISA, Taiji and Tianqin can
accurately measure almost parameters except $\dot{f}_{0}$ and $\dot{e}_{0}$.
Here when we say ‘accurately’ we mean the relative error is less than one
percent. Regarding to $\phi_{0}$, it’s value takes order 0.1 as the angle
$\iota_{0}$ although the assumed value is 0 here. Regarding to
$\dot{\iota}_{0}$, if the effect like (54) comes in, its value will be order
$10^{-8}$. Relative to these values, the measurement error bar is small.

The well estimated $\dot{\gamma}_{0}$ can be used to judge whether the matter
effect is strong or not based on (56). If the matter effect is ignorable, the
well estimated $\dot{\gamma}_{0}$ can be used to constrain gravity theory
beyond GR based on (43). Note that
$\frac{\Delta\dot{\gamma}_{0}}{\dot{\gamma}_{0}}\lesssim 10^{-4}$, we can
constrain

$\displaystyle\left|\frac{\mathcal{P}}{\mathcal{G}}-1\right|\lesssim 10^{-4},$
(83) $\displaystyle\left|2\gamma-\beta-1\right|\lesssim 10^{-4}.$ (84)

If we can further make sure the source is two black holes, we can get

$\displaystyle\left|\mathcal{G}-1\right|\lesssim 10^{-4},$ (85)
$\displaystyle\left|\xi\right|\lesssim 10^{-4},$ (86)
$\displaystyle\left|\omega_{\rm BD}\right|\gtrsim 10^{4}.$ (87)

If everything indicates that only GR dominates, we can plug $\mathcal{M}$ and
$D_{L}$ into the the Eqs. (18)-(27) and do the parameters estimation again.
Then the chirp mass $\mathcal{M}$ can be well estimated Willems et al. (2008);
Wolz et al. (2020). With the information of $\dot{\gamma}_{0}$, $e_{0}$ and
$f_{0}$, the Eq. (27) gives us the information of total mass $M$. Combining
$M$ and $\mathcal{M}$ we can deduce mass ratio. Consequently the individual
mass can be well estimated. And more we can use detail waveform model
according to specific gravity theory and plug in component masses at this
stage. Then the involved parameters may be constrained more strictly.

If the matter of the compact stars is very stiff, the two components may be
very close. Possibly the frequency $f_{0}$ may be as high as $10^{-2}$Hz. We
investigate such case in the second group of Table. 1. This set of parameters
also corresponds to stellar origin binary black hole or binary neutron star.
If the mass transfer and the tidal interaction between two white dwarfs are
very strong, the binary’s frequency may also fall in this group. Firstly we
can see LISA and Tianqin get roughly the same measurement accuracy. This is
because the sensitivity for LISA and Tianqin roughly equals at $10^{-2}$Hz.
Compared to the first group, the measurement accuracy improves roughly one
order. This is due to that the detector sensitivity at $10^{-2}$Hz is about 10
times better than that at $10^{-3}$Hz. Consequently the afore mentioned
parameter constrain and gravity theory constrain may be improved one order
based on this kind of sources.

Compared to the first group, now we consider more eccentric binary with
$e_{0}=0.6$ in the third group. Along with the increasing of the eccentricity,
the waveform becomes more complicated. As shown in Fig. 3, higher tones appear
clearly Han et al. (2017); Liu et al. (2022). As expected, higher eccentricity
improves much the measurement accuracy Sun et al. (2015); Ma et al. (2017);
Pan et al. (2019); Ma and Yunes (2019); Yang et al. (2022, 2023). Compared to
the first group, the measurement accuracy improves one order. This means more
eccentric binary even facilitate more the study of the gravity theory and the
nuclear matter property.

In the fourth group we consider higher $\dot{f}_{0}$, $\dot{e}_{0}$ and
$\dot{\gamma}_{0}$ compared to the first group. This may happen if the tidal
interaction and the mass transfer between the two binary components take
effect. We find that the measurement accuracy is roughly the same to the ones
got in the first group. This result indicates only $A$, $f_{0}$ and $e_{0}$
affect the measurement accuracy strongly. At the same time we can conclude
that the current planning space-based gravitational wave detectors can
determine the post-Keplerian parameters as well as that shown in the first
group of Table. 1. If the signal is stronger (larger $A$), and/or the
frequency is higher (larger $f_{0}$), and/or the orbit is more eccentric
(larger $e_{0}$), we can get even better parameter estimation. Consequently we
can constrain gravity theory tight and give deep insight to the nuclear
matter.

##  V Conclusion and discussion

Compact object binary provides ones a valuable laboratory for fundamental
physics and astronomy. Pulsar timing detection has made a good use of binary
pulsars, Existing gravitational wave detection has made a good use of
coalescence of binary black hole, binary neutron star and neutron star-black
hole binary. In the near future, the space-based gravitational wave detectors
will facilitate us to make early inspiraling compact object binaries also
become good laboratories. To do so, feasible waveform model is important.

In the current work, we propose a post-Keplerian waveform model for early
inspiraling compact object binaries. Such a waveform model takes all kinds of
gravity theories and all kinds of matter dynamics involved in the binary into
consideration. This post-Keplerian waveform model can be looked as a
unification of all previous existing waveform models for early inspiraling
compact object binaries.

The post-Keplerian expansion parameters of the binary orbit are the waveform
model parameters instead of binary component mass and other matter parameters.
Based on this waveform model we can constrain gravity theory without the
assumption of specific gravity theory. This trick is quite similar to that
used in pulsar timing experiments. Based on the post-Keplerian expansion, we
have constructed a ready-to-use unified waveform model of binary compact
object for space-based detectors.

We have also checked the parameter estimation accuracy for the post-Keplerian
parameters. Based on the current planning space-based gravitational wave
detectors including LISA, Taiji and Tianqin, we find that the post-Keplerian
parameters can be determined accurately. Consequently gravitational
interaction dominating or matter dynamics dominating can be well
distinguished. Following that the model independent constrain on gravity
theory can be well done for gravitational dominated binary. And the nuclear
matter properties can be well studied for matter dynamics dominated binary.
Detail parameter estimation accuracy has been shown in Table. 1. Using our
waveform model, ones can not only catch the GW signal much easier but also
distinguish different physical situations and even constraint different
gravity theories based on LISA, Taiji or Tianqin detectors. When specific
gravity theory or matter dynamics is determined, more specific waveform model
can be used to fine tune the involved parameters.

Phenomenologically the effects of spin precession and orbital eccentricity
will result in similar waveforms. Like event GW190521, it is hard to
distinguish spin precession and orbital eccentricity for ground-based
detectors Romero-Shaw et al. (2023). In the GW190521 case, we are limited by
the small number of observed GW cycles. For the space-based detectors we
considered in the current work, this will not be an issue. This is because the
observation time is 5 years and many wave cycles will be accumulated.

In conclusion, our post-Keplerian waveform model provides a good toolkit for
the future space-based gravitational wave detectors to study early inspiraling
compact object binaries.

## Acknowledgments

This work was supported in part by the National Key Research and Development
Program of China Grant No. 2021YFC2203001 and the CAS Project for Young
Scientists in Basic Research YSBR-006.

##  Appendix A Waveform derivative respect to post-Keplerian parameters

Based on the setting in the main text, the derivative respect to the $i$-th
parameter can be written as

$\displaystyle\frac{\partial h}{\partial
p_{i}}=\frac{\sqrt{3}}{2}\sum_{n=1}^{\infty}[F_{+}\frac{\partial
h_{n}^{+}}{\partial p_{i}}+F_{\times}\frac{\partial h_{n}^{\times}}{\partial
p_{i}}]$ (88)

Consequently we have

$\displaystyle\int_{0}^{T}\frac{\partial h}{\partial p_{i}}\frac{\partial
h}{\partial p_{j}}dt$ $\displaystyle=$
$\displaystyle\frac{3}{4}\sum_{n,m=0}^{\infty}\int_{0}^{T}[F_{+}\frac{\partial
h_{n}^{+}}{\partial p_{i}}+F_{\times}\frac{\partial h_{n}^{\times}}{\partial
p_{i}}][F_{+}\frac{\partial h_{m}^{+}}{\partial
p_{j}}+F_{\times}\frac{\partial h_{m}^{\times}}{\partial p_{j}}]dt$
$\displaystyle=$
$\displaystyle\frac{3}{4}\sum_{n,m=0}^{\infty}[F^{2}_{+}\int_{0}^{T}\frac{\partial
h_{n}^{+}}{\partial p_{i}}\frac{\partial h_{m}^{+}}{\partial
p_{j}}dt+F^{2}_{\times}\int_{0}^{T}\frac{\partial h_{n}^{\times}}{\partial
p_{i}}\frac{\partial h_{m}^{\times}}{\partial p_{j}}dt$
$\displaystyle+F_{+}F_{\times}\int_{0}^{T}(\frac{\partial h_{n}^{+}}{\partial
p_{i}}\frac{\partial h_{m}^{\times}}{\partial p_{j}}+\frac{\partial
h_{n}^{\times}}{\partial p_{i}}\frac{\partial h_{m}^{+}}{\partial p_{j}})dt]$
(89)

Derivative respect to $A$

$\displaystyle\frac{\partial h_{n}^{+}}{\partial
A}=\\{(1+\cos^{2}\iota)a_{n}(e)\cos[n\phi(t)]\cos(2\gamma)$
$\displaystyle-(1+\cos^{2}\iota)b_{n}(e)\sin[n\phi(t)]\sin(2\gamma)$
$\displaystyle+(\sin^{2}\iota)c_{n}(e)\cos[n\phi(t)]\\},$ (90)
$\displaystyle\frac{\partial h_{n}^{\times}}{\partial
A}=-2\cos\iota\\{b_{n}(e)\sin[n\phi(t)]\cos(2\gamma)$
$\displaystyle+a_{n}(e)\cos[n\phi(t)]\sin(2\gamma)\\}.$ (91)

Derivative respect to $\phi_{0}$

$\displaystyle\frac{\partial
h_{n}^{+}}{\partial\phi_{0}}=-A\\{(1+\cos^{2}\iota)a_{n}(e)n\sin[n\phi(t)]\cos(2\gamma)$
$\displaystyle+(1+\cos^{2}\iota)b_{n}(e)n\cos(n\phi(t))\sin(2\gamma)]$
$\displaystyle+(\sin^{2}\iota)c_{n}(e)n\sin[n\phi(t)]\\},$ (92)
$\displaystyle\frac{\partial
h_{n}^{\times}}{\partial\phi_{0}}=-2A\cos\iota\\{b_{n}(e)n\cos[n\phi(t)]\cos(2\gamma)$
$\displaystyle-a_{n}(e)n\sin[n\phi(t)]\sin(2\gamma)\\}.$ (93)

Due to $\phi(t)$ dependence we have $\frac{\partial h}{\partial f_{0}}=2\pi
t\frac{\partial h}{\partial\phi_{0}}$ and $\frac{\partial
h}{\partial\dot{f}_{0}}=2\pi t^{2}\frac{\partial h}{\partial\phi_{0}}$.
Derivative respect to $\iota_{0}$

$\displaystyle\frac{\partial
h_{n}^{+}}{\partial\iota_{0}}=-A\\{a_{n}(e)\sin(2\iota)\cos[n\phi(t)]\cos(2\gamma)$
$\displaystyle-b_{n}(e)\sin(2\iota)\sin[n\phi(t)]\sin(2\gamma)$
$\displaystyle-c_{n}(e)\sin(2\iota)\cos[n\phi(t)]\\},$ (94)
$\displaystyle\frac{\partial
h_{n}^{\times}}{\partial\iota_{0}}=2A\sin\iota\\{a_{n}(e)\cos[n\phi(t)]\sin(2\gamma)$
$\displaystyle+b_{n}(e)\sin[n\phi(t)]\cos(2\gamma)\\}.$ (95)

Due to $\iota$ dependence we have $\frac{\partial
h}{\partial\dot{\iota}_{0}}=t\frac{\partial h}{\partial\iota_{0}}$. Derivative
respect to $e_{0}$

$\displaystyle\frac{\partial h_{n}^{+}}{\partial
e_{0}}=A\\{(1+\cos^{2}\iota)a^{\prime}_{n}(e)\cos[n\phi(t)]\cos(2\gamma)$
$\displaystyle-(1+\cos^{2}\iota)b^{\prime}_{n}(e)\sin(n\phi(t))\sin(2\gamma)]$
$\displaystyle+(\sin^{2}\iota)c^{\prime}_{n}(e)\cos[n\phi(t)]\\},$ (96)
$\displaystyle\frac{\partial h_{n}^{\times}}{\partial
e_{0}}=-2A\cos\iota\\{b^{\prime}_{n}(e)\sin[n\phi(t)]\cos(2\gamma)$
$\displaystyle+a^{\prime}_{n}(e)\cos[n\phi(t)]\sin(2\gamma)\\},$ (97)
$\displaystyle a^{\prime}_{n}(e)=\frac{n^{2}}{2}[J_{n-3}(ne)-2eJ_{n-2}(ne)$
$\displaystyle-(\frac{2}{n}+1)J_{n-1}(ne)+4eJ_{n}(ne)$
$\displaystyle+(\frac{2}{n}-1)J_{n+1}(ne)-2eJ_{n+2}+J_{n+3}],$ (98)
$\displaystyle
b^{\prime}_{n}(e)=-\frac{2ne}{\sqrt{1-e^{2}}}[J_{n-2}(ne)-2J_{n}(ne)+J_{n+2}(ne)]$
$\displaystyle+\frac{n^{2}\sqrt{1-e^{2}}}{2}[J_{n-3}(ne)-3J_{n-1}(ne)$
$\displaystyle+3J_{n+1}(ne)-J_{n+3}(ne)],$ (99) $\displaystyle
c^{\prime}_{n}(e)=\frac{n}{2}[J_{n-1}(ne)-J_{n+1}(ne)].$ (100)

Due to $e$ dependence we have $\frac{\partial
h}{\partial\dot{e}_{0}}=t\frac{\partial h}{\partial e_{0}}$. Derivative
respect to $\dot{\gamma}_{0}$

$\displaystyle\frac{\partial
h_{n}^{+}}{\partial\dot{\gamma}_{0}}=-2At(1+\cos^{2}\iota)\\{a_{n}(e)\cos[n\phi(t)]\sin(2\gamma)$
$\displaystyle+b_{n}(e)\sin[n\phi(t)]\cos(2\gamma)\\},$ (101)
$\displaystyle\frac{\partial
h_{n}^{\times}}{\partial\dot{\gamma}_{0}}=-4A\cos\iota\\{a_{n}(e)\cos[n\phi(t)]\cos(2\gamma)$
$\displaystyle-b_{n}(e)\sin[n\phi(t)]\sin(2\gamma)\\}.$ (102)

##  Appendix B Calculation of the inner product

In order to calculate the inner product (71), we need calculate the integrate
$\int_{0}^{T}g(t)h(t)dt$. Here functions $g(t)$ and $h$ take form

$\displaystyle g(t)=g(\phi(t);\vec{\xi}(t))$ (103)

$\vec{\xi}$ are adiabatic parameters set including $(\iota,e,\gamma)$ which
slowly change respect to time $t$. Consequently we have

$\displaystyle g(t)\approx g(\phi(t);\vec{\xi}_{0})+\frac{\partial
g}{\partial\vec{\xi}}\cdot\dot{\vec{\xi}}\ t,$ (104)
$\displaystyle\int_{0}^{T}g(t)h(t)dt\approx\int_{0}^{T}g(\phi(t);\vec{\xi}_{0})h(\phi(t);\vec{\xi}_{0})dt+$
$\displaystyle\dot{\vec{\xi}}\cdot\int_{0}^{T}t\left[g(\phi(t);\vec{\xi}_{0})\frac{\partial
h}{\partial\vec{\xi}}+h(\phi(t);\vec{\xi}_{0})\frac{\partial
g}{\partial\vec{\xi}}\right]dt,$ (105)
$\displaystyle\int_{0}^{T}t\left[g(\phi(t);\vec{\xi}_{0})\frac{\partial
h}{\partial\vec{\xi}}+h(\phi(t);\vec{\xi}_{0})\frac{\partial
g}{\partial\vec{\xi}}\right]dt\approx$
$\displaystyle\frac{T}{2}\int_{0}^{T}\left[g(\phi(t);\vec{\xi}_{0})\frac{\partial
h}{\partial\vec{\xi}}+h(\phi(t);\vec{\xi}_{0})\frac{\partial
g}{\partial\vec{\xi}}\right]dt.$ (106)

Related to the Fisher information matrix calculation we will met following
form $\int_{0}^{T}t^{2}g(\phi(t);\vec{\xi}_{0})h(\phi(t);\vec{\xi}_{0})dt$. We
approximate it as

$\displaystyle\int_{0}^{T}t^{2}g(\phi(t);\vec{\xi}_{0})h(\phi(t);\vec{\xi}_{0})dt\approx$
$\displaystyle\frac{T^{2}}{3}\int_{0}^{T}g(\phi(t);\vec{\xi}_{0})h(\phi(t);\vec{\xi}_{0})dt.$
(107)

In addition we have relations

$\displaystyle\int_{0}^{T}\sin(n\phi)\sin(m\phi)dt$
$\displaystyle\approx\int_{0}^{\Phi}\sin(n\phi)\sin(m\phi)\frac{d\phi}{2\pi
f_{0}}$ $\displaystyle\approx\delta_{nm}\frac{\Phi}{4\pi
f_{0}}=\frac{T}{2}\delta_{nm},$ (108)
$\displaystyle\int_{0}^{T}\cos(n\phi)\cos(m\phi)dt\approx\frac{T}{2}\delta_{nm}$
(109) $\displaystyle\int_{0}^{T}\cos(n\phi)\sin(m\phi)dt\approx 0,$ (110)

which will be used in the Fisher information matrix calculation.

According to the above approximation we have

$\displaystyle\int_{0}^{T}\frac{\partial h_{n}^{+}}{\partial
A}(\phi(t);\vec{\xi}_{0})\frac{\partial h_{m}^{+}}{\partial
A}(\phi(t);\vec{\xi}_{0})dt\approx$
$\displaystyle\int_{0}^{T}\\{(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos[n\phi(t)]\cos(2\gamma_{0})$
$\displaystyle-(1+\cos^{2}\iota_{0})b_{n}(e_{0})\sin[n\phi(t)]\sin(2\gamma_{0})$
$\displaystyle+(\sin^{2}\iota_{0})c_{n}(e_{0})\cos[n\phi(t)]\\}\times$
$\displaystyle\\{(1+\cos^{2}\iota_{0})a_{m}(e_{0})\cos[m\phi(t)]\cos(2\gamma_{0})$
$\displaystyle-(1+\cos^{2}\iota_{0})b_{m}(e_{0})\sin[m\phi(t)]\sin(2\gamma_{0})$
$\displaystyle+(\sin^{2}\iota_{0})c_{m}(e_{0})\cos[m\phi(t)]\\}dt$ (111)
$\displaystyle\approx\frac{T\delta_{nm}}{2}\\{[(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos(2\gamma_{0})+(\sin^{2}\iota_{0})c_{n}(e_{0})]^{2}$
$\displaystyle+(1+\cos^{2}\iota_{0})^{2}b^{2}_{n}(e_{0})\sin^{2}(2\gamma_{0})\\},$
(112) $\displaystyle\int_{0}^{T}t\frac{\partial h_{n}^{+}}{\partial
A}(\phi(t);\vec{\xi}_{0})\frac{\partial}{\partial\iota}\frac{\partial
h_{m}^{+}}{\partial A}(\phi(t);\vec{\xi}_{0})dt\approx$
$\displaystyle\frac{T}{2}\int_{0}^{T}\\{(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos[n\phi(t)]\cos(2\gamma_{0})$
$\displaystyle-(1+\cos^{2}\iota_{0})b_{n}(e_{0})\sin[n\phi(t)]\sin(2\gamma_{0})$
$\displaystyle+(\sin^{2}\iota_{0})c_{n}(e_{0})\cos[n\phi(t)]\\}\times$
$\displaystyle\\{-\sin(2\iota_{0})a_{m}(e_{0})\cos[m\phi(t)]\cos(2\gamma_{0})$
$\displaystyle+\sin(2\iota_{0})b_{m}(e_{0})\sin[m\phi(t)]\sin(2\gamma_{0})$
$\displaystyle+\sin(2\iota_{0})c_{m}(e_{0})\cos[m\phi(t)]\\}dt$ (113)
$\displaystyle\approx\frac{T^{2}\delta_{nm}}{4}\\{\sin(2\iota_{0})[c_{n}(e_{0})-a_{n}(e_{0})\cos(2\gamma_{0})]\cdot$
$\displaystyle[(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos(2\gamma_{0})+(\sin^{2}\iota_{0})c_{n}(e_{0})]$
$\displaystyle-(1+\cos^{2}\iota_{0})\sin(2\iota_{0})b^{2}_{n}(e_{0})\sin^{2}(2\gamma_{0})\\},$
(114) $\displaystyle\int_{0}^{T}t\frac{\partial h_{n}^{+}}{\partial
A}(\phi(t);\vec{\xi}_{0})\frac{\partial}{\partial e}\frac{\partial
h_{m}^{+}}{\partial A}(\phi(t);\vec{\xi}_{0})dt\approx$
$\displaystyle\frac{T}{2}\int_{0}^{T}\\{(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos[n\phi(t)]\cos(2\gamma_{0})$
$\displaystyle-(1+\cos^{2}\iota_{0})b_{n}(e_{0})\sin[n\phi(t)]\sin(2\gamma_{0})$
$\displaystyle+(\sin^{2}\iota_{0})c_{n}(e_{0})\cos[n\phi(t)]\\}\times$
$\displaystyle\\{(1+\cos^{2}\iota_{0})a^{\prime}_{n}(e_{0})\cos[m\phi(t)]\cos(2\gamma_{0})$
$\displaystyle-(1+\cos^{2}\iota_{0})b^{\prime}_{n}(e_{0})\sin[m\phi(t)]\sin(2\gamma_{0})$
$\displaystyle+(\sin^{2}\iota_{0})c^{\prime}_{n}(e_{0})\cos[m\phi(t)]\\}dt$
(115)
$\displaystyle\approx\frac{T^{2}\delta_{nm}}{4}\\{[(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos(2\gamma_{0})+(\sin^{2}\iota_{0})c_{n}(e_{0})]\cdot$
$\displaystyle[(1+\cos^{2}\iota_{0})a^{\prime}_{n}(e_{0})\cos(2\gamma_{0})+(\sin^{2}\iota_{0})c^{\prime}_{n}(e_{0})]$
$\displaystyle+(1+\cos^{2}\iota_{0})^{2}b_{n}(e_{0})b^{\prime}_{n}(e_{0})\sin^{2}(2\gamma_{0})\\},$
(116) $\displaystyle\int_{0}^{T}t\frac{\partial h_{n}^{+}}{\partial
A}(\phi(t);\vec{\xi}_{0})\frac{\partial}{\partial\gamma}\frac{\partial
h_{m}^{+}}{\partial A}(\phi(t);\vec{\xi}_{0})dt\approx$
$\displaystyle\frac{T}{2}\int_{0}^{T}\\{(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos[n\phi(t)]\cos(2\gamma_{0})$
$\displaystyle-(1+\cos^{2}\iota_{0})b_{n}(e_{0})\sin[n\phi(t)]\sin(2\gamma_{0})$
$\displaystyle+(\sin^{2}\iota_{0})c_{n}(e_{0})\cos[n\phi(t)]\\}\times$
$\displaystyle\\{-2(1+\cos^{2}\iota_{0})a_{m}(e_{0})\cos[m\phi(t)]\sin(2\gamma_{0})$
$\displaystyle-2(1+\cos^{2}\iota_{0})b_{m}(e_{0})\sin[m\phi(t)]\cos(2\gamma_{0})\\}dt$
(117)
$\displaystyle\approx-\frac{T^{2}}{2}\delta_{nm}\\{[(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos(2\gamma_{0})+(\sin^{2}\iota_{0})c_{n}(e_{0})]\cdot$
$\displaystyle(1+\cos^{2}\iota_{0})a_{n}(e_{0})\sin(2\gamma_{0})$
$\displaystyle+(1+\cos^{2}\iota_{0})^{2}b^{2}_{n}(e_{0})\sin(2\gamma_{0})\cos(2\gamma_{0})\\},$
(118) $\displaystyle\int_{0}^{T}\frac{\partial h_{n}^{+}}{\partial
A}\frac{\partial h_{m}^{+}}{\partial A}dt\approx$
$\displaystyle\frac{T\delta_{nm}}{2}\\{[(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos(2\gamma_{0})+(\sin^{2}\iota_{0})c_{n}(e_{0})]^{2}$
$\displaystyle+(1+\cos^{2}\iota_{0})^{2}b^{2}_{n}(e_{0})\sin^{2}(2\gamma_{0})\\}+$
$\displaystyle\frac{T^{2}\dot{\iota}_{0}\delta_{nm}}{2}\\{\sin(2\iota_{0})[c_{n}(e_{0})-a_{n}(e_{0})\cos(2\gamma_{0})]\cdot$
$\displaystyle[(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos(2\gamma_{0})+(\sin^{2}\iota_{0})c_{n}(e_{0})]$
$\displaystyle-(1+\cos^{2}\iota_{0})\sin(2\iota_{0})b^{2}_{n}(e_{0})\sin^{2}(2\gamma_{0})\\}+$
$\displaystyle\frac{T^{2}\dot{e}_{0}\delta_{nm}}{2}\\{[(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos(2\gamma_{0})+(\sin^{2}\iota_{0})c_{n}(e_{0})]\cdot$
$\displaystyle[(1+\cos^{2}\iota_{0})a^{\prime}_{n}(e_{0})\cos(2\gamma_{0})+(\sin^{2}\iota_{0})c^{\prime}_{n}(e_{0})]$
$\displaystyle+(1+\cos^{2}\iota_{0})^{2}b_{n}(e_{0})b^{\prime}_{n}(e_{0})\sin^{2}(2\gamma_{0})\\}$
$\displaystyle-T^{2}\dot{\gamma}_{0}\delta_{nm}\\{[(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos(2\gamma_{0})+(\sin^{2}\iota_{0})c_{n}(e_{0})]\cdot$
$\displaystyle(1+\cos^{2}\iota_{0})a_{n}(e_{0})\sin(2\gamma_{0})$
$\displaystyle+(1+\cos^{2}\iota_{0})^{2}b^{2}_{n}(e_{0})\sin(2\gamma_{0})\cos(2\gamma_{0})\\}.$
(119) $\displaystyle\int_{0}^{T}\frac{\partial h_{n}^{\times}}{\partial
A}(\phi(t);\vec{\xi}_{0})\frac{\partial h_{m}^{\times}}{\partial
A}(\phi(t);\vec{\xi}_{0})dt\approx$
$\displaystyle\int_{0}^{T}4\cos^{2}\iota_{0}\\{b_{n}(e_{0})\sin[n\phi(t)]\cos(2\gamma_{0})$
$\displaystyle+a_{n}(e_{0})\cos[n\phi(t)]\sin(2\gamma_{0})\\}\times$
$\displaystyle\\{b_{m}(e_{0})\sin[m\phi(t)]\cos(2\gamma_{0})$
$\displaystyle+a_{m}(e_{0})\cos[m\phi(t)]\sin(2\gamma_{0})\\}dt$ (120)
$\displaystyle\approx
2T\cos^{2}\iota_{0}\delta_{nm}[b^{2}_{n}(e_{0})\cos^{2}(2\gamma_{0})+a^{2}_{n}(e_{0})\sin^{2}(2\gamma_{0})],$
(121) $\displaystyle\int_{0}^{T}t\frac{\partial h_{n}^{\times}}{\partial
A}(\phi(t);\vec{\xi}_{0})\frac{\partial}{\partial\iota}\frac{\partial
h_{m}^{\times}}{\partial A}(\phi(t);\vec{\xi}_{0})dt\approx$
$\displaystyle\frac{T}{2}\int_{0}^{T}-4\sin\iota_{0}\cos\iota_{0}\\{b_{n}(e_{0})\sin[n\phi(t)]\cos(2\gamma_{0})$
$\displaystyle+a_{n}(e_{0})\cos[n\phi(t)]\sin(2\gamma_{0})\\}\times$
$\displaystyle\\{b_{m}(e_{0})\sin[m\phi(t)]\cos(2\gamma_{0})$
$\displaystyle+a_{m}(e_{0})\cos[m\phi(t)]\sin(2\gamma_{0})\\}dt$ (122)
$\displaystyle\approx-T^{2}\sin\iota_{0}\cos\iota_{0}\delta_{nm}[b^{2}_{n}(e_{0})\cos^{2}(2\gamma_{0})$
$\displaystyle+a^{2}_{n}(e_{0})\sin^{2}(2\gamma_{0})],$ (123)
$\displaystyle\int_{0}^{T}t\frac{\partial h_{n}^{\times}}{\partial
A}(\phi(t);\vec{\xi}_{0})\frac{\partial}{\partial e}\frac{\partial
h_{m}^{\times}}{\partial A}(\phi(t);\vec{\xi}_{0})dt\approx$
$\displaystyle\frac{T}{2}\int_{0}^{T}4\cos^{2}\iota_{0}\\{b_{n}(e_{0})\sin[n\phi(t)]\cos(2\gamma_{0})$
$\displaystyle+a_{n}(e_{0})\cos[n\phi(t)]\sin(2\gamma_{0})\\}\times$
$\displaystyle\\{b^{\prime}_{m}(e_{0})\sin[m\phi(t)]\cos(2\gamma_{0})$
$\displaystyle+a^{\prime}_{m}(e_{0})\cos[m\phi(t)]\sin(2\gamma_{0})\\}dt$
(124) $\displaystyle\approx
T^{2}\cos^{2}\iota_{0}\delta_{nm}[b_{n}(e_{0})b^{\prime}_{n}(e_{0})\cos^{2}(2\gamma_{0})$
$\displaystyle+a_{n}(e_{0})a^{\prime}_{n}(e_{0})\sin^{2}(2\gamma_{0})],$ (125)
$\displaystyle\int_{0}^{T}t\frac{\partial h_{n}^{\times}}{\partial
A}(\phi(t);\vec{\xi}_{0})\frac{\partial}{\partial\gamma}\frac{\partial
h_{m}^{\times}}{\partial A}(\phi(t);\vec{\xi}_{0})dt\approx$
$\displaystyle\frac{T}{2}\int_{0}^{T}4\cos^{2}\iota_{0}\\{b_{n}(e_{0})\sin[n\phi(t)]\cos(2\gamma_{0})$
$\displaystyle+a_{n}(e_{0})\cos[n\phi(t)]\sin(2\gamma_{0})\\}\times$
$\displaystyle\\{-2b_{m}(e_{0})\sin[m\phi(t)]\sin(2\gamma_{0})$
$\displaystyle+2a_{m}(e_{0})\cos[m\phi(t)]\cos(2\gamma_{0})\\}dt$ (126)
$\displaystyle\approx
2T^{2}\cos^{2}\iota_{0}\sin(2\gamma_{0})\cos(2\gamma_{0})\delta_{nm}[a^{2}_{n}(e_{0})+b^{2}_{n}(e_{0})],$
(127) $\displaystyle\int_{0}^{T}\frac{\partial h_{n}^{\times}}{\partial
A}\frac{\partial h_{m}^{\times}}{\partial A}dt\approx$ $\displaystyle
2T\cos^{2}\iota_{0}\delta_{nm}[b^{2}_{n}(e_{0})\cos^{2}(2\gamma_{0})+a^{2}_{n}(e_{0})\sin^{2}(2\gamma_{0})]$
$\displaystyle-2\dot{\iota}_{0}T^{2}\sin\iota_{0}\cos\iota_{0}\delta_{nm}[b^{2}_{n}(e_{0})\cos^{2}(2\gamma_{0})$
$\displaystyle+a^{2}_{n}(e_{0})\sin^{2}(2\gamma_{0})]$
$\displaystyle+2\dot{e}_{0}T^{2}\cos^{2}\iota_{0}\delta_{nm}[b_{n}(e_{0})b^{\prime}_{n}(e_{0})\cos^{2}(2\gamma_{0})$
$\displaystyle+a_{n}(e_{0})a^{\prime}_{n}(e_{0})\sin^{2}(2\gamma_{0})]$
$\displaystyle+4\dot{\gamma}_{0}T^{2}\cos^{2}\iota_{0}\sin(2\gamma_{0})\cos(2\gamma_{0})\delta_{nm}[a^{2}_{n}(e_{0})+b^{2}_{n}(e_{0})],$
(128) $\displaystyle\int_{0}^{T}\frac{\partial h_{n}^{+}}{\partial
A}(\phi(t);\vec{\xi}_{0})\frac{\partial h_{m}^{\times}}{\partial
A}(\phi(t);\vec{\xi}_{0})dt\approx$
$\displaystyle-\int_{0}^{T}\\{(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos[n\phi(t)]\cos(2\gamma_{0})$
$\displaystyle-(1+\cos^{2}\iota_{0})b_{n}(e_{0})\sin[n\phi(t)]\sin(2\gamma_{0})$
$\displaystyle+(\sin^{2}\iota_{0})c_{n}(e_{0})\cos[n\phi(t)]\\}\times$
$\displaystyle 2\cos(\iota_{0})\\{b_{m}(e_{0})\sin[m\phi(t)]\cos(2\gamma_{0})$
$\displaystyle+a_{m}(e_{0})\cos[m\phi(t)]\sin(2\gamma_{0})\\}dt$ (129)
$\displaystyle\approx-T\delta_{nm}\cos(\iota_{0})\\{a_{n}(e_{0})\sin(2\gamma_{0})\times$
$\displaystyle[(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos(2\gamma_{0})+(\sin^{2}\iota_{0})c_{n}(e_{0})]-$
$\displaystyle(1+\cos^{2}\iota_{0})b^{2}_{n}(e_{0})\sin(2\gamma_{0})\cos(2\gamma_{0})\\},$
(130) $\displaystyle\int_{0}^{T}t\frac{\partial}{\partial\iota}\frac{\partial
h_{n}^{+}}{\partial A}(\phi(t);\vec{\xi}_{0})\frac{\partial
h_{m}^{\times}}{\partial A}(\phi(t);\vec{\xi}_{0})dt\approx$
$\displaystyle-\frac{T}{2}\int_{0}^{T}\\{-2\cos\iota_{0}\sin\iota_{0}a_{n}(e_{0})\cos[n\phi(t)]\cos(2\gamma_{0})$
$\displaystyle+2\cos\iota_{0}\sin\iota_{0}b_{n}(e_{0})\sin[n\phi(t)]\sin(2\gamma_{0})$
$\displaystyle+2\cos\iota_{0}\sin\iota_{0}c_{n}(e_{0})\cos[n\phi(t)]\\}\times$
$\displaystyle 2\cos(\iota_{0})\\{b_{m}(e_{0})\sin[m\phi(t)]\cos(2\gamma_{0})$
$\displaystyle+a_{m}(e_{0})\cos[m\phi(t)]\sin(2\gamma_{0})\\}dt$ (131)
$\displaystyle\approx-T^{2}\delta_{nm}\cos^{2}(\iota_{0})\sin\iota_{0}\sin(2\gamma_{0})\\{a_{n}(e_{0})\times$
$\displaystyle[c_{n}(e_{0})-a_{n}(e_{0})\cos(2\gamma_{0})]+b^{2}_{n}(e_{0})\cos(2\gamma_{0})\\},$
(132) $\displaystyle\int_{0}^{T}t\frac{\partial h_{n}^{+}}{\partial
A}(\phi(t);\vec{\xi}_{0})\frac{\partial}{\partial\iota}\frac{\partial
h_{m}^{\times}}{\partial A}(\phi(t);\vec{\xi}_{0})dt\approx$ $\displaystyle
T\int_{0}^{T}\\{(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos[n\phi(t)]\cos(2\gamma_{0})$
$\displaystyle-(1+\cos^{2}\iota_{0})b_{n}(e_{0})\sin[n\phi(t)]\sin(2\gamma_{0})$
$\displaystyle+(\sin^{2}\iota_{0})c_{n}(e_{0})\cos[n\phi(t)]\\}\times$
$\displaystyle\sin(\iota_{0})\\{b_{m}(e_{0})\sin[m\phi(t)]\cos(2\gamma_{0})$
$\displaystyle+a_{m}(e_{0})\cos[m\phi(t)]\sin(2\gamma_{0})\\}dt$ (133)
$\displaystyle\approx\frac{T^{2}\sin(\iota_{0})\delta_{nm}}{2}\\{[(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos(2\gamma_{0})+$
$\displaystyle(\sin^{2}\iota_{0})c_{n}(e_{0})]a_{n}(e_{0})\sin(2\gamma_{0})-$
$\displaystyle(1+\cos^{2}\iota_{0})b^{2}_{n}(e_{0})\sin(2\gamma_{0})\cos(2\gamma_{0})\\},$
(134) $\displaystyle\int_{0}^{T}t\frac{\partial}{\partial e}\frac{\partial
h_{n}^{+}}{\partial A}(\phi(t);\vec{\xi}_{0})\frac{\partial
h_{m}^{\times}}{\partial A}(\phi(t);\vec{\xi}_{0})dt\approx$
$\displaystyle-\frac{T}{2}\int_{0}^{T}\\{(1+\cos^{2}\iota_{0})a^{\prime}_{n}(e_{0})\cos[n\phi(t)]\cos(2\gamma_{0})$
$\displaystyle-(1+\cos^{2}\iota_{0})b^{\prime}_{n}(e_{0})\sin[n\phi(t)]\sin(2\gamma_{0})$
$\displaystyle+(\sin^{2}\iota_{0})c^{\prime}_{n}(e_{0})\cos[n\phi(t)]\\}\times$
$\displaystyle 2\cos(\iota_{0})\\{b_{m}(e_{0})\sin[m\phi(t)]\cos(2\gamma_{0})$
$\displaystyle+a_{m}(e_{0})\cos[m\phi(t)]\sin(2\gamma_{0})\\}dt$ (135)
$\displaystyle\approx-\frac{T^{2}}{2}\delta_{nm}\cos(\iota_{0})\\{a_{n}(e_{0})\sin(2\gamma_{0})\times$
$\displaystyle[(1+\cos^{2}\iota_{0})a^{\prime}_{n}(e_{0})\cos(2\gamma_{0})+(\sin^{2}\iota_{0})c^{\prime}_{n}(e_{0})]-$
$\displaystyle(1+\cos^{2}\iota_{0})b_{n}(e_{0})b^{\prime}_{n}(e_{0})\sin(2\gamma_{0})\cos(2\gamma_{0})\\},$
(136) $\displaystyle\int_{0}^{T}t\frac{\partial h_{n}^{+}}{\partial
A}(\phi(t);\vec{\xi}_{0})\frac{\partial}{\partial e}\frac{\partial
h_{m}^{\times}}{\partial A}(\phi(t);\vec{\xi}_{0})dt\approx$
$\displaystyle-\frac{T}{2}\int_{0}^{T}\\{(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos[n\phi(t)]\cos(2\gamma_{0})$
$\displaystyle-(1+\cos^{2}\iota_{0})b_{n}(e_{0})\sin[n\phi(t)]\sin(2\gamma_{0})$
$\displaystyle+(\sin^{2}\iota_{0})c_{n}(e_{0})\cos[n\phi(t)]\\}\times$
$\displaystyle
2\cos(\iota_{0})\\{b^{\prime}_{m}(e_{0})\sin[m\phi(t)]\cos(2\gamma_{0})$
$\displaystyle+a^{\prime}_{m}(e_{0})\cos[m\phi(t)]\sin(2\gamma_{0})\\}dt$
(137)
$\displaystyle\approx-\frac{T^{2}}{2}\delta_{nm}\cos(\iota_{0})\\{a^{\prime}_{n}(e_{0})\sin(2\gamma_{0})\times$
$\displaystyle[(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos(2\gamma_{0})+(\sin^{2}\iota_{0})c_{n}(e_{0})]-$
$\displaystyle(1+\cos^{2}\iota_{0})b_{n}(e_{0})b^{\prime}_{n}(e_{0})\sin(2\gamma_{0})\cos(2\gamma_{0})\\},$
(138) $\displaystyle\int_{0}^{T}t\frac{\partial}{\partial\gamma}\frac{\partial
h_{n}^{+}}{\partial A}(\phi(t);\vec{\xi}_{0})\frac{\partial
h_{m}^{\times}}{\partial A}(\phi(t);\vec{\xi}_{0})dt\approx$ $\displaystyle
T\int_{0}^{T}\\{(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos[n\phi(t)]\sin(2\gamma_{0})$
$\displaystyle+(1+\cos^{2}\iota_{0})b_{n}(e_{0})\sin[n\phi(t)]\cos(2\gamma_{0})\\}\times$
$\displaystyle 2\cos(\iota_{0})\\{b_{m}(e_{0})\sin[m\phi(t)]\cos(2\gamma_{0})$
$\displaystyle+a_{m}(e_{0})\cos[m\phi(t)]\sin(2\gamma_{0})\\}dt$ (139)
$\displaystyle\approx
T^{2}\delta_{nm}(1+\cos^{2}\iota_{0})\cos(\iota_{0})\times$
$\displaystyle[a^{2}_{n}(e_{0})\sin^{2}(2\gamma_{0})+b^{2}_{n}(e_{0})\cos^{2}(2\gamma_{0})],$
(140) $\displaystyle\int_{0}^{T}t\frac{\partial h_{n}^{+}}{\partial
A}(\phi(t);\vec{\xi}_{0})\frac{\partial}{\partial\gamma}\frac{\partial
h_{m}^{\times}}{\partial A}(\phi(t);\vec{\xi}_{0})dt\approx$
$\displaystyle-\frac{T}{2}\int_{0}^{T}\\{(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos[n\phi(t)]\cos(2\gamma_{0})$
$\displaystyle-(1+\cos^{2}\iota_{0})b_{n}(e_{0})\sin[n\phi(t)]\sin(2\gamma_{0})$
$\displaystyle+(\sin^{2}\iota_{0})c_{n}(e_{0})\cos[n\phi(t)]\\}\times$
$\displaystyle
2\cos(\iota_{0})\\{-2b_{m}(e_{0})\sin[m\phi(t)]\sin(2\gamma_{0})$
$\displaystyle+2a_{m}(e_{0})\cos[m\phi(t)]\cos(2\gamma_{0})\\}dt$ (141)
$\displaystyle\approx-T^{2}\delta_{nm}\cos(\iota_{0})\\{[(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos(2\gamma_{0})+$
$\displaystyle(\sin^{2}\iota_{0})c_{n}(e_{0})]a_{n}(e_{0})\cos(2\gamma_{0})+$
$\displaystyle(1+\cos^{2}\iota_{0})b^{2}_{n}(e_{0})\sin^{2}(2\gamma_{0})\\},$
(142) $\displaystyle\int_{0}^{T}\frac{\partial h_{n}^{+}}{\partial
A}\frac{\partial h_{m}^{\times}}{\partial A}dt\approx$
$\displaystyle-T\delta_{nm}\cos(\iota_{0})\\{a_{n}(e_{0})\sin(2\gamma_{0})\times$
$\displaystyle[(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos(2\gamma_{0})+(\sin^{2}\iota_{0})c_{n}(e_{0})]-$
$\displaystyle(1+\cos^{2}\iota_{0})b^{2}_{n}(e_{0})\sin(2\gamma_{0})\cos(2\gamma_{0})\\}$
$\displaystyle-T^{2}\dot{\iota}_{0}\delta_{nm}[\cos^{2}(\iota_{0})\sin\iota_{0}\sin(2\gamma_{0})\\{a_{n}(e_{0})\times$
$\displaystyle[c_{n}(e_{0})-a_{n}(e_{0})\cos(2\gamma_{0})]+b^{2}_{n}(e_{0})\cos(2\gamma_{0})\\}$
(143)
$\displaystyle-\frac{\sin(\iota_{0})}{2}\\{[(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos(2\gamma_{0})+$
$\displaystyle(\sin^{2}\iota_{0})c_{n}(e_{0})]a_{n}(e_{0})\sin(2\gamma_{0})-$
$\displaystyle(1+\cos^{2}\iota_{0})b^{2}_{n}(e_{0})\sin(2\gamma_{0})\cos(2\gamma_{0})\\}]$
$\displaystyle-\frac{T^{2}}{2}\dot{e}_{0}\delta_{nm}\cos(\iota_{0})\\{a_{n}(e_{0})\sin(2\gamma_{0})\times$
$\displaystyle[(1+\cos^{2}\iota_{0})a^{\prime}_{n}(e_{0})\cos(2\gamma_{0})+(\sin^{2}\iota_{0})c^{\prime}_{n}(e_{0})]-$
$\displaystyle(1+\cos^{2}\iota_{0})b_{n}(e_{0})b^{\prime}_{n}(e_{0})\sin(2\gamma_{0})\cos(2\gamma_{0})$
$\displaystyle+a^{\prime}_{n}(e_{0})\sin(2\gamma_{0})\times$
$\displaystyle[(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos(2\gamma_{0})+(\sin^{2}\iota_{0})c_{n}(e_{0})]-$
$\displaystyle(1+\cos^{2}\iota_{0})b_{n}(e_{0})b^{\prime}_{n}(e_{0})\sin(2\gamma_{0})\cos(2\gamma_{0})\\}$
$\displaystyle+T^{2}\dot{\gamma}_{0}\delta_{nm}\cos(\iota_{0})\\{(1+\cos^{2}\iota_{0})\times$
$\displaystyle[a^{2}_{n}(e_{0})\sin^{2}(2\gamma_{0})+b^{2}_{n}(e_{0})\cos^{2}(2\gamma_{0})]$
(144) $\displaystyle-\\{[(1+\cos^{2}\iota_{0})a_{n}(e_{0})\cos(2\gamma_{0})+$
$\displaystyle(\sin^{2}\iota_{0})c_{n}(e_{0})]a_{n}(e_{0})\cos(2\gamma_{0})+$
$\displaystyle(1+\cos^{2}\iota_{0})b^{2}_{n}(e_{0})\sin^{2}(2\gamma_{0})\\}\\}.$
(145)

The calculation of other elements of the Fisher information matrix is similar.

## Data availability

The data that support the plots within this paper and the other findings of
this study are available from the corresponding author upon request.

## References

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