# Gravitational waves for eccentric extreme mass ratio inspirals of self-dual
spacetime

Yunlong Liu  Department of Physics, South China University of Technology,
Guangzhou 510641, China  Xiangdong Zhang111Corresponding author.
<EMAIL_ADDRESS>Department of Physics, South China University of
Technology, Guangzhou 510641, China

###### Abstract

In this paper, we calculate the frequencies of geodesic orbits in self-dual
spacetime on the equatorial plane and obtain the leading-order effects of loop
quantum parameters $P$ on the energy flux and angular momentum flux in
eccentric extreme mass ratio inspirals. The gravitational waveform under
different eccentricity is carried out by improved “analytic-kludge” method.
Through the calculation of waveform mismatches for the LISA detector, the
constraints on loop quantum parameters will be improved by 1 to 2 orders of
magnitude, compared to the weak field experiments in the solar system, and can
reach the level of $10^{-8}$.

##  I Introduction

General relativity (GR) has been developed for over a century, and so far the
observed data have not exceeded the predictions of general relativity.
However, there are irreconcilable problems between GR and quantum mechanics
(QM). Hence, establishing a unified theory of GR and QM is the primary task in
theoretical physics. Today, various quantum gravity theories have been
proposed. Among them, loop quantum gravity (LQG) is notable due to its
background independence and non-perturbative nature [1, 2, 3, 4].

The application of LQG to cosmology, known as loop quantum cosmology (LQC),
now becomes a fruitful field. The most notable feature of LQC is that it
successfully uses the “Big Bounce” to replace the classical big bang
singularity [5, 8, 6, 7, 9, 10].

Due to the success of LQC, and note that the interior of Schwarzschild black
hole is isometric to the Kantowski-Sachs model [7, 11], the attempt to apply
quantization techniques to black holes to address the issue of black hole
singularity is a very intriguing idea. By utilizing the techniques developed
in LQC and through different quantization schemes, various LQG black hole
models can be constructed [12]. Among these different models, one famous and
well studied method is to set the loop quantum regularization parameters as
constants[7, 13]. By using the technique of holonomy correction, a so-called
self-dual solution of loop quantum black holes without curvature singularities
has been constructed [14, 15].

One natural question for a quantum black hole model is that does LQG
parameters can generate any observable features, or whether these quantum
parameters can be constrained through current or future experiments and
observations? A lot of effects have been made towards this direction. For
example, in the solar system [16], or in strong field regimes observations of
binary pulsars [17, 18, 19], imaging of black hole shadows [20, 21], and
direct detection of gravitational wave(GW) [22]. Based on this, lots of
properties and observational constraints in self-dual spacetimes have been
studied [23, 24, 25, 26]. As a result, the best constraints on LQG parameters
are provided within the solar system experiments [24], with a limit of quantum
parameter $|\delta|<0.0199$ (or $P<5.5\times 10^{-6}$).

To obtain with more tighter constraints on parameters, more precise
experimental tests are necessary. The detection of GWs presents an opportunity
for this. The next generation of space-based detectors, such as LISA [27],
Tianqin [28], and Taiji [30, 29], will expand the types and spectra of GW
signals to unprecedented precision, providing possibilities to testing LQG and
constrain the quantum parameters.

Among various GW signals, an important one comes from the extreme mass ratio
inspiral (EMRI) system composed of a massive black hole ($M\approx 10^{5\sim
7}M_{\odot}$) and a small compact object ($m\approx 10^{0\sim 2}M_{\odot}$)
[31]. Analyzing EMRIs requires the establishment of corresponding waveform
models [32, 33, 34, 35, 36, 37]. For example, the high-precision kludge method
[33] has been proposed. However, these methods come with a high computational
cost. To save computational resources, Barack and Cutler [32] proposed the
“analytic kludge” (AK) model. This model can quickly obtain a GW template of
EMRI, and can reflect the main physical properties of the real EMRI
waveform[33]. However, when this method approaches the innermost stable orbit
(ISO), the phase error will increase rapidly. In order to solve this problem,
Chua et al. proposed an improved AK waveform method [34, 35] with “map” way,
and obtained more accurate waveforms.

In this paper, we propose a new improvement on the AK method differs from the
“map” way of Chua [34, 35]. We used the inverse series way and obtain the high
precision perihelion precession angle, which encodes the major cumulative
orbital error in spherically symmetric spacetime when energy and angular
momentum fluxes of GWs are not considered. As a result, we calculate the
impact of LQG parameter on GWs and explore the detection capabilities of
space-based GW detectors for the parameter of LQG. We find that the constraint
ability on loop quantum parameters will be greatly improved compared with the
experiments in the solar system.

The structure of this paper is as follows: In Sec. II, we introduce the
geodesic equations on the self-dual spacetime and calculate two mean orbital
frequencies on the equatorial plane, and give the leading-order correction of
fluxes due to the LQG effects. In Sec. III, we provide two improved adiabatic
evolution equations for orbitals and also present the quadrupole moment
waveform in the self-dual spacetime. Then, in Sec. IV, we obtain the numerical
solutions of ODEs from the previous section and corresponding gravitational
waveforms. With all this, we analyze the detection capability of LISA for the
parameter of LQG with mismatch in Sec. V. Finally, the conclusions and
perspectives are provided in Sec. VI.

##  II The background metric and motion equation

###  II.1 background metric

The metric of static spherically symmetric self-dual black hole is given
by[15, 12]:

$\displaystyle ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+D(r)d\Omega^{2},$ (1)

where

$\displaystyle A(r)$ $\displaystyle=$
$\displaystyle\frac{\left(r-r_{+}\right)\left(r-r_{-}\right)\left(r-r_{*}\right)}{r^{4}+a_{0}^{2}},$
(2) $\displaystyle B(r)$ $\displaystyle=$
$\displaystyle\frac{\left(r+r_{+}\right)^{2}\left(r^{4}+a_{0}^{2}\right)}{\left(r-r_{+}\right)\left(r-r_{-}\right)r^{4}},$
(3) $\displaystyle D(r)$ $\displaystyle=$ $\displaystyle
r^{2}+\frac{a_{0}^{2}}{r^{2}}.$ (4)

Here, $r_{+}=2M/(1+P)^{2}$, $r_{-}=2MP^{2}/(1+P)^{2}$, and
$r_{*}=\sqrt{r_{+}r_{-}}$. It should be noted that $a_{0}$ depends on the
minimal area element $\Delta$ and $P$ corresponds to the regularization
parameter $\delta$ in LQG as

$\displaystyle a_{0}=\frac{\Delta}{8\pi};\quad
P=\frac{\sqrt{1+\gamma^{2}\delta^{2}}-1}{\sqrt{1+\gamma^{2}\delta^{2}}+1};$
(5)

When $a_{0}=0$ and $P=0$, the spacetime can return to the Schwarzschild case.
Since the parameter $a_{0}$ usually appears at higher order terms in the
Taylor series expansion while the gravitational wave we concern in this paper
only involves quadrupole which is irrelevant with higher order terms and hence
we can safely set $a_{0}=0$ in the following. Moreover, we adopt geometric
units $G=c=1$.

###  II.2 Equations of motion in the spherically symmetric metric

Now we turn to the equations of motion. By using the Killing vectors, we can
define two conserved quantities, energy $E$ and angular momentum $L$ as

$\displaystyle E$ $\displaystyle:=$ $\displaystyle mA(r)\frac{dt}{d\tau},$ (6)
$\displaystyle L$ $\displaystyle:=$ $\displaystyle
m\sin^{2}(\theta)D(r)\frac{d\phi}{d\tau}.$ (7)

Through the Lagrangian and separating variables, we can obtain the equations
of motion of this spacetime as follows:

$\displaystyle\frac{dt}{d\tau}=\frac{E}{mA(r)},$ (8)
$\displaystyle(\frac{dr}{d\tau})^{2}=\frac{B(r)}{D(r)}\left(-D(r)+\frac{D(r)}{A(r)}\frac{E^{2}}{m^{2}}-\frac{C}{m^{2}}\right),$
(9)
$\displaystyle(\frac{d\theta}{d\tau})^{2}=\frac{C}{m^{2}}-\frac{L^{2}}{m^{2}\sin^{2}(\theta)},$
(10) $\displaystyle\frac{d\phi}{d\tau}=\frac{L}{m\sin^{2}(\theta)D(r)}.$ (11)

Moreover, to facilitate calculations, we perform a variable transformation
from the original coordinate system $(t,r,\theta,\phi)$ to the new one
$(t,\psi,\theta,\phi)$, where

$\displaystyle r(\psi)=\frac{Mp}{1+e\cos(\psi)},$ (12)

where $p$ is the dimensionless semilatus rectum and $e$ is the eccentricity.

Note that the distances of the orbit’s pericenter and apocenter are
respectively given by:

$\displaystyle r_{\text{apo}}=\frac{Mp}{1-e},$ (13) $\displaystyle
r_{\text{peri}}=\frac{Mp}{1+e}.$ (14)

For $0<e<1$, the radial velocity of the small body equal zero ($dr/d\tau=0$)
at the pericenter or apocenter, where
$\psi=\mathbb{N}\pi,\mathbb{N}=1,2,3,...$. Hence, $dr/d\tau$ and $\sin(\psi)$
change sign on $\psi=\mathbb{N}\pi$ at the same time. Take the derivative of
Eq. (12), we get

$\displaystyle\frac{d\psi}{d\tau}=\frac{(1+e\cos(\psi))^{2}}{eMp}\frac{1}{\sin(\psi)}\frac{dr}{d\tau}.$
(15)

Therefore, $d\psi/d\tau$ always retains the same sign. This feature enables us
to set $d\psi/d\tau\geq 0$ which indicates $\psi$ monotonically increases with
proper time $\tau$. For equatorial plane orbits, we have $\theta=\pi/2$ and
$d\theta/d\tau=0$ . From Eq. (10), we have $C=L^{2}$. The equations of motion
can be reduced to:

$\displaystyle\frac{dt}{d\tau}=\frac{E}{mA(r(\psi))},$ (16)
$\displaystyle\frac{d\psi}{d\tau}=\frac{(1+e\cos(\psi))^{2}}{eMp\sqrt{\sin(\psi)^{2}}}\sqrt{\frac{1}{B(r(\psi))}\left(-1+\frac{1}{A(r(\psi))}\frac{E^{2}}{m^{2}}-\frac{1}{D(r(\psi))}\frac{L^{2}}{m^{2}}\right)},$
(17) $\displaystyle\frac{d\theta}{d\tau}=0,$ (18)
$\displaystyle\frac{d\phi}{d\tau}=\frac{L}{mD(r(\psi))}.$ (19)

Next, let’s further analyze the relationship between $(p,e)$ and $(E,L)$.
Since we have $dr/d\tau=0$ at $r_{\text{apo}}$ and $r_{\text{peri}}$, we can
obtain the relationship between $(p,e)$ and $(E,L)$ from Eq. (10) as follows:

$\displaystyle
E=\frac{m\sqrt{A(r_{\text{apo}})}\sqrt{A(r_{\text{peri}})}\sqrt{D(r_{\text{apo}})-D(r_{\text{peri}})}}{\sqrt{A(r_{\text{peri}})D(r_{\text{apo}})-A(r_{\text{apo}})D(r_{\text{peri}})}},$
(20) $\displaystyle
L=\frac{m\sqrt{A(r_{\text{apo}})-A(r_{\text{peri}})}\sqrt{D(r_{\text{apo}})}\sqrt{D(r_{\text{peri}})}}{\sqrt{A(r_{\text{peri}})D(r_{\text{apo}})-A(r_{\text{apo}})D(r_{\text{peri}})}}.$
(21)

It should be note that when $e=0$, Eqs. (15) and (17) are not valid, but we
can do a similar analysis. Since in this case, we have $dr/d\tau=0$,
$dr^{2}/d\tau^{2}=0$ and $r_{\text{apo}}=r_{\text{peri}}=Mp$, hence we can
obtain:

$\displaystyle E=\frac{mA\left[r_{\text{apo
}}\right]\sqrt{D^{\prime}\left[r_{\text{apo
}}\right]}}{\sqrt{-D\left[r_{\text{apo }}\right]A^{\prime}\left[r_{\text{apo
}}\right]+A\left[r_{\text{apo }}\right]D^{\prime}\left[r_{apo}\right]}},$ (22)
$\displaystyle L=\frac{mD\left[r_{\text{apo
}}\right]\sqrt{A^{\prime}\left[r_{\text{apo
}}\right]}}{\sqrt{-D\left[r_{\text{apo }}\right]A^{\prime}\left[r_{\text{apo
}}\right]+A\left[r_{\text{apo }}\right]D^{\prime}\left[r_{\text{apo
}}\right]}}.$ (23)

Expanding the energy $E$ and angular momentum $L$ in terms of the
dimensionless semilatus rectum $p$, we have:

$\displaystyle\frac{E}{m}$ $\displaystyle=$ $\displaystyle
1+\frac{(-1+e^{2})(1-P)^{2}}{2(1+P)^{2}p}+\frac{3(-1+e^{2})^{2}(1-P)^{4}}{8(1+P)^{4}p^{2}}+\mathcal{O}(p^{-3}),$
(24) $\displaystyle\frac{L}{Mm}$ $\displaystyle=$
$\displaystyle\frac{1-P}{1+P}p^{1/2}-\frac{3+e^{2}(1-P)^{4}-4P+10P^{2}-4P^{3}+3P^{4}}{2(1-P)(1+P)^{3}}p^{-1/2}+\mathcal{O}(p^{-3/2}).$
(25)

It is easy to see that when $P=0$, the above result reduced to the
Schwarzschild case, while $e=0$, the case of circular orbit will be recovered.

###  II.3 Orbital frequencies

We assume that the trajectory is on the equatorial plane $\theta=\pi/2$, it
means that the polar angle frequency $\Omega_{\theta}$ can be neglected. For
bounded orbits, we are left with two frequencies $\Omega_{r}$ and
$\Omega_{\phi}$ that correspond to the parameters $r(\psi)$ and $\phi$. The
coordinate time spent by the test body in moving from one pericenter to the
next one is given by [38]:

$\displaystyle\Lambda_{t}=\int_{0}^{t_{0}}dt=\int_{0}^{2\pi}\frac{dt}{d\psi}d\psi.$
(26)

The corresponding change in the azimuthal angle $\phi$ reads:

$\displaystyle\Lambda_{\phi}=\int_{0}^{\phi_{0}}d\phi=\int_{0}^{2\pi}\frac{d\phi}{d\psi}d\psi.$
(27)

For this, the average angular velocity of $r(\psi)$ and $\phi$ over each
period can be expressed as

$\displaystyle\Omega_{r}=\frac{2\pi}{\Lambda_{t}};\quad\Omega_{\phi}=\frac{\Lambda_{\phi}}{\Lambda_{t}}.$
(28)

By combining the geodesic equations (16) and (19), we can obtain the following
expansion:

$\displaystyle\Omega_{\phi}$
$\displaystyle=\frac{(1-P)X^{3/2}}{(1+P)Mp^{3/2}}-\frac{X^{3/2}}{(1-P)(1+P)^{3}Mp^{5/2}}$
(29)
$\displaystyle\times\Bigl{(}-3+4P-6P^{2}+4P^{3}-3P^{4}+(3-8P+10P^{2}-8P^{3}+3P^{4})X\Bigr{)}$
$\displaystyle-\frac{X^{3/2}}{4(1-P)^{3}(1+P)^{5}Mp^{7/2}}\Bigl{(}-57-4(1-P)^{4}X^{2}(3+\mathcal{P}_{2}{})^{2}$
$\displaystyle+2(1-P)^{4}X^{3/2}(15+\mathcal{P}_{4}{})+(1-P)^{2}X(63+\mathcal{P}_{6})+\mathcal{P}_{8}\Bigr{)}+\mathcal{O}(p^{-9/2}),$
$\displaystyle\Omega_{r}$
$\displaystyle=\frac{(1-P)X^{3/2}}{(1+P)Mp^{3/2}}-\frac{(1-P)X^{5/2}(3+\mathcal{P}_{2}{})}{(1+P)^{3}Mp^{5/2}}$
(30)
$\displaystyle-\frac{(1-P)X^{5/2}}{2(1+P)^{5}Mp^{7/2}}\left(12(1+3P^{2}+P^{4})-2X(3+\mathcal{P}_{2}{})^{2}+X^{1/2}(15+\mathcal{P}_{4}{})\right)+\mathcal{O}(p^{-9/2}),$

where

$\displaystyle\mathcal{P}_{2}=$ $\displaystyle-2P+3P^{2},$ (31)
$\displaystyle\mathcal{P}_{4}=$ $\displaystyle-12P+26P^{2}-12P^{3}+15P^{4},$
(32) $\displaystyle\mathcal{P}_{6}=$
$\displaystyle-122P+233P^{2}-284P^{3}+233P^{4}-122P^{5}+63P^{6},$ (33)
$\displaystyle\mathcal{P}_{8}=$ $\displaystyle
200P-364P^{2}+568P^{3}-662P^{4}+568P^{5}-364P^{6}+200P^{7}-57P^{8}.$ (34)

Thus, we can obtain the frequency of perihelion precession
$\dot{\hat{\gamma}}$ as:

$\displaystyle\dot{\hat{\gamma}}$ $\displaystyle=$
$\displaystyle\frac{(-3+4P-6P^{2}+4P^{3}-3P^{4})X^{3/2}-(1-P)(2P-5P^{2}+3P^{3}+(1-P)\mathcal{P}_{2})X^{5/2}}{-(1-P)(1+P)^{3}Mp^{5/2}}$
(35)
$\displaystyle+\frac{X^{5/2}\bigl{(}-24(1-P)^{4}(1+3P^{2}+P^{4})+(1-P)^{2}(63+\mathcal{P}_{6})\bigr{)}+X^{3/2}(-57+\mathcal{P}_{8})}{-4(1-P)^{3}(1+P)^{5}Mp^{7/2}}+\mathcal{O}(p^{-9/2}),$

where $X=1-e^{2}$. Here, it can be easily verified that when $P=0$, the result
returns to the Schwarzschild case. In our code, in order to obtain a higher
precision geodesic orbit, we use a higher order series expansion formula of
$\dot{\hat{\gamma}}$. However, due to the increased complexity, we will not
show it here.

###  II.4 Fluxes

In the AK waveform on the equatorial plane, in addition to the two motion
equations obtained from geodesics, there are also adiabatic evolution for
frequency $\nu$ and eccentricity $e$ caused by energy and angular momentum
flux of GW.

We use the quadrupole approximation to calculate the energy flux and angular
momentum flux[39, 40] as

$\displaystyle\frac{dE}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{5}\left\langle\dddot{\mathcal{Q}}_{ij}\dddot{\mathcal{Q}}_{ij}\right\rangle,$
(36) $\displaystyle\frac{dL^{i}}{dt}$ $\displaystyle=$
$\displaystyle\frac{2}{5}\epsilon^{ikl}\left\langle\ddot{\mathcal{Q}}_{ka}\dddot{\mathcal{Q}}_{la}\right\rangle,$
(37)

where the quadrupole moment $\mathcal{Q}$ can be described in terms of the
mass moment $\mathcal{M}$ as follows:

$\displaystyle\mathcal{Q}^{ij}$ $\displaystyle=$
$\displaystyle\mathcal{M}^{ij}-\frac{1}{3}\delta^{ij}\mathcal{M}_{kk},$ (38)
$\displaystyle\mathcal{M}^{ij}$ $\displaystyle=$ $\displaystyle\int
d^{3}xT^{00}(t,\mathbf{x})x^{i}x^{j},$ (39)

where $T^{ab}$ is the stress-energy tensor and $\mathbf{x}$ is the position
vector. In the point particle approximation and in the center-of-mass
coordinates, the second mass moment can be expressed as:

$\displaystyle\mathcal{M}^{ij}=\mu x_{0}^{i}x_{0}^{j},$ (40)

where $\mu=mM/(m+M)$ is the reduced mass, and $\mathbf{x}_{0}$ is the relative
position between the small body and the central black hole. In the weak field
approximation, $x_{0}^{i}=(r\cos\phi,r\sin\phi,0)$, the leading-order energy
flux and angular momentum flux are given as follows:

$\displaystyle\frac{1}{m}\frac{dE}{dt}$ $\displaystyle=$
$\displaystyle-\frac{(1-e^{2})^{3/2}(96+292e^{2}+37e^{4})(-1+P)^{6}\mu^{2}}{15(1+P)^{6}M^{2}mp^{5}}+\mathcal{O}(p^{-6}),$
(41) $\displaystyle\frac{1}{mM}\frac{dL}{dt}$ $\displaystyle=$
$\displaystyle-\frac{4(1-e^{2})^{3/2}(8+7e^{2})(1-P)^{5}\mu^{2}}{5(1+P)^{5}M^{2}mp^{7/2}}+\mathcal{O}(p^{-9/2}).$
(42)

##  III Orbital evolution equations

Now we have the expressions for energy flux and angular momentum flux in terms
of a series expansion in Eq.(41)-(42), the next step is to convert this series
expansion in terms of the Keplerian frequency $\nu$. Additionally, in
equations (29) and (30), there are two frequencies involved, $\Omega_{r}$ and
$\Omega_{\phi}$. In the conventional approach of the $AK$ method, it is
usually assumed that $\Omega_{r}\approx 2\pi\nu$ [34, 35]. However, this
approximation introduces significant errors, especially when approaching the
innermost stable orbit(ISO) [34, 35]. A detailed comparison of $\Omega_{r}$,
$\Omega_{\phi}$ and $2\pi\nu$ is provided in Appendix A.1. Since this paper
only considers the spherically symmetric case, we do not use the “map” method
proposed in [34] to address the large discrepancies between the orbital and
precession frequency $\\{f_{orb},f_{peri}\\}$ and the approximations of
$\\{\Omega_{r}/2\pi,(\Omega_{\phi}-\Omega_{r})/2\pi\\}$. Instead, we will
directly employ higher order series expansions of $\dot{\hat{\gamma}}$ to
resolve this issue.

###  III.1 Geodesic Equations

In this part, we will use both frequencies, $\Omega_{r}=2\pi\nu_{r}$ and
$\Omega_{\phi}=2\pi\nu_{\phi}$, as base frequencies to obtain the approximate
equations of geodesics.

####  III.1.1 The large eccentricity method (eL method)

We assume $\Omega_{r}=2\pi\nu_{r}$, and by calculating the inverse series
expansion of Eq. (29), we obtain the relationship between $p$ and the radial
frequency $\nu_{r}$:

$\displaystyle p$ $\displaystyle=$
$\displaystyle\left(\frac{1-P}{1+P}\right)^{2/3}\frac{X}{(2\pi
M\nu_{r})^{2/3}}-\frac{2(3+P(-2+3P))X}{3(1+P)^{2}}$ (43) $\displaystyle+(2\pi
M\nu_{r})^{2/3}\left(\frac{1+P}{1-P}\right)^{2/3}\frac{1}{9(1+P)^{4}}\Bigl{(}-36\left(1+3P^{2}+P^{4}\right)$
$\displaystyle+3\left(-15+12P-26P^{2}+12P^{3}-15P^{4}\right)\sqrt{X}+\left(3-2P+3P^{2}\right)^{2}X\Bigr{)}+\mathcal{O}((2\pi
M\nu_{r})^{4/3}).$

If we substitute Eq. (43) into Eq. (29), we can obtain
$\Omega_{r}=2\pi\nu_{r}+\mathcal{O}(\nu_{r}^{7/3})$. By substituting the
expression (43) into Eq. (35), evolution equation determined by geodesic is
given by

$\displaystyle\frac{d\Psi}{dt}$ $\displaystyle=$
$\displaystyle\Omega_{r}=2\pi\nu_{r},$ (44) $\displaystyle\frac{d\Phi}{dt}$
$\displaystyle=$ $\displaystyle\Omega_{r}+\frac{d\hat{\gamma}}{dt},$ (45)
$\displaystyle\frac{d\hat{\gamma}}{dt}$ $\displaystyle=$
$\displaystyle\frac{\left(1+P^{2}\right)\left(3-4P+3P^{2}\right)}{(1-P)^{8/3}(1+P)^{4/3}}\frac{(2\pi
M\nu_{r})^{5/3}}{\left(1-e^{2}\right)M}$ (46)
$\displaystyle+\frac{1}{(1-P)^{16/3}(1+P)^{8/3}}\frac{(2\pi
M\nu_{r})^{7/3}}{12M\left(1-e^{2}\right)^{2}}\left(2\left(117+\mathscr{P}_{2\gamma
0}^{8}\right)-e^{2}\left(63+\mathscr{P}_{\gamma
2}^{8}\right)\right)+\mathcal{O}((2\pi M\nu_{r})^{3}),$

where $\Psi$ represents mean anomaly for $\psi$ and $\Phi$ denotes mean
“longitude” for $\phi$ with

$\displaystyle\mathscr{P}_{2\gamma 0}^{8}$ $\displaystyle=$
$\displaystyle-432P+860P^{2}-1360P^{3}+1582P^{4}-1360P^{5}+860P^{6}-432P^{7}+117P^{8},$
(47) $\displaystyle\mathscr{P}_{\gamma 2}^{8}$ $\displaystyle=$
$\displaystyle\left(7-6P+7P^{2}\right)\left(3-5P+5P^{2}-3P^{3}\right)^{2}-63.$
(48)

####  III.1.2 The small eccentricity method(eS method)

Analogously, we set $\Omega_{\phi}=2\pi\nu_{\phi}$. By computing the inverse
series of Eq. (30), we can obtain the relation between $p$ and the “longitude”
frequency $\nu_{\phi}$:

$\displaystyle p$ $\displaystyle=$
$\displaystyle\left(\frac{1-P}{1+P}\right)^{2/3}\frac{X}{(2\pi
M\nu_{\phi})^{2/3}}-\frac{2(3+P(-2+3P))X}{3(1+P)^{2}}+\frac{2\left(1+P^{2}\right)(3+P(-4+3P))}{3\left(-1+P^{2}\right)^{2}}$
(49) $\displaystyle+F[P,X](2\pi M\nu_{\phi})^{2/3}+\mathcal{O}\left((2\pi
M\nu_{\phi})^{4/3}\right).$

Here,

$\displaystyle F[P,X]$ $\displaystyle=$
$\displaystyle\frac{(1-P)^{10/3}}{9(1+P)^{10/3}}\Big{(}{\left(81-360P+572P^{2}-984P^{3}+1126P^{4}-984P^{5}+572P^{6}-360P^{7}+81P^{8}\right)}X^{-1}+$
(50)
$\displaystyle\left(-9+6P+P^{2}+132P^{3}+P^{4}+6P^{5}-9P^{6}\right)(1-P)^{2}$
$\displaystyle-6\left(15-12P+26P^{2}-12P^{3}+15P^{4}\right)(1-P)^{4}\sqrt{X}+2\left(3-2P+3P^{2}\right)^{2}(1-P)^{4}X\Big{)}.$

Substituting Eq.(49) into Eq. (30), we get
$\Omega_{\phi}=2\pi\nu_{\phi}+\mathcal{O}(\nu_{\phi}^{7/3})$. By plugging the
Eq. (49) into Eq. (35), equations of motion can be obtained as

$\displaystyle\frac{d\Psi}{dt}$ $\displaystyle=$
$\displaystyle\Omega_{r}=\Omega_{\phi}-\frac{d\hat{\gamma}}{dt},$ (51)
$\displaystyle\frac{d\Phi}{dt}$ $\displaystyle=$
$\displaystyle\Omega_{\phi}=2\pi\nu_{\phi},$ (52)
$\displaystyle\frac{d\hat{\gamma}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{\left(1+P^{2}\right)\left(3-4P+3P^{2}\right)}{(1-P)^{8/3}(1+P)^{4/3}}\frac{(2\pi
M\nu_{\phi})^{5/3}}{\left(1-e^{2}\right)M}$ (53)
$\displaystyle-\frac{1}{(1-P)^{16/3}(1+P)^{8/3}}\frac{(2\pi
M\nu_{\phi})^{7/3}}{12M\left(1-e^{2}\right)^{2}}\left(2\left(27+\mathscr{P}_{1\gamma
0}^{8}\right)-e^{2}\left(63+\mathscr{P}_{\gamma
2}^{8}\right)\right)+\mathcal{O}((2\pi M\nu_{r})^{3}),$

where

$\displaystyle\mathscr{P}_{1\gamma 0}^{8}$ $\displaystyle=$
$\displaystyle-192P+340P^{2}-640P^{3}+722P^{4}-640P^{5}+340P^{6}-192P^{7}+27P^{8}.$
(54)

Along these lines, we can extand the approximate Eq. (46) or Eq. (53) to
higher orders. In the following part of the paper, whenever we talk about
using Eq. (46) or Eq. (53) up to $(n,m)$ orders, it means that the
approximation of $d\hat{\gamma}/dt$ will be calculated to $n$-th orders
($\mathcal{O}((2\pi M\nu_{\kappa})^{(2n+3)/3}),\ \kappa=r,\phi$) in
Schwarzschild part and $m$-th orders in LQG part. For example,the expressions
in Eq. (46) or Eq.(53) is up to $(2,2)$ orders.

####  III.1.3 Orbits

As the extreme mass ratio parameter $m/M$ approaches zero, the energy and
angular momentum flux also tend to zero, and the orbit can be described by a
timelike geodesic and the information of orbit is completely encoded in the
Eqs. (44)-(46) or (51)-(53). However, the azimuthal angle $\Psi$ and $\Phi$
are the mean of true angel $\psi$ and $\phi$. The relationship between the
true anomaly $\psi$ and the mean anomaly $\Psi$ can be derived using classical
methods, and it is as follows[40]:

$\displaystyle\psi$ $\displaystyle\approx$
$\displaystyle\psi_{Ap}=2\arctan\left(\left(\frac{1+e}{1-e}\right)^{1/2}\tan\left(\frac{u}{2}\right)\right)+2\pi\mathrm{IntegerPart}(\frac{\Psi+\pi}{2\pi}),$
(55) $\displaystyle u$ $\displaystyle=$ $\displaystyle\Psi+e\sin(u).$ (56)

The third order approximation for $u$ is:

$\displaystyle u=\Psi+e\sin(\Psi+e\sin((\Psi+e\sin(\Psi))).$ (57)

Moreover, $\phi$ can be approximated as:

$\displaystyle\phi\approx\phi_{Ap}=\psi_{Ap}(1+\frac{\hat{\gamma}}{\Psi}).$
(58)

The approximate orbit is given by $r^{i}=(r\cos(\phi),r\sin(\phi),0)$, where
$r$ is defined in Eq. (12). In Eq. (58) , the use of the mean precession angle
$\hat{\gamma}$ and mean anomaly $\Psi$ introduces periodic errors in the
orbit. In addition, from Eqs. (44) - (46) or (51) -(53), we know that the main
and cumulative error is mainly in $\hat{\gamma}$. Hence, the analysis of the
error of $\hat{\gamma}$ is helpful for us to estimate the orbit accuracy which
well be shown in Appendix A.2.

Note that, to reduce this error accumulates over time, we usually employing
higher orders of $d\hat{\gamma}/dt$ in Eq. (53) or (46) when calculating
numerical solutions. By numerically solving the system of Eqs. (16)-(19), Eqs.
(44)-(46) up to $(10,5)$ orders, or Eqs. (51) -(53) up to $(11,6)$ orders, we
obtain the following orbital results as shown in the Fig. 1:

(a) $e=0.1$ (b) $e=0.8$

Figure 1:  Equatorial orbits: the solid black line represents high precision
numerical result for Eqs. (16)-(19), while the dotted lines with different
colors represent the approximate orbits obtained using eL and eS method. The
mass of supermassive black hole is set to be $M=10^{6}M_{\odot}$, and the loop
quantum parameter is chosen as $P=1/1000$. The frequency of $\phi$ is
$\Omega_{\phi}/(2\pi)=0.2\mathrm{mHz}$, corresponding to an approximate
orbital frequency of $\Omega_{r}/(2\pi)=0.14331\mathrm{mHz}$. The small yellow
circle at the center represents the minimum periastron of the innermost stable
orbit, which is $r_{\mathrm{ISO}}=p_{\mathrm{ISO}}/(1+e)$.

In Figure 1, we can see that the three methods for obtaining the orbit
essentially overlap. The equations (44)-(46) or equations (51) -(53) provide
an approximate orbit, which accumulates errors over time. The approximation
methods used in equations (55) and (58) introduce periodic errors between the
actual orbit and the numerically solved geodesic orbit.

By comparing the obtained orbit with the numerically solved orbit, we can
determine that the approximate equations (44)-(46) or equations (51) -(53) we
used are reliable. For further error analysis, please refer to Appendix A.2.

###  III.2 Adiabatic evolution of frequency and eccentricity

We adopt the following approximation for the Keplerian orbital frequency
$\nu$.

$\displaystyle p=\left(\frac{1-P}{1+P}\right)^{2/3}\frac{X}{(2\pi
M\nu)^{2/3}}.$ (59)

Combining equations (24), (25), (41), and (42), we can obtain the leading-
order equations for the evolution of orbital frequency $\nu$ and eccentricity
$e$.

$\displaystyle\frac{d\nu}{dt}$ $\displaystyle=$
$\displaystyle\frac{\mu^{2}}{M^{3}m}\frac{(1-P)^{6}}{(1+P)^{6}}\frac{(2\pi
M\nu)^{11/3}}{\left(1-e^{2}\right)^{7/2}}\frac{1}{10\pi}\left(96+292e^{2}+37e^{4}\right)+\mathcal{O}((2\pi
M\nu)^{13/3}),$ (60) $\displaystyle\frac{de}{dt}$ $\displaystyle=$
$\displaystyle-\frac{\mu^{2}}{15mM^{2}}\frac{(1-P)^{5}}{(1+P)^{6}}\frac{(2\pi
M\nu)^{8/3}}{\left(1-e^{2}\right)^{5/2}}\frac{1}{e}\left(304e^{2}+121e^{4}-\left(192+280e^{2}-47e^{4}\right)P\right)+\mathcal{O}((2\pi
M\nu)^{10/3}).$ (61)

It should be noted that, as $e\rightarrow 0$,
$\Omega_{\phi}=2\pi\nu_{\phi}\rightarrow 2\pi\nu$, which means
$\nu\approx\nu_{\phi}$. Similarly, as $e\rightarrow 1$,
$\Omega_{r}=2\pi\nu_{r}\rightarrow 2\pi\nu$, which means $\nu\approx\nu_{r}$.
The detailed analysis can be found in Appendix A.1.

If $P$ is a small quantity less than or equals to $(2\pi Mv)^{2/3}$, then the
leading order correction for $P$ is equal to or less than $3.5PN$, the
evolution of $\nu$ and $e$ become:

$\displaystyle\frac{d\nu}{dt}$ $\displaystyle=$
$\displaystyle\frac{\mu^{2}}{M^{3}m}\frac{(1-P)^{6}}{(1+P)^{6}}\frac{(2\pi
M\nu)^{11/3}}{\left(1-e^{2}\right)^{7/2}}\frac{1}{10\pi}\left(96+292e^{2}+37e^{4}\right)$
(62) $\displaystyle+\frac{96m}{10\pi M^{3}}\frac{(2\pi
M\nu)^{13/3}}{\left(1-e^{2}\right)^{9/2}}\left(\frac{1273}{336}-\frac{2561}{224}e^{2}-\frac{3885}{128}e^{4}-\frac{13147}{5376}e^{6}\right)+\mathcal{O}((2\pi
M\nu)^{5}),$ $\displaystyle\frac{de}{dt}$ $\displaystyle=$
$\displaystyle-\frac{\mu^{2}}{15mM^{2}}\frac{(1-P)^{5}}{(1+P)^{6}}\frac{(2\pi
M\nu)^{8/3}}{\left(1-e^{2}\right)^{5/2}}e\left(304+121e^{2}-\left(192+280e^{2}-47e^{4}\right)\frac{P}{e^{2}}\right)$
(63) $\displaystyle-\frac{m}{15M^{2}}\frac{(2\pi
M\nu)^{10/3}}{\left(1-e^{2}\right)^{7/2}}e\left(\frac{1}{56}\left(70648-231960e^{2}-106523e^{4}\right)\right)+\mathcal{O}((2\pi
M\nu)^{4}).$

Through Eq. (63), we can see that the leading order effect of the $P$ is
positive. This means that the correction of LQG tends to increase the orbital
eccentricity. As the eccentricity decreases, this effect becomes more
significant. Moreover, when the eccentricity approaches

$\displaystyle
e_{B}=2\sqrt{\frac{-38+35P+\sqrt{1444-1208P+1789P^{2}}}{121+47P}},$ (64)

the leading-order effect of $P$ will reach the same order as that in the
Schwarzschild case. At this point, for the evolution Eq. (63), the precision
will degrade to the $2.5PN$ order.

###  III.3 Waveforms

In this paper, based on equations (55) and (58), we use the quadrupole
waveform [33, 40] as follows:

$\displaystyle h^{jk}=\frac{2}{r}\ddot{\mathcal{Q}}^{jk}.$ (65)

In the previous section, we assumed that the orbit is located in the
equatorial plane, which means that the orbital angular momentum direction is
$(\theta_{L}=0,\phi_{L}=0)$. Combining equations (38), (8), (11), and
performing a Taylor series expansion, the leading-order gravitational waveform
can be expressed as:

$\displaystyle h^{11}$ $\displaystyle=$
$\displaystyle\mathcal{A}_{h}\Big{(}2e^{2}+2e\cos(\psi(t))-5e\cos(\psi(t)-2\phi(t))\Big{.}$
(66)
$\displaystyle\Big{.}-2e^{2}\cos(2\psi(t)-2\phi(t))-4\cos(2\phi(t))-e\cos(\psi(t)+2\phi(t))\Big{)},$
$\displaystyle h^{22}$ $\displaystyle=$
$\displaystyle\mathcal{A}_{h}\Big{(}2e^{2}+2e\cos(\psi(t))+5e\cos(\psi(t)-2\phi(t))\Big{.}$
(67)
$\displaystyle\Big{.}+2e^{2}\cos(2\psi(t)-2\phi(t))+4\cos(2\phi(t))+e\cos(\psi(t)+2\phi(t))\Big{)},$
$\displaystyle h^{12}$ $\displaystyle=$
$\displaystyle\mathcal{A}_{h}\Big{(}5e\sin(\psi(t)-2\phi(t))+2e^{2}\sin(2\psi(t)-2\phi(t))$
(68) $\displaystyle-4\sin(2\phi(t))-e\sin(\psi(t)+2\phi(t))\Big{)},$
$\displaystyle h^{13}$ $\displaystyle=$ $\displaystyle h^{23}=h^{33}=0,$ (69)

where, $\mathcal{A}_{h}={m(1-P)^{2}}/{(2p(1+P)^{2})}$. In the actual
calculation, we computed up to the second order with $p$. The traceless gauge
of the waveform is given by[33, 40]:

$\displaystyle
h_{\mathrm{TT}}^{jk}=\frac{1}{2}\left(\begin{array}[]{ccc}0&0&0\\\
0&h^{\theta_{S}\theta_{S}}-h^{\phi_{S}\phi_{S}}&2h^{\theta_{S}\phi_{S}}\\\
0&2h^{\theta_{S}\phi_{S}}&h^{\phi_{S}\phi_{S}}-h^{\theta_{S}\theta_{S}}\end{array}\right),$
(73)

where

$\displaystyle h^{\theta_{S}\theta_{S}}$ $\displaystyle=$
$\displaystyle\cos^{2}\theta_{S}\left(h^{11}\cos^{2}\phi_{S}+h^{12}\sin
2\phi_{S}+h^{22}\sin^{2}\phi_{S}\right),$ (74) $\displaystyle
h^{\theta_{S}\phi_{S}}$ $\displaystyle=$
$\displaystyle\cos\theta_{S}\left(-\frac{1}{2}h^{11}\sin 2\phi_{S}+h^{12}\cos
2\phi_{S}+\frac{1}{2}h^{22}\sin 2\phi_{S}\right),$ (75) $\displaystyle
h^{\phi_{S}\phi_{S}}$ $\displaystyle=$ $\displaystyle
h^{11}\sin^{2}\phi_{S}-h^{12}\sin 2\phi_{S}+h^{22}\cos^{2}\phi_{S}.$ (76)

The polarized gravitational waveforms correspond to:
$h_{+}(t)=h^{\theta_{S}\theta_{S}}-h^{\phi_{S}\phi_{S}}$ and
$h_{\times}(t)=2h^{\theta_{S}\phi_{S}}$.

##  IV Results of Waveforms

###  IV.1 Waveforms under eS method

In calculations, we primarily used the ordinary differential equations (ODEs)
given by Eqs. (51)-(53) up to $(11,6)$ orders together with (62) and (63).

(a) $e=0.05$

(b) $e=0.1$

(c) $e=0.2$

Figure 2: The evolution of $e$ and $\nu_{\phi}$ for different initial
eccentricities $(e_{0}=0.2,0.1,0.05)$ and an initial frequency of
$\nu_{\phi}=1\mathrm{mHz}$. The different horizontal lines in the background
represent the critical eccentricity $e_{B}$. The red dot indicates the
position of ISO.

As shown in Fig. 2, by numerically solving equations (62) and (63), we can
obtain the evolution of eccentricity $e$ and frequency $\nu_{\phi}$ with time.
In the figure, the red dot represents the orbit reaching ISO, where the ISO is
calculated in Appendix B. It can be observed that with increasing $P$, both
the decay rate of $e$ and the growth rate of $\nu_{\phi}$ decrease.
Additionally, as shown in Fig. 2(a), when eccentricity $e$ is smaller than the
limit $e_{B}$ in Eq. (64), the leading order term in Eq. (63) becomes
positive. When the leading order dominates, the eccentricity gradually
increases during evolution. It is important to note that when the eccentricity
evolves close to or below the dashed line, the accuracy of eccentricity
evolution changes from $3.5PN$ to $2.5PN$. On the other hand, with increasing
$P$, The growth rate of $\nu_{\phi}$ slows down. This implies that the quantum
effect of the inspiraling binary plays a role similar to repulsion(reducing
the impact of massive black holes).

(a) $e_{0}=0.05$

(b) $P=1/1000$

Figure 3: The accumulated orbital phase difference
$\Delta\Phi=\Phi(P)-\Phi(P=0)$ under different values of $P$ or $e_{0}$ with
$M=10^{6}M_{\odot}$ and $m=10M_{\odot}$.

Fig. 3(a) shows the evolution of accumulated phase difference
$\Delta\Phi=\Phi(P)-\Phi(P=0)$ with time for various values of initial
eccentricities $e_{0}$ and $P$. It can be seen that with $e_{0}=0.1$, as $P$
increases, $\Delta\Phi$ increases faster. When $P=1/1000$, $\Delta\Phi$ grows
more rapidly with increasing eccentricity, although the change is not
significant.

After performing the above calculations, we obtained the corresponding
evolutions of $(\Psi,\nu_{\phi},e_{0},\gamma)$. Combining this with quadrupole
waveform, we can obtain the gravitational waveforms for different values of
$e_{0}$ and $P$, as shown in Fig. 4.

(a) $e_{0}=0.1$ , $P=1/10000$ (b) $e_{0}=0.2$ , $P=1/10000$ (c) $e_{0}=0.1$ ,
$P=1/1000$ (d) $e_{0}=0.2$ , $P=1/1000$

Figure 4: The gravitational waveforms under $e_{0}=\\{1/10,1/5\\}$ and
$P=\\{1/10000,1/1000\\}$. The solid black line represents the waveform with
$P=0$. The left side of each plot shows the initial waveforms, while the right
side shows the ones after a period of evolution.

In Fig. 4, it can be observed that when $e_{0}=0.1$ and $P=1/10000$, it takes
around 100 days for the gravitational waveforms to show a significant
difference. On the other hand, when $P=1/1000$, the difference becomes
apparent in just 30 days. This is consistent with the conclusions in Fig.
3(a). Similarly, when $P$ is fixed, the variation in eccentricity does not
have a significant impact at which the gravitational waveforms exhibit
noticeable phase differences. Moreover, each vertical line in the Fig. 4
represents the time interval $T_{\phi}=1/\nu_{\phi}(t=0)$. We can find the GW
frequency $\nu_{GW}(t=0)\approx 2\nu_{\phi}(t=0)$, which is consistent with
Ref. [34].

###  IV.2 Waveforms under eL method

The gravitational waveforms obtained through the large eccentricity method,
described by equations (44)-(46) up to $(10,5)$ orders with (62) and (63), can
be used to generate waveforms for different values of $e$ and $P$.

(a) $e_{0}=0.6$ , $P=1/10000$ (b) $e_{0}=0.7$ , $P=1/10000$ (c) $e_{0}=0.6$ ,
$P=1/1000$ (d) $e_{0}=0.7$ , $P=1/1000$

Figure 5: The gravitational waveforms under different eccentricities
$e_{0}=\\{0.6,0.7\\}$ and different loop quantum parameters
$P=\\{1/10000,1/1000\\}$. The left side of each plot represents the initial
waveforms, while the right side displays the ones after a period of evolution.

Waveforms are illustrated in Fig. 5. It can be observed that when $e_{0}$ is
fixed, a larger $P$ leads to a shorter time for significant differences to
appear in the gravitational waveforms. Similarly, when $P$ is fixed, a larger
$e_{0}$ results in a shorter time for noticeable differences in the waveforms.
This trend is consistent with the evolution of $\Delta\Phi$ observed at
smaller eccentricities.

##  V Parameter estimation

For a three arm spacebased GW detector, different sets of Michelson
interferometers can be constructed by choosing different optical paths. The
strain amplitude of the detector can be described by the unified expression:

$\displaystyle h(t)$ $\displaystyle=$
$\displaystyle\frac{1}{D}\frac{\sqrt{3}}{2}\left(F^{+}(t)h_{+}(t)+F^{\times}(t)h_{\times}(t)\right),$
(77)

where the distance $D$ represents the separation between the observer and the
GW source, while $F^{+}(t)$ and $F^{\times}(t)$ denote the antenna functions.
These functions depend on both the direction of the source
$(\theta_{S},\phi_{S})$ and the orbital angular momentum direction
$(\theta_{L}=0,\phi_{L}=0)$. The specific mathematical expressions can be
found in references [41, 32], where it is stated that these antenna functions
have a period of one year ($T_{ap}=1\mathrm{year}$), with initial parameter
angles set to $\bar{\alpha}_{0}=\bar{\phi}_{0}=0$. Therefore, the GW signal is
completely determined by the parameters
$\lambda=(M,m=10M_{\odot},P,\Phi_{0}\approx 0,e_{0},\nu_{0}(or\
t_{0}),\hat{\gamma}_{0}=0,\theta_{S}=\pi/2,\phi_{S}=\pi/2,\theta_{L}=0,\phi_{L}=0,D=1Gpc)$
where $e_{0}$ is the initial eccentricity of the orbit, and $\nu_{0}$ is the
initial frequency of the orbit, corresponding to the chosen time range of the
data. For example, in our data sampling, we select the data within
$t_{0}=1\mathrm{year}$ before the plunge $p_{\text{plunge}}=p+\delta p$, where
a fixed shift $\delta p=0.1$ is applied. The corresponding $\nu_{0}$ will be
obtained. For the chosen source azimuthal angles
$(\theta_{S}=\pi/2,\phi_{S}=\pi/2)$, this implies that $h_{+}=h^{11}$ and
$h_{\times}=0$. In the subsequent calculations, we mainly consider the varying
parameters $(M,P,e_{0})$, while the remaining parameters are set to the
aforementioned values.

To assess the effects of the varying parameters, the noise weighted inner
product between two templates need to be introduce as [42, 43]:

$\displaystyle\langle h_{e}\mid
h_{m}\rangle=2\int_{0}^{\infty}\frac{h_{e}^{*}(f)h_{m}(f)+h_{e}(f)h_{m}^{*}(f)}{S_{n}(f)}df,$
(78)

where $h_{e}(f)$ and $h_{m}(f)$ are the frequency domain results of the
$h_{e}(t)$ and $h_{m}(t)$ after Discrete Fourier transformation(DFT),
respectively. $*$ denotes complex conjugation. $S_{n}(f)$ is the noise power
spectral density(PSD) of the space-borne GW detector, such as LISA [27, 44].
Note that a Tukey window function with a cosine fraction of $0.01$ is used
before DFT and the sampling frequencies of the GW signals are analyzed in
Appendix C.

$\rho$ represents the signal-to-noise ratio (SNR) of a signal, given by
$\rho=\sqrt{\langle h\mid h\rangle}$. We can define faithfulness as:

$\displaystyle\mathcal{F}\left[P_{e},P_{m}\right]=\frac{\left\langle
h_{P=P_{e}}\mid h_{P=P_{m}}\right\rangle}{\sqrt{\left\langle h_{P=P_{e}}\mid
h_{P=P_{e}}\right\rangle\left\langle h_{P=P_{m}}\mid
h_{P=P_{m}}\right\rangle}}.$ (79)

The corresponding mismatch is defined as:

$\displaystyle\mathcal{O}\left[P_{e},P_{m}\right]=1-\mathcal{F}\left[P_{e},P_{m}\right].$
(80)

When the mismatch $\mathcal{O}$ is larger than $\sim\mathcal{D}/(2\rho^{2})$
($\mathcal{D}$ represents the parameter space of the model, which is
approximately $10$), it indicates significant differences between two GW
signals and they cannot faithfully describe each other. For a signal with an
SNR of 30, this requires a mismatch $\mathcal{O}\gtrsim 0.012$
($\log_{10}\mathcal{O}\lesssim-1.92$) [42, 43].

(a) $e_{0}=0.05$

(b) $e_{0}=0.1$

Figure 6:  Mismatch of LISA under different values of $M$ and $P$. The red
line represents a boundary where mismatch $\mathcal{O}=0.012$
($\log_{10}\mathcal{O}=-1.92$).

Fig. 6 shows the constraint capability of the LISA detector on $P$ for
eccentricities $e_{0}={0.05,0.1}$ and different central black hole masses.
From these figures, it can be seen that for one year of GW detection, the
constraint capability of LISA on $P$ increases as the mass decreases within
the range of $M=1\times 10^{5}M_{\odot}\sim 1\times 10^{7}M_{\odot}$. This is
because as $M$ decreases from $1\times 10^{7}M_{\odot}$ to $1\times
10^{5}M_{\odot}$, the frequency $\nu_{\phi}$ around ISO goes from about $\sim
0.2\mathrm{mHz}$ to $\sim 20\mathrm{mHz}$, which is close to the LISA
detector’s most sensitive frequency $\sim 10\mathrm{mHz}$. At the same time,
the energy and angular momentum flux(fluxes?) increase as $M$ decreases.

(a) $M=10^{5}M_{\odot}$

(b) $M=10^{6}M_{\odot},$

Figure 7: Mismatch as a function of $P$ with different values of $e$ for
LISA.The Black line represents a boundary where mismatch $\mathcal{O}=0.012$

Fig. 7 presents mismatch as a function of $P$ with different values of $e$ for
LISA, in more detail. From Fig. 7(a), it can be seen that for one year of GW
detection with $M=1\times 10^{5}M_{\odot}$, LISA achieves a constraint
capability of $3\times 10^{-8}\sim 5\times 10^{-8}$. When $M=1\times
10^{6}M_{\odot}$, as shown in Figure 7(b), LISA achieves a constraint
capability of $2\times 10^{-7}\sim 3\times 10^{-7}$.

##  VI Conclusion

In this paper, we consider the modified AK waveform in the EMRI system under
the background of self-dual spacetime in loop quantum gravity. First, we
calculate the motion equations of the test body in a self-dual spacetime and
obtain the two orbital frequencies on the equatorial plane in the weak field
limit. Then, combining the energy and angular momentum flux carried by the
gravitational waves under the quadrupole moment approximation, we obtain the
adiabatic orbit of the test body. Using different fundamental frequencies
$\nu_{r}$ or $\nu_{\phi}$, we get two methods applied to different values of
$e$.

Through numerical calculations, we found that slight perturbations of $P$ will
be amplified after one year of orbit evolution, and the final waveform results
are significantly different from those without the perturbation of $P$. Among
them, the smaller the mass $M$ of the central black hole, the stronger the
constraint ability of LISA on the loop quantum parameters, in the range of
$M\in[1\times 10^{5}M_{\odot},1\times 10^{7}M_{\odot}]$. Specially, the
ability to constrain $P$ will be approached the upper limit of about $3\times
10^{-8}\sim 5\times 10^{-8}$ with $M=1\times 10^{5}M_{\odot}$ for small
changes in $e$ near zero. Overall, compared with the best constraint from the
Cassini experiment in the solar system, $P\leq 5.5\times 10^{-6}$ [24], the
constraint capability will be improved by 1 to 2 orders of magnitude.

We only consider the self-dual spacetime in this paper, however, the methods
described in the current paper can also be generalised to other loop quantum
black hole models [7, 14, 45, 46, 12] and obtain the constraint on the
relevant quantum parameter. We would like to leave these interesting topics
for future study.

###### Acknowledgements.

This work is supported by National Natural Science Foundation of China with
No. 12275087.

##  Appendix A Error analysis

###  A.1 Error of Frequency

Note that when $P=0$, equations (29) and (30) reduce to the Schwarzschild
case. We obtain:

$\displaystyle\Omega_{\phi}=\frac{X^{3/2}}{Mp^{3/2}}+\left(\frac{3X^{3/2}}{M}-\frac{3X^{5/2}}{M}\right)\frac{1}{p^{5/2}}+\left(\frac{57X^{3/2}}{4M}-\frac{63X^{5/2}}{4M}-\frac{15X^{3}}{2M}+\frac{9X^{7/2}}{M}\right)\frac{1}{p^{7/2}}$
$\displaystyle+\left(\frac{315X^{3/2}}{4M}-\frac{96X^{5/2}}{M}-\frac{60X^{3}}{M}+\frac{237X^{7/2}}{4M}+\frac{45X^{4}}{M}-\frac{27X^{9/2}}{M}\right)\frac{1}{p^{9/2}}+\mathcal{O}(\frac{1}{p^{11/2}}),$
(81)
$\displaystyle\Omega_{r}=\frac{X^{3/2}}{Mp^{3/2}}-\frac{3X^{5/2}}{Mp^{5/2}}+\left(-\frac{6X^{5/2}}{M}-\frac{15X^{3}}{2M}+\frac{9X^{7/2}}{M}\right)\frac{1}{p^{7/2}}$
$\displaystyle+\left(-\frac{24X^{5/2}}{M}-\frac{75X^{3}}{2M}+\frac{30X^{7/2}}{M}+\frac{45X^{4}}{M}-\frac{27X^{9/2}}{M}\right)\frac{1}{p^{9/2}}+\mathcal{O}(\frac{1}{p^{11/2}}).$
(82)

Interestingly, even if we calculate $\Omega_{\phi}$ and $\Omega_{r}$ up to
$\mathcal{O}(p^{-23/2})$, we can still obtain the following approximation:

$\displaystyle\Omega_{\phi}\rightarrow\frac{X^{3/2}}{Mp^{3/2}},\quad When\quad
e\rightarrow 0(X\rightarrow 1);$ (83)
$\displaystyle\Omega_{r}\rightarrow\frac{X^{3/2}}{Mp^{3/2}},\quad When\quad
e\rightarrow 1(X\rightarrow 0).$ (84)

(a) $e=0.1$ , $M=10^{6}M_{\odot}$ (b) $e=0.8$ , $M=10^{6}M_{\odot}$

Figure 8:  Relationship between the Kepler frequency
$\nu={X^{3/2}}/{Mp^{3/2}}$, the anomaly frequency $\Omega_{r}/(2\pi)$ and the
“longitude” frequency $\Omega_{\phi}/(2\pi)$ with respect to $p$. Among them,
$\Omega_{r}$ and $\Omega_{\phi}$ are both accurate to $\mathcal{O}(p^{-21/2})$
order.

From the Fig. 8, we can see that when $e\rightarrow 0$, assuming
$\Omega_{\phi}=2\pi\nu_{\phi}\approx\nu={X^{3/2}}/{Mp^{3/2}}$ can achieve
better accuracy.

When the parameter $P$ is small, it is only a small correction to the Eqs.
(81)-(82) , as shown below:

$\displaystyle\Omega_{\phi}\rightarrow\frac{(1-P)X^{3/2}}{(1+P)Mp^{3/2}}+O(P),\quad
When\quad e\rightarrow 0(X\rightarrow 1);$ (85)
$\displaystyle\Omega_{r}\rightarrow\frac{(1-P)X^{3/2}}{(1+P)Mp^{3/2}}+O(P),\quad
When\quad e\rightarrow 1(X\rightarrow 0).$ (86)

Therefore, we can assume the leading order to be

$\displaystyle\Omega_{\phi}=2\pi\nu_{\phi}\rightarrow 2\pi\nu,\quad When\quad
e\rightarrow 0(X\rightarrow 1);$ (87)
$\displaystyle\Omega_{r}=2\pi\nu_{r}\rightarrow 2\pi\nu,\quad When\quad
e\rightarrow 1(X\rightarrow 0).$ (88)

In addition, when $e\rightarrow 0$, $\nu_{r}(p)$ is not a monotonic function
outside the ISO, causing problems with the series expansion of its inverse
function $p(\nu_{r})$, which is a multivalued one. When $e\rightarrow 1$, the
inverse series expansion $p(\nu_{\phi})$ in Eq. (49) of $\nu_{\phi}(p)$
converges very slowly near the ISO.

In summary, we have $\nu_{\phi}\approx\nu$ when $e\rightarrow 0$ and
$\nu_{r}\approx\nu$ while $e\rightarrow 1$.

###  A.2 Error of Geodesic orbit

In the AK equation, when the energy flux and angular momentum flux tend to
zero, the accumulated error of geodesic orbit is mostly concentrated in the
accumulated error of $\hat{\gamma}$. Therefore, the error of the orbital
approximation can be determine by Eqs. (44)-(46) or Eqs. (51)-(53) together
with Eqs. (16)-(19).

Now, let us consider the case when

$\\{e=0.1(or\ 0.8),\nu_{\phi}=0.2mHz,M=10^{6}M_{\odot},P=1/1000\\}$

Thus, $\dot{\hat{\gamma}}$ can be expanded in a series using $(2\pi
M\nu_{\phi})^{2/3}$(or $(2\pi M\nu_{r})^{2/3}$), and the effects of different
orders in the expansion are shown in Fig. 9 and 10:

(a) $e=0.1$ (b) $e=0.8$

Figure 9: Influence of different order under eS method.

(a) $e=0.1$ (b) $e=0.8$

Figure 10: Influence of different order under eL method.

The black dotted line represents the magnitude of each order in the series
expansion of $\dot{\hat{\gamma}}$. The red square dotted line represents the
magnitude of each order caused by the LQG effect. It can be seen that both
methods achieve high accuracy. The convergence rate of the eS method is faster
overall, while the eL one is more stable.

(a) $e=0.1$ and $M=10^{6}M_{\odot}$

(b) $e=0.8$ and $M=10^{6}M_{\odot}$

Figure 11:  Error of $\hat{\gamma}$ for 1 year.

In Fig. 11, the difference between the precessional angle ,
$\gamma(t)=\phi(t)-\psi(t)$, obtained by numerically solving Eqs. (16)-(19)
and the approximate one in the AK method, $\gamma_{Ap}=\phi_{Ap}-\psi_{Ap}$
(see equations 55 and 58), is shown as $\delta\hat{\gamma}$.

It can be seen from Fig. 11(a) that when $e\rightarrow 0$ such as $e=0.1$,
after one year of evolution, the value of $\delta\hat{\gamma}$ is still much
smaller than $1\ \mathrm{rad}$, and the eS method is better than the eL one.
when $e\rightarrow 1$ such as $e=0.8$, the $\delta\hat{\gamma}$ from eL method
increases rapidly, which is due to the slow convergence as shown in Fig.
10(b). For example, the accuracy at the 10th order is approximately $5\times
10^{-7}$, which means that the error $\delta\hat{\gamma}$ after one year is
approximately $5\times 10^{-7}(\mathrm{year}/\mathrm{s})\
\mathrm{rad}=15\mathrm{rad}$, consistent with Fig. 11(b). This further
confirms the previous conclusion. Note that the error is not a line but is
caused by the periodic error of $\gamma_{Ap}$. In conclusion, when the initial
eccentricity $e$ is relatively small, we apply the eS method, using
$\nu_{\phi}$ as the fundamental frequency, for the calculation and analysis of
gravitational waveforms.

##  Appendix B Innermost stable orbit

Note that $a_{0}$ appears in the higher order terms and is a very small
quantity that can be set to zero. To find the ISO, we can make
$d((dr/d\tau)^{2})/dr=0$ at the pericenter $r_{\text{peri}}$[47, 48].
Combining equations (9), (20), and (21), we obtain the separatrix equation

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
512e(1+e)^{3}\bigl{(}3+e(-2+3e)\bigr{)}P^{7}-32e(1+e)^{2}P^{5}(1+P)^{2}\mathcal{S}_{1}p$
(89)
$\displaystyle+16e(1+e)P^{3}(1+P)^{4}\mathcal{S}_{2}p^{2}+8e(1+e)P^{2}(1+P)^{6}\mathcal{S}_{3}p^{3}$
$\displaystyle+4e(1+e)(-1+P)^{2}P(1+P)^{8}\Bigl{(}15+8e+e^{2}-2\bigl{(}5+e(2+e)\bigr{)}P+(3+e)(5+e)P^{2}\Bigr{)}p^{4}$
$\displaystyle+2e(1+e)(-1+P)^{2}(1+P)^{10}\bigl{(}3+e-(5+e)P+(3+e)P^{2}\bigr{)}p^{5}$
$\displaystyle-e(1+e)(-1+P)^{2}(1+P)^{12}p^{6},$

where

$\displaystyle\mathcal{S}_{1}$ $\displaystyle=$ $\displaystyle
37+8e^{3}(-1+P)^{2}+e^{4}(-1+P)^{2}-16eP+P(-50+37P)+6e^{2}\bigl{(}3+P(-2+3P)\bigr{)},$
(90) $\displaystyle\mathcal{S}_{2}$ $\displaystyle=$ $\displaystyle
15+e^{4}(-1+P)^{2}\bigl{(}1+(-1+P)P\bigr{)}$ (91)
$\displaystyle+2e^{3}(-1+P)^{2}\bigl{(}3+P(-2+3P)\bigr{)}-6e^{2}P\bigl{(}5+P(-6+5P)\bigr{)}$
$\displaystyle+P\Bigl{(}-71+P\bigl{(}88+P(-71+15P)\bigr{)}\Bigr{)}+2e\biggl{(}5+P\Bigl{(}-36+P\bigl{(}38+P(-36+5P)\bigr{)}\Bigr{)}\biggr{)},$
$\displaystyle\mathcal{S}_{3}$ $\displaystyle=$ $\displaystyle
27+e^{3}(-1+P)^{4}+e^{2}(-1+P)^{2}\bigl{(}1+(-14+P)P\bigr{)}+P\Bigl{(}-64+P\bigl{(}98+P(-64+27P)\bigr{)}\Bigr{)}$
(92)
$\displaystyle+e\biggl{(}19+P\Bigl{(}-44+P\bigl{(}74+P(-44+19P)\bigr{)}\Bigr{)}\biggr{)},$

Assuming $P$ is small and accurate up to second order of $P$, the approximate
solution to the equation is obtained as

$\displaystyle
p=6+2e-(12+4e)P+\frac{6\bigl{(}35+35e+9e^{2}+e^{3}\bigr{)}}{(3+e)^{2}}P^{2}.$
(93)

Figure 12: For different values of $e$, the $p$ of ISO respect to $P$, where
the thick line is the numerical solution of Eq. (89), while the thin lines are
the approximate solutions of Eq. (93).

From the Fig. 12, when P is less than 0.01, the approximate solution can be
obtained with high precision. For example, when the parameters are
$\\{P=1/100,e=0.2\\}$, we take the approximate $p=6.27448$, and the
corresponding numerical solution to Eq. (89) is $p=6.27445$, with a relative
error of approximately $5.8\times 10^{-6}$.

##  Appendix C Discrete Fourier transform

(a) $h(f)$ when $e\rightarrow 0$, initial frequency
$\nu_{\phi}=1\mathrm{mHz}$. The vertical line is
$\nu=2\nu_{\phi}+\mathbf{n}\nu_{r},\mathbf{n}=0,\pm 1,\pm 2,...$ for $e=0.05$
(b) $h(f)$ when $e\rightarrow 1$, initial frequency
$\nu_{\phi}=0.2\mathrm{mHz}$. The vertical line is
$\nu=\mathbf{n}\nu_{\phi},\mathbf{n}=0,\pm 1,\pm 2,...$ for $e=0.6$

Figure 13:  $h(f)$ from DFT of $h(t)$ for $e={0.05,0.1,0.2}$ with $P=0.001$.

Fig. 13 shows $h(f)$ from DFT of $h(t)$. It can be seen that as the
eccentricity decreases, the contribution of the GW frequency
$\nu=2\nu_{\phi}+\mathbf{n}\nu_{r},\mathbf{n}=\pm 1,\pm 2,...$ becomes more
and more significant. In particular, the contribution of $\nu=2\nu_{\phi}$
reaches its maximum value. This is consistent with the fact that as
$e\rightarrow 0$, the GW frequency approaches $2\nu_{\phi}$. Moreover, for
$e=0.1$, the contributions of frequencies $\nu=2\nu_{\phi}+2\nu_{r}$ cannot be
neglected. When the signal is sampled, the sampling frequency can be selected
as $f_{Ny}=2\nu_{\phi}+2\nu_{r}<4\nu_{\phi}$.

For example, in the case of $M=10^{6}M_{\odot}$, with an initial eccentricity
of $e=0.2$, after 1 year of evolution to the innermost stable orbit, the
eccentricity is still approximately $e\approx 0.1$ and corresponds to
$\nu_{\phi}\approx 2\mathrm{mHz}$ and $\nu_{r}\approx 0.5\mathrm{mHz}$. At
this point, the sampling frequency $f_{S}=2.5(2\nu_{\phi}+2\nu_{r})\approx
12.5\mathrm{mHz}$.

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### Index

  * ’t Hooft limit §C.2
  * (A)SD YM
    * —connection §1.2, §1.4.2, §3.3
    * —equation §1.2, §1.3.2
  * $\mathsf{A}$-function §5.2.1, §5.2.1
  * $\mathsf{A}$-operator §6.2.6
    * —charge §7.6.1
    * elliptic— §8.3.3
    * fractional— §7.4.3
  * Adams operation §1.8.4, §2.2.3, §2.4.2
  * ADHM construction §1.4, §2.3.1
    * ALE space §2.3.2
    * supergroup §3.3
  * AGT relation Chapter 0, §6.2.2, §8.2
  * ALE space §2.3.2, §4.5.3
  * Baker–Akhiezer function §C.4.1, §6.2.5
  * Baker–Campbell–Hausdorff formula §6.2.1
  * Bethe equation §5.8.1, §7.4.10
  * Bianchi identity §1.2
  * BPS spectrum §4.1.3, §4.2.5
  * BPS/CFT correspondence §6.2.2
  * Cartan matrix
    * classical— §2.2.3
    * full— §2.2.3
    * half— §2.2.3
    * inverse of— §6.2.3
    * symmetrized— §2.4.2
  * Coulomb moduli §C.3.1, §4.1.1, §4.2.3, §4.2.3, §4.4, §5.4.3, §5.6.1, §5.8.2
    * dual— §4.1.2
  * $\mathcal{D}$-module §5.7.1
  * delta function §C.2.1
    * multiplicative— §1.9.5
  * doubling trick §8.1.1
  * dual Dirac operator §1.4.2, §3.3
  * effective twisted superpotential §5.3.2, §5.6.1
  * equivariant
    * —action §1.6.3, §1.7.2, §1.9.5, §2.1.2, §2.3.2
    * —cohomology §1.6.1, §1.7
    * —form §1.6.1, §1.7
    * —index §1.8, §2.2.1, §2.4.2, §3.4.1
    * —localization §1.6.2, §1.7.2, §2.3.2, §3.4
  * equivariantly compact §1.6, §1.6.2
  * framing §1.4
    * —bundle §1.8.2, §2.1.1, §3.4
  * fugacity (instanton) §1.3.1, §1.7.2, §2.2.1, §4.2.2
  * gauge origami §2.2.2, §2.3.3, §5.5
  * holomorphic deformation §6.1
  * instanton bundle §1.8.2, §2.1.1, §3.4
  * iWeyl reflection §5.2.1, §6.2.6, §7.4.4
  * Lax matrix §4.3.1, §4.3.2, §4.3.4, §5.7.1, §5.7.4, §7.4.10
  * LMNS formula §1.7.2, §1.9.5
    * —with flavor §1.7.3
    * fractional quiver gauge theory §2.4.2
    * gauge origami §5.5.1
    * quiver gauge theory §2.2.2, §7.6.1
    * supergroup §3.4.3
  * loop equation §C.5.1
  * Mathai–Quillen formalism §1.7
  * matter bundle §1.8.6, §2.1.1
  * McKay correspondence §2.3.2
  * moduli space of vacua
    * Coulomb branch §4.1.1, §4.1.2, §4.5.4, §4.6.1
    * Higgs branch §4.1.1, §4.5.4
  * normal ordering §6.2.1
  * operator product expansion (OPE) §6.2.1
    * $\mathsf{A}$ and $\mathsf{A}$ §6.2.6
    * $\mathsf{A}$ and $S$ §6.3
    * $\mathsf{A}$ and $\mathsf{V}$ §6.2.6, §8.3.3
    * $\mathsf{A}$ and $\mathsf{Y}$ §6.2.6
    * $S$ and $S$ §6.2.1, §8.1.2
    * $S$ and $\mathsf{V}$ §6.2.3, §8.3.1
    * $S$ and $\mathsf{Y}$ §6.2.5, §7.4.2, §8.3.2
    * $\mathsf{T}$ and $\mathsf{T}$ ($A_{1}$) §7.3.1
    * $\mathsf{T}$ and $\mathsf{T}$ ($A_{2}$) §7.3.2
    * $\mathsf{T}$ and $\mathsf{T}$ ($BC_{2}$) §7.4.6
    * $\mathsf{T}$ and $\mathsf{T}$ (elliptic $A_{1}$) §8.4.1
    * $\mathsf{T}$ and $\mathsf{T}$ (elliptic $A_{2}$) §8.4.2
  * operator-state correspondence §6.2, §8.1.3
  * pit condition §1.9.2
  * potential term §C.1, §6.1, §6.2.1
  * prepotential §C.3.1, §4.1.2, §5.3.1, §5.6.1, §6.1
  * $q$-character §5.3.2, §5.4.1, §5.7.1, §5.7.2, §5.7.3, §5.7.4, §7.4.10
  * $\mathsf{Q}$-function §5.7.1, §5.8.2
  * $qq$-character §5.2.1, §5.5.2, §7.6.2
    * $A_{1}$ §5.4.1, §7.3.1, §8.4.1
    * $A_{2}$ §5.4.2, §7.3.2, §8.4.2
    * $A_{p}$ §7.3.3
    * $B_{3}$ §7.4.7
    * $BC_{2}$ §7.4.6
    * $C_{3}$ §7.4.8
    * $D_{4}$ §7.3.4
    * $G_{2}$ §7.4.9
    * $\widehat{A}_{0}$ §5.4.3, §7.5.1
  * quantum curve §C.4, §5.7.1, §5.7.2, §5.7.3, §5.7.4, §7.4.10
  * quiver Chapter 2
    * —variety §2.3, §2.3.2, §5.5.2, §7.6
    * —W-algebra Chapter 7
    * fractional— §2.4
    * fractional—W-algebra §7.4
  * radial ordering §6.2.1, §6.2.4, §8.1.3
  * resolvent §C.2, §C.2.2, §C.4
  * root of Higgs branch §1.9.2, §4.5.4, §5.8.2
  * S-duality §4.6.1, §4.6.1
  * $\mathscr{S}$-function §1.7.2, §1.9.5, §2.2.2, §2.4.2
  * saddle point analysis §C.2, §5.3.1, §5.8.1, §7.4.10
  * screening current §C.5.5, §6.2.1
    * elliptic— §8.1.2
    * fractional— §7.4.1
  * Seiberg–Witten curve
    * 5d theory §4.6.1
    * 6d theory §4.6.2
    * $A_{2}$ quiver §4.3.2
    * $A_{3}$ quiver §4.3.3
    * SQCD §4.2.5
    * SU(2) §4.2.3
    * SU($n$) §4.2.4
    * supergroup §4.4
  * stability condition §1.5.1, §1.6.3, §2.3, §3.3.2
  * $\mathsf{T}$-operator, see $qq$-character
  * T-system §5.4.1
  * Taub–NUT space §4.5.3
  * time variable §6.1, §8.2
  * topological twist §1.3.2
  * TQ-relation
    * Toda chain §5.7.1
    * XXX spin chain §5.7.2, §5.8.2
  * Uhlenbeck compactification §1.5.1
  * $\mathsf{V}$-operator §6.2.3
    * elliptic— §8.3.1
  * vacuum expectation value (vev) §4.1.1
  * Vandermonde determinant §C.1.1
  * Virasoro constraint §C.5.2
  * $\mathsf{Y}$-function §5.1.1
  * $\mathsf{Y}$-operator §6.2.5
    * elliptic— §8.3.2
    * fractional— §7.4.2
  * Yang–Mills action §1.1
    * supergroup— §3.1.2
  * $Z$-state §C.5.5, §6.2