# Detection of scalar fields by Extreme Mass Ratio Inspirals with Kerr black
hole

Hong Guo<EMAIL_ADDRESS>Center for Gravitation and Cosmology, College of
Physical Science and Technology, Yangzhou University, Yangzhou 225009, China
Shanghai Frontier Research Center for Gravitational Wave Detection, Shanghai
Jiao Tong University, Shanghai 200240, China Yunqi Liu<EMAIL_ADDRESS>(corresponding author) Center for Gravitation and Cosmology, College of
Physical Science and Technology, Yangzhou University, Yangzhou 225009, China
School of Aeronautics and Astronautics, Shanghai Jiao Tong University,
Shanghai 200240, China Chao Zhang<EMAIL_ADDRESS>School of Physics,
Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
Yungui Gong  0000-0001-5065-2259<EMAIL_ADDRESS>School of Physics,
Huazhong University of Science and Technology, Wuhan, Hubei 430074, China Wei-
Liang Qian<EMAIL_ADDRESS>Escola de Engenharia de Lorena, Universidade de São
Paulo, 12602-810, Lorena, SP, Brazil Faculdade de Engenharia de Guaratinguetá,
Universidade Estadual Paulista, 12516-410, Guaratinguetá, SP, Brazil Center
for Gravitation and Cosmology, College of Physical Science and Technology,
Yangzhou University, Yangzhou 225009, China Rui-Hong Yue<EMAIL_ADDRESS>Center for Gravitation and Cosmology, College of Physical Science and
Technology, Yangzhou University, Yangzhou 225009, China

###### Abstract

We study extreme mass ratio inspirals occurring in modified gravity, for which
the system is modeled by a small compact object with scalar charge spiraling
into a supermassive Kerr black hole. Besides the tensorial gravitational waves
arising from the metric perturbations, radiations are also induced by the
scalar field. The relevant metric and scalar perturbations are triggered by
the orbital motion of the small object, which give rise to a system of
inhomogeneous differential equations under the adiabatic approximation. Such a
system of equations is then solved numerically using Green’s function
furnished by the solutions of the corresponding homogeneous equations. To
explore the present scenario from an observational perspective, we investigate
how the pertinent observables are dependent on specific spacetime
configurations. In this regard, the energy fluxes and the gravitational wave
dephasing accumulated during the process are evaluated, as functions of the
scalar charge, mass ratio, and spin of the central supermassive black hole. In
particular, the presence of additional scalar emission leads to a more
significant rate of overall energy loss, which, in turn, decreases the total
number of orbital cycles before the small object plunges into the central
black hole. Moreover, for a central black hole with a higher spin, the
imprints of the scalar charge on the resultant gravitational radiations are
found to be more significant, which indicates the possibility of detecting the
scalar charge.

##  I Introduction

Since the first gravitational wave (GW) detection in 2015 [1], the Laser
Interferometer Gravitational Wave Observatory (LIGO) Scientific Collaboration
[2, 3], the Virgo Collaboration [4] and the Kamioka Gravitational Wave
Detector (KAGRA) Collaboration [5, 6] have detected tens of coalescing events
of binary systems consisting of binary black holes, binary neutron stars, and
neutron-star black hole (BH) binaries [7, 8, 9]. GW detections provide a new
window to measure the properties of compact binaries and test new theories of
gravity in the strong-field and nonlinear regimes [10, 11, 12, 13, 14, 15, 16,
17, 18, 19]. The success of the ground-based GW projects has raised further
confidence and expectation in space-based GW detectors such as the Laser
Interferometer Space Antenna (LISA) [20, 21], TianQin [22], Taiji [23], and
DECIGO [24]. A recent review on the concepts and status of space-borne
gravitational wave detection can be found in [25].

Extreme mass ratio inspirals (EMRIs) are among the most promising sources for
space-based GW detectors [26]. An EMRI occurs when a stellar-mass object (the
secondary) is trapped in a sufficiently tight orbit of a supermassive BH (the
primary). Due to gravitational radiation, the orbital energy decays, and
subsequently, the secondary body will spiral until it eventually plunges into
the primary BH. Typically, the timescale of the inspiral is much longer than a
single orbital period, and the entire process lasts for tens to hundreds of
years. As a result, although the instantaneous strength of the radiation is
less significant when compared with binary merger events of supermassive
objects, the accumulated signal can possibly be extracted using matched
filtering. Also, the emitted GW is expected to locate in the sensitive band of
space-based GW detector like LISA [20].

When calculating the energy loss of the orbital energy due to gravitational
radiation absorbed by the primary BH or escaping to the spatial infinity, one
needs to sum up the contributions from all possible radiation channels. In
general relativity (GR), astrophysical binary systems only emit gravitational
waves with tensor polarizations, and the lowest radiative multipole moment is
the quadrupole moment. However, in alternative theories of gravity, additional
emission channels may exist. For example, in Brans-Dicke theory as well as
some scalar-tensor theories, the additional scalar field activates the scalar
dipole radiation [27, 28, 29, 30, 31, 32]. Moreover, the dynamics of the
binary system will be modified by the presence of the dipole gravitational
radiation [33, 34, 35, 10], which significantly deviates from that determined
by GR. In fact, even one assumes that the strength of the additional mode is
insignificant compared with the tensorial ones, the extensive duration of the
process is subjected to accumulated energy loss that eventually becomes
observable. The latter may demonstrate themselves in terms of imprint on the
orbital dynamics, as well as the GW frequency [27, 36, 37]. Therefore, one may
discriminate between different theories of gravity regarding the information
on the additional degree of freedom extracted from empirical observations of
EMRIs. The present study is focused on the EMRI signals in the modified
gravity, where an additional scalar field is non-minimally coupled to the
metric.

For most EMRIs, the secondary object can be considered a test particle that
moves in a fixed background generated by the central supermassive BH.
Therefore, the test particle follows a geodesic of the background on each time
slice. The geodesic parameters, namely, the particle’s orbital energy and
angular momentum, will change adiabatically due to the gravitational
radiation. In an alternative theory of gravity, the presence of an additional
scalar degree of freedom, together with the metric fields governing the
gravitational interaction, may violate the strong equivalence principle. To be
specific, an object’s internal structure may depend on the local value of the
scalar field. Therefore, as the object changes its location, its internal
structure varies, which, in turn, affects the object’s motion. Eardley [33]
first proposed an effective action to explore such an effect in Brans-Dicke’s
theory by assuming that the mass of a compact object is a general function of
the scalar field. In the so-called “skeletonized approach” [38, 39, 40, 41], a
compact object is treated as a point particle with the mass $m(\phi)$, and the
action reads

$S_{\rm p}=-\int m(\phi)ds=-\int m(\phi)\sqrt{g_{\mu\nu}\frac{dy_{\rm
p}^{\mu}}{d\lambda}\frac{dy_{\rm p}^{\nu}}{d\lambda}}d\lambda\,,$ (1)

where $s$ is the proper time of the particle along the worldline $y_{\rm
p}^{\mu}(\lambda)$ at a given coordinate, $\lambda$ is an affine parameter.
The effective action (1) has been extensively employed to investigate the
scalar dipole emission in alternative theories of gravity [38, 28, 35, 42, 32,
43, 44, 45]. Furthermore, it has also been applied to the case of binaries
consisting of hairy BHs [46, 47, 48, 49, 50, 51, 52, 53, 54, 14, 12, 13, 55,
56, 18, 57].

In [43], the authors investigated an EMRI in an alternative theory of gravity
in which the central supermassive BH provides an approximate Schwarzschild
background, and the secondary body with a scalar charge is described by the
action (1). It is shown that the scalar field emission has a significant
cumulative effect on the dynamics of the EMRI. In particular, the additional
energy loss depends on the scalar charge of the secondary body and is
relatively insensitive to the specific theory of gravity. Ref. [58] extended
the primary BH to a Kerr one but for chosen values of the mass ratio $10^{-5}$
and dimensionless spin $0.9$. The authors concluded that LISA could detect the
scalar charge of the secondary body in EMRIs.

Considering that astrophysical black holes have different scales and angular
momenta, it is of great interest to further generalize the study of [58] to
the entire parameters space. We are interested in examining in which parameter
range the accumulation of scalar field emission is more significant. In
dynamical Chern-Simons gravity [59], it was disclosed that the correction to
the number of gravitational wave cycles depends on the mass ratio. With
increasing mass ratio, the correction was found to initially rise to a maximum
value and then fall back. Following this line of thought, it is meaningful to
study the effects of various model parameters. In particular, it would be
intriguing to identify whether the impact on the relevant observables is
monotonous or some optimal configuration can be encountered. In this regard,
we aim to explore the entire parameters space, consisting of the BH spin, mass
ratio, and especially the charge of the scalar field. The relevant observable
quantities include the EMRI radiation flux of different modes and the
accumulated dephasing. Among others, we expect to find a combined range of
parameters, for which the accumulated effect of the scalar effect is possibly
more substantial than that found in [58]. It is also noted that instead of one
year, we simulate the EMRI up to four years before the binary merge occurs.
This is because the difference in the energy emission exhibited in the EMRIs
between GR and the modified gravity is mostly insignificant, and therefore, a
more extensive observation period is desirable. Besides, the LISA mission is
proposed for a four-year duration, according to its science plan [20, 60].
Therefore, by running the simulations for its entire lifespan, one might give
a reasonable estimation about the lightest scalar charge that is
experimentally accessible by the LISA project.

This paper is organized as follows. Sec. II introduces the theoretical
framework of the present study. Under appropriate approximations, in Sec. III
we present the equations of motion that govern the perturbations of the metric
and scalar field. The particular profiles of the latter are obtained via
Green’s function, which is constructed using the asymptotical solutions of the
corresponding homogeneous differential equation. We present the numerical
results in Sec. IV, where we study the effects of various model parameters on
the energy flux and dephasing. The last section is devoted to further
discussions and the conclusion.

##  II Setup of the background

The action is [43]

$S[g_{\mu\nu},\phi,\Psi]=S_{0}[g_{\mu\nu},\phi]+\alpha
S_{c}[g_{\mu\nu},\phi]+S_{m}[g_{\mu\nu},\phi,\Psi],$ (2)

where

$S_{0}=\int
d^{4}x\frac{\sqrt{-g}}{16\pi}\left(R-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi\right),$
(3)

$R$ is the Ricci scalar and $\phi$ is a scalar field. The second term $\alpha
S_{c}[g_{\mu\nu},\phi]$ denotes the non-minimal coupling between the metric
$g_{\mu\nu}$ and scalar field $\phi$, and $\alpha$ is a coupling constant with
dimensions $[\alpha]=(mass)^{n}$ which characterizes the deviation from GR.
The action for the matter field $\Psi$ is $S_{m}[g_{\mu\nu},\phi,\Psi]$. For
EMRIs, the matter field is the secondary body with the mass $m(\phi)$ and its
action is Eq. (1).

The equations of motion are obtained by varying the action (2)

$\displaystyle
G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=T_{\mu\nu}^{\text{scal}}+\alpha
T_{\mu\nu}^{c}+T_{\mu\nu}^{p},$ (4)
$\displaystyle\square\phi+\frac{16\pi\alpha}{\sqrt{-g}}\frac{\delta
S_{c}}{\delta\phi}=16\pi\int
m^{\prime}(\phi)\frac{\delta^{(4)}\left(x-y_{p}(\lambda)\right)}{\sqrt{-g}}d\lambda,$
(5)

where $m^{\prime}(\phi)=dm(\phi)/d\phi$, the stress-energy tensor of the
scalar field is
$T_{\mu\nu}^{\text{scal}}=\frac{1}{2}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{4}g_{\mu\nu}(\partial\phi)^{2}$,
$T_{\mu\nu}^{c}$ is the stress-energy of the coupling term and the stress-
energy tensor of the test particle is given by

$T_{\mathrm{p}}^{\alpha\beta}=8\pi\int
m(\phi)\frac{\delta^{(4)}\left(x-y_{p}(\lambda)\right)}{\sqrt{-g}}\frac{dy_{\rm
p}^{\alpha}}{d\lambda}\frac{dy_{\rm p}^{\beta}}{d\lambda}d\lambda.$ (6)

Based on the discussions in Refs. [43, 58], both the first two terms on the
right-hand side of Eq. (4) and the second term on the left-hand side of Eq.
(5) can be neglected. Far away from all the sources, the scalar field
approximates to $\phi=\phi_{0}+\frac{m_{\rm p}d}{r}+...$, where $\phi_{0}$
stands for the background value of the scalar field, $d$ is the dimensionless
scalar charge of the body with mass $m(\phi_{0})=m_{\rm p}$, the ellipsis
$...$ stands for higher-order infinitesimal quantity. The $\phi$-dependent
mass could be expanded as $m(\phi)=m_{\rm p}+m^{\prime}(\phi_{0})\frac{m_{\rm
p}d}{r}+...$. Subsituting this expansion into Eq. (5), one can obtain the
relation $m^{\prime}(\phi_{0})/m(\phi_{0})=-d/4$.

Finally, to the lowest order, the perturbed field equations are

$\displaystyle G_{\mu\nu}$ $\displaystyle=T^{\rm p}_{\mu\nu}=8\pi
m_{p}\int\frac{\delta^{(4)}(x-y_{p}(\lambda))}{\sqrt{-g}}\frac{dy_{\rm
p}^{\alpha}}{d\lambda}\frac{dy_{\rm p}^{\beta}}{d\lambda}d\lambda,$ (7)
$\displaystyle\square\phi$ $\displaystyle=-4\pi d\,m_{\rm
p}\int\frac{\delta^{(4)}(x-y_{\rm p}(\lambda))}{\sqrt{-g}}d\lambda\,.$ (8)

Equation (7) for the metric perturbation takes the same form as that in GR,
which is sourced by the inspiral of a secondary body with mass $m_{\rm p}$.
Thus the GW emission determined by the familiar tensor polarizations is the
same as GR. The additional equation (8) governs the scalar perturbation
sourced by the scalar charge $d$. Since the secondary body accelerates in a
fixed background, it generates scalar radiation. Therefore, besides the
familiar GW emission by tensor modes, there exists scalar radiation in EMRIs.

##  III EMRIs in the Teukolsky framework

We focus on a quasi-circular orbital evolution of the secondary body spiraling
into the supermassive Kerr BH adiabatically on the equatorial plane. The Kerr
BH provides the background on each time slice. The test particle follows a
geodesic in the background, and the geodesic parameters, i.e., the orbital
energy and angular momentum of the particle, change adiabatically due to
gravitational radiation. This section introduces the particle’s orbital
motion, including the essential elements we will use in our following
calculation, derives and solves the perturbation equations in the Teukolsky
framework and calculates the gravitational and scalar energy fluxes. When
numerically solving the perturbation equations, we utilize the codes of BH
Perturbation ToolKit [61, 62].

We write the background Kerr BH spacetime in the Boyer-Lindquist coordinate,

$ds^{2}=-\left(1-\frac{2Mr}{\Sigma}\right)dt^{2}+\frac{\Sigma}{\Delta}dr^{2}-\frac{4Mar\sin^{2}\theta}{\Sigma}dtd\varphi+\Sigma
d\theta^{2}+\frac{\sin^{2}\theta}{\Sigma}\left(\varpi^{4}-a^{2}\Delta\sin^{2}\theta\right)d\varphi^{2},$
(9)

where $\Sigma\equiv r^{2}+a^{2}\cos^{2}\theta$, $\Delta\equiv
r^{2}-2Mr+a^{2}$, $\varpi\equiv\sqrt{r^{2}+a^{2}}$, $M$ is the mass of the
central BH, the mass ratio is $q=m_{\rm p}/M$, $a$ is the spin of the
supermassive BH and satisfies $|a|\leq M$. Without loss of generality, we
assume $a$ is positive and aligned to the z-axis. The inner and outer horizons
of the BH are defined as $r_{\pm}=M^{2}\pm\sqrt{M^{2}-a^{2}}$. And here we
define the tortoise coordinate $dr/dr^{*}=\Delta/(r^{2}+a^{2})\equiv g(r)$.

###  III.1 Orbital motion

Geodesic motion in Kerr spacetime is completely integrable; this leads to
three constants of motion: the specific energy $E$, the angular momentum $L$,
and the Carter constant $Q$. The geodesic equations are

$\displaystyle m_{\rm p}\Sigma_{w}\frac{dt_{w}}{d\tau}$ $\displaystyle=$
$\displaystyle
E\frac{\varpi^{4}}{\Delta}+aL\left(1-\frac{\varpi^{2}}{\Delta}\right)-a^{2}E\sin^{2}\theta,$
(10) $\displaystyle m_{\rm p}\Sigma_{w}\frac{dr_{w}}{d\tau}$ $\displaystyle=$
$\displaystyle\pm\sqrt{V_{r}\left(r_{\rm p}\right)},$ (11) $\displaystyle
m_{\rm p}\Sigma_{w}\frac{d\theta_{w}}{d\tau}$ $\displaystyle=$
$\displaystyle\pm\sqrt{V_{\theta}\left(\theta_{\rm p}\right)},$ (12)
$\displaystyle m_{\rm p}\Sigma_{w}\frac{d\varphi_{w}}{d\tau}$ $\displaystyle=$
$\displaystyle
aE\left(\frac{\varpi^{2}}{\Delta}-1\right)-\frac{a^{2}L}{\Delta}+L\csc^{2}\theta,$
(13)

where the subscript $w$ stands for a function evaluated on the worldline of
the secondary body, the radial and polar potentials are

$\displaystyle V_{r}(r)$
$\displaystyle=\left(E\varpi^{2}-aL\right)^{2}-\Delta\left(r^{2}+\left(L-aE\right)^{2}+Q\right),$
(14) $\displaystyle V_{\theta}(\theta)$
$\displaystyle=Q-L^{2}\cot^{2}\theta-a^{2}\left(1-E^{2}\right)\cos^{2}\theta.$
(15)

In the adiabatic approximation, for a quasi-circular orbit on the equatorial
plane, the coordinates $r$ and $\theta$ are considered as constants, then Eqs.
(10) and (13) are the remaining equations. For this orbit, the conserved
constants $E$ and $L$ are expressed as [63]

$\displaystyle E/m_{\rm p}$ $\displaystyle=$ $\displaystyle\frac{r_{\rm
p}^{3/2}-2Mr_{\rm p}^{1/2}\pm aM^{1/2}}{r_{\rm p}^{3/4}\left(r_{\rm
p}^{3/2}-3Mr_{\rm p}^{1/2}\pm 2aM^{1/2}\right)^{1/2}},$ (16) $\displaystyle
L/m_{\rm p}$ $\displaystyle=$ $\displaystyle\frac{\pm M^{1/2}(r_{\rm p}^{2}\mp
2aM^{1/2}r_{\rm p}^{1/2}+a^{2})}{r_{\rm p}^{3/4}\left(r_{\rm p}^{3/2}-3Mr_{\rm
p}^{1/2}\pm 2aM^{1/2}\right)^{1/2}}.$ (17)

The orbital angular frequency, measured by a distant observer, is given by
[63]

$\Omega\equiv\frac{d\varphi}{dt}=\frac{\pm M^{1/2}}{r_{\rm p}^{3/2}\pm
aM^{1/2}}.$ (18)

In Eqs. (16) and (18) the upper sign refers to the case co-rotating with the
central BH while the lower one refers to the counter-rotating case. In the
following discussion, we consider the co-rotating motion without loss of
generality.

###  III.2 Metric perturbations

We use the Teukolsky formalism to calculate the energy flux of GW emission. At
spatial infinity, the metric perturbation is related to the Newman-Penrose
quantity $\psi_{4}$ as

$\psi_{4}=\frac{1}{2}\frac{\partial^{2}}{\partial t^{2}}(h_{+}-ih_{\times}).$
(19)

The Newman-Penrose variable $\psi_{4}$ can be expanded as

$\psi_{4}=\rho^{4}\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}\int_{-\infty}^{\infty}d\hat{\omega}R_{\ell
m\hat{\omega}}(r)_{-2}S_{\ell
m}^{a\hat{\omega}}(\theta)e^{i(m\varphi-\hat{\omega}\hat{t})},$ (20)

where $\rho=(r-ia\cos\theta)^{-1}$, ${}_{-2}S_{\ell
m}^{a\hat{\omega}}(\theta)$ is the $s=-2$ spin weighted orthonormal spheroidal
harmonics, and its eigenvalue is $\lambda_{G}$. The radial function $R_{\ell
m\hat{\omega}}$ obeys the radial Teukolsky equation

$\Delta^{2}\frac{d}{dr}\left(\frac{1}{\Delta}\frac{dR_{\ell
m\omega}}{dr}\right)-V_{G}(r)R_{\ell m\hat{\omega}}(r)=\mathcal{T}^{T}_{\ell
m\hat{\omega}},$ (21)

where the source term $\mathcal{T}^{T}_{\ell m\hat{\omega}}$ is constructed
from the stress-energy tensor in Eq. (7) [64, 65]. The potential $V_{G}(r)$
takes the form

$\displaystyle V_{G}(r)$ $\displaystyle=$
$\displaystyle-\frac{K^{2}+4i(r-1)K}{\Delta}+8i\hat{\omega}r+\lambda_{G},$
(22)

where $K=\left(r^{2}+a^{2}\right)\hat{\omega}-am$. The gravitational energy
flux at the horizon and at infinity are given by [66, 64]

$\displaystyle\dot{E}_{T}^{H}$ $\displaystyle=$
$\displaystyle\sum_{\ell=2}^{\infty}\sum_{m=1}^{\ell}\alpha_{\ell
m}\frac{\left|\mathcal{A}_{\ell
m\hat{\omega}}^{\infty}\right|^{2}}{2\pi(m\Omega)^{2}},$ (23)
$\displaystyle\dot{E}_{T}^{\infty}$ $\displaystyle=$
$\displaystyle\sum_{\ell=2}^{\infty}\sum_{m=1}^{\ell}\frac{\left|\mathcal{A}_{\ell
m\hat{\omega}}^{H}\right|^{2}}{2\pi(m\Omega)^{2}},$ (24)

where the subscript “$T$" stands for tensor modes, superscripts “$H$" and
“$\infty$" are for horizon and infinity, respectively. The coefficients are

$\alpha_{\ell
m}=\frac{256\left(2r_{+}\right)^{5}\hat{\kappa}\left(\hat{\kappa}^{2}+4\epsilon^{2}\right)\left(\hat{\kappa}^{2}+16\epsilon^{2}\right)(m\Omega)^{3}}{\left|C_{\ell
m}\right|^{2}},$ (25) $\displaystyle\left|C_{\ell m}\right|^{2}$
$\displaystyle=\left[\left(\lambda_{G}+2\right)^{2}+4\hat{a}(m\Omega)-4a^{2}(m\Omega)^{2}\right]\times\left[\lambda_{G}^{2}+36ma(m\Omega)-36a^{2}(m\Omega)^{2}\right]$
(26)
$\displaystyle+\left(2\lambda_{G}+3\right)\left[96a^{2}(m\Omega)^{2}-48ma(m\Omega)\right]+144(m\Omega)^{2}\left(1-a^{2}\right)$

with $\epsilon=\sqrt{1-a^{2}}/(4r_{+})$ and
$\hat{\kappa}=\hat{\omega}-ma/(2r_{+})$. The function $\mathcal{A}_{\ell
m\hat{\omega}}^{H,\infty}$ comes from the solution of Eq. (21),

$\displaystyle Z_{\ell
m\hat{\omega}}^{H,\infty}=\delta(\hat{\omega}-m\Omega)\mathcal{A}_{\ell
m\hat{\omega}}^{H,\infty},$ (27) $\displaystyle R_{\ell
m\hat{\omega}}\left(r\rightarrow r_{+}\right)=Z_{\ell
m\hat{\omega}}^{\infty}\Delta^{2}e^{-i\hat{\kappa}r^{*}},$ (28) $\displaystyle
R_{\ell m\hat{\omega}}(r\rightarrow\infty)=Z_{\ell
m\hat{\omega}}^{H}r^{3}e^{i\hat{\omega}r^{*}}.$ (29)

###  III.3 Scalar perturbations

To solve Eq. (8), we expand the scalar perturbation $\phi$ in scalar
spheroidal harmonics as

$\phi(t,r,\theta,\varphi)=\sum_{\ell,m}\int
d\hat{\omega}~{}e^{i(m\varphi-\hat{\omega}t)}\frac{X_{\ell
m\hat{\omega}}(r)}{\sqrt{r^{2}+a^{2}}}{}_{0}S_{\ell m}(\theta),$ (30)

where ${}_{0}S_{\ell m}(\theta)$ is the $s=0$ orthnormal spheroidal harmonics
with the eigenvalue $\lambda_{s}$.

The radial perturbation of the scalar field reads

$\left[\frac{d^{2}}{dr_{*}^{2}}+V_{s}(r)\right]X_{\ell
m\hat{\omega}}(r)=\frac{\Delta}{\left(r^{2}+a^{2}\right)^{3/2}}\mathcal{T}^{s}_{\ell
m\hat{\omega}},$ (31)

and the effective potential

$V_{s}=\left(\hat{\omega}-\frac{am}{\sigma}\right)^{2}-\frac{\Delta}{\sigma^{4}}\left[\lambda_{s}~{}\sigma^{2}+2Mr^{3}+a^{2}\left(r^{2}-4Mr+a^{2}\right)\right],$
(32)

where $\sigma=r^{2}+a^{2}$, and $\mathcal{T}^{s}_{\ell m\hat{\omega}}$ is
constructed from the source term on the right hand side of Eq. (8).

Following the discussion of [62], we rewrite the homogeneous radial
perturbation equation

$\frac{d^{2}}{~{}dr^{2}}X_{\ell m\hat{\omega}}+F(r)\frac{d}{dr}X_{\ell
m\hat{\omega}}+G(r)X_{\ell m\hat{\omega}}=0$ (33)

with $F(r)\equiv g^{\prime}(r)/g(r)$ and $G(r)\equiv V_{s}/g(r)^{2}$. Eq. (33)
has two linearly independent solutions which satisfy pure ingoing boundary
condition at the horizon and pure outgoing boundary condition at infinity,

$\begin{array}[]{ll}X_{\ell m\hat{\omega}}^{\rm
in}\sim\begin{cases}e^{-i\hat{\kappa}r^{*}},&r\rightarrow r_{+},\\\ A_{\ell
m\hat{\omega}}^{\rm out}e^{i\hat{\omega}r^{*}}+A_{\ell m\hat{\omega}}^{\rm
in}e^{-i\hat{\omega}\hat{r}^{*}},&r\rightarrow\infty,\end{cases}\\\ X_{\ell
m\hat{\omega}}^{\rm up}\sim\begin{cases}C_{\ell m\hat{\omega}}^{\rm
out}e^{i\hat{\kappa}r^{*}}+C_{\ell m\hat{\omega}}^{\rm
in}e^{-i\hat{\kappa}r^{*}},&r\rightarrow r_{+},\\\
e^{i\hat{\omega}r^{*}},&r\rightarrow\infty,\end{cases}\end{array}$ (34)

where $\hat{\kappa}(\hat{\omega})\equiv\hat{\omega}-ma/(2r_{+})$. Through the
Green’s function method, the solution is expressed as

$X_{\ell m\hat{\omega}}(r)=X_{\ell m\hat{\omega}}^{\rm
up}(r)\int_{r_{+}}^{r}~{}dr^{\prime}\frac{X_{\ell m\hat{\omega}}^{\rm
in}\left(r^{\prime}\right)\mathcal{T}^{s}_{\ell
m\hat{\omega}}\left(r^{\prime}\right)}{W_{r}}+X_{\ell m\hat{\omega}}^{\rm
in}(r)\int_{r}^{\infty}dr^{\prime}\frac{X_{\ell m\hat{\omega}}^{\rm
up}\left(r^{\prime}\right)\mathcal{T}^{s}_{\ell
m\hat{\omega}}\left(r^{\prime}\right)}{W_{r}},$ (35)

where the Wronskian is $W_{r}\equiv X^{\rm in}_{\ell m\hat{\omega}}dX^{\rm
up}_{\ell m\hat{\omega}}/dr^{*}-X^{\rm up}_{\ell m\hat{\omega}}dX^{\rm
in}_{\ell m\hat{\omega}}/dr^{*}$. Since the inhomogeneous solution is purely
ingoing at the horizon and outgoing at infinity, it can be rewritten as

$\displaystyle X_{\ell m\hat{\omega}}\left(r\rightarrow r_{+}\right)$
$\displaystyle=\mathcal{Z}_{\ell
m\hat{\omega}}^{\infty}e^{-i\hat{\kappa}r^{*}},$ (36) $\displaystyle X_{\ell
m\hat{\omega}}(r\rightarrow\infty)$ $\displaystyle=\mathcal{Z}_{\ell
m\hat{\omega}}^{H}e^{i\hat{\omega}r^{*}},$ (37)

where the coefficients are

$\mathcal{Z}_{\ell m\hat{\omega}}^{H,\infty}=-4\pi dq\frac{X_{\ell
m\hat{\omega}}^{\rm in,up}\left(r_{\rm p}\right)}{W_{r}\ u^{t}}\
\frac{{}_{0}S_{\ell m}^{*}(\pi/2)}{\sqrt{r_{\rm p}^{2}+a^{2}}},$ (38)

${}_{0}S_{\ell m}(\theta)^{*}$ is the complex conjugation of ${}_{0}S_{\ell
m}(\theta)$, $u^{\mu}$ is the four velocity of the particle, and $r_{\rm p}$
is the orbital radius. The scalar energy flux at the horizon and at infinity
is [66, 14]

$\displaystyle\dot{E}_{s}^{H}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi}\sum_{\ell=1}^{\infty}\sum_{m=-\ell}^{\ell}m~{}\Omega~{}\hat{\kappa}(m\Omega)\left|\mathcal{Z}_{lm\omega}^{\infty}\right|^{2},$
(39) $\displaystyle\dot{E}_{s}^{\infty}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi}\sum_{\ell=1}^{\infty}\sum_{m=-\ell}^{\ell}m^{2}~{}\Omega^{2}\left|\mathcal{Z}_{\ell
m\omega}^{H}\right|^{2}.$ (40)

##  IV Numerical results

This section presents the numerical results. In our simulation, we sum all the
multipole contributions up to $\ell=18$. The total energy flux is given by

$\mathcal{F}_{\rm
tot}=\dot{E}_{GR}+\delta\dot{E}_{d}=\dot{E}^{H}_{G}+\dot{E}^{\infty}_{G}+\dot{E}^{H}_{s}+\dot{E}^{\infty}_{s},$
(41)

where $\dot{E}_{GR}$ is for the total energy flux of tensor modes and
$\delta\dot{E}_{d}$ is for the energy flux of scalar emission.

Figure 1: The normalized value of the scalar energy flux $\delta\dot{E}_{d}$
as the function of the radial orbit velocity $v=(M\Omega)^{1/3}$ with
different scalar charge $d$. We take $a=0,\ 0.6M,\ 0.9M$ and $0.99M$.

Fig. 1 shows the dependence of the normalized scalar energy flux
$\delta\hat{E}_{d}/q^{2}$ on the orbital velocity $v=(M\Omega)^{1/3}$, the
mass ratio $q$ is set to be $2\times 10^{-5}$, the region of the horizontal
$v$-axis corresponds to the orbital radius starting from $r=80M$ and ending at
the innermost stable circular orbit (ISCO). We see that the scalar energy flux
increases monotonously as the secondary object inspiralling into the central
BH until it plunges into the ISCO. For a fixed velocity of the secondary body,
as the scalar charge $d$ increases, the scalar energy flux increases. Note
that for an EMRI with a high spinning primary BH, as the secondary body
inspirals near the ISCO, the scalar energy flux increases significantly. For
example, in the case of the primary BH with the spin $a=0.99M$, the strength
of the scalar energy flux is almost two orders of magnitude larger than that
with $a=0$.

Figure 2: The relative ratio $\delta\dot{E}_{d}/\dot{E}_{GR}$ as the function
of the radial orbital velocity with different scalar charge $d$. We take
$a=0,\ 0.6M,\ 0.9M$ and $0.99M$.

In Fig. 2, we show the relative ratio $\delta\dot{E}_{d}/\dot{E}_{GR}$ as a
function of the orbital velocity $v=(M\Omega)^{1/3}$. A more significant
scalar charge increases the relative ratio because the source term of the
scalar perturbation in Eq. (5) is proportional to the scalar charge $d$. For
different spin values $a$ of supermassive BHs, the ratio
$\delta\dot{E}_{d}/\dot{E}_{GR}$ decreases as the orbital velocity
$v=(M\Omega)^{1/3}$, this is because gravitational energy flux grows more
rapidly than the scalar energy flux in the high-frequency region. For small
scalar charges, the relative ratio is small; for example, when the scalar
charge $d=0.01$, the relative ratio is smaller than $10^{-4}$. However, the
secondary body can orbit the central BH for more than $10^{5}$ cycles before
the coalescence, so the tiny deviation accumulates and eventually becomes
detectable by the future space-based GW detector.

To show this point, we study the dephasing of the gravitational waveform
caused by the additional energy loss during inspirals. The obital angular
frequency $\Omega$ is

$\frac{d\varphi}{dt}=\Omega(r(t)),$ (42)

where $\varphi$ is referred to as the orbital phase $\varphi_{\rm orb}$ and
the GW frequency $f=\Omega/\pi$. We take four years before the merger for the
observation time,

$T_{\text{obs}}=\int^{f_{\rm max}}_{f_{\rm min}}\frac{1}{\dot{f}}df=4\
\text{years},$ (43)

with

$f_{\text{max}}=min(f_{\rm ISCO},f_{\rm
up}),~{}~{}~{}~{}~{}~{}f_{\text{min}}=max(f_{\rm low},f_{\rm start}),$ (44)

where $f_{\rm ISCO}$ is the frequency that the test particle reaches the ISCO
at $t=t_{\rm end}$ and $f_{\rm start}$ is the initial frequency at $t=0$,
$f_{\rm low}=10^{-4}$Hz and $f_{up}=1$ Hz. The time derivative of the
frequency is

$\dot{f}=\frac{df}{dr}\frac{dr}{dt},$ (45)

where $dr/dt$ is determined from the energy conservation

$\frac{dr}{dt}=-\mathcal{F}_{\rm tot}(t)\left(\frac{dE}{dr}\right)^{-1}.$ (46)

Since

$\frac{d\varphi}{df}=\frac{\pi f}{\dot{f}},$ (47)

so the total phase before the merger can also be calculated by

$\varphi_{\rm orb}(t_{\rm end})=\pi\int_{f_{\rm start}}^{f_{\rm
max}}\frac{f}{\dot{f}}df.$ (48)

If the scalar wave emission vanishes, $\mathcal{F}_{\rm tot}$ is determined
entirely by the gravitational radiation of tensor modes, then the model
reduces to its counterpart in GR and Eq. (48) gives the phase $\varphi_{\rm
orb}^{GR}(t_{\rm end})$ in GR. To discuss the effect of scalar field on
dephasing, we separate the total orbital phase into two parts $\varphi_{\rm
orb}(t)=\varphi_{\rm orb}^{GR}(t)-\delta\varphi_{\rm orb}(t)$. The quantity
$\delta\varphi_{\rm orb}(t)$ denotes the deviation in the orbital phase from
GR. Compared with the case in GR, the additional scalar emission increases the
energy flux and causes the orbit to shrink faster, so the small object
experiences fewer cycles before the coalescence. For the dominant mode, the
dephasing accumulated till the coalescence due to the scalar wave emission is
$\delta\varphi_{GW}(t_{\rm end})=2(\varphi_{\rm orb}^{GR}(t_{\rm
end})-\varphi_{\rm orb}(t_{\rm end}))$.

Figure 3: Four years observation before the plunge. Left: fixing $a=0.9M$, the
dephasing $\delta\varphi_{GW}$ as the function of the time with different
scalar charge $d$; Right: by setting $d=0.3$, the dephasing
$\delta\varphi_{GW}$ as the function of the time with different BH spin $a$.
Both of the data are under the assumption that $m_{\rm p}=10M_{\odot}$ and
$q=2\times 10^{-5}$.

Fig. 3 shows the dephasing $\delta\varphi_{GW}(t)$, measured in radians, for
different scalar charge $d$ and spin $a$ of the supermassive BH. In the left
figure, we take $a=0.9M$ and show that the dephasing grows with time. For all
the values of $d$ in the left figure, the final dephasing
$\delta\varphi_{GW}(t_{\rm end})$ is bigger than 1 radian which are
detectable. In the right panel of Fig. 3, by setting the scalar charge $d=0.3$
and varying the spin of the central BH, we show the dephasing
$\delta\varphi_{GW}(t)$ as a function of time. The dephasing is cumulative
with time for EMRIs, and it reaches the maximum value at $r=r_{\rm ISCO}$. A
central BH with a bigger spin has a smaller ISCO, and the test particle can
spiral closer to the central BH where the radiated energy flux is larger (see
Fig. 2). Therefore, for the same observation time before the plunge, the final
value of $\delta\varphi_{GW}(t_{\rm end})$ is larger for bigger $a$ as shown
in Fig. 3. This means that the observation of EMRIs with a larger-spin Kerr BH
can detect the scalar field easier.

Figure 4: The dephasing $\delta\varphi_{GW}(t_{\rm end})$ with $m_{\rm
p}=10M_{\odot},\ M\in[10^{5},10^{8}]M_{\odot}$ under different value of the
scalar charge $d$, for example, we set $a=0,\ 0.6M,\ 0.9M,\ 0.99M$
correspondingly. The black dashed line represents $\delta\varphi_{GW}(t_{\rm
end})=1$ radian.

Figure 5: Fixing $q$, each panel shows the dependence of dephasing
$\delta\varphi_{GW}(t_{\rm end})$ on $M/M_{\odot}$ with different values of
$a=0,\ 0.6M,\ 0.9M,\ 0.99M$.  Figure 6: Contour plot of gravitational-wave
dephasing $\delta\varphi_{GW}(t_{\rm end})$ in the two-dimensional parameter
space of $a$ and $d$. The red, blue, and green contour lines label the
isocurve with $\delta\varphi_{GW}(t_{\rm end})=1,2$ and $3$ radians,
respectively. The contour line of one radian is the standard conservative
value for a detectable dephasing.

Figs. 4 and 5 show the dephasing as a function of mass of supermassive BHs.
The mass of the secondary body is fixed as $m_{\rm p}=10M_{\odot}$. The black
dashed vertical lines in Fig. 4 label one radian’s position, which is
considered as the conservative value for a detectable dephasing. For each
panel, we see that when a BH becomes more massive, the dephasing
$\delta\varphi_{GW}(t_{\rm end})$ decreases monotonically. For a chosen mass
ratio, Fig. 4 shows $\delta\varphi_{GW}(t_{\rm end})$ increases significantly
as $d$ increases, while Fig. 5 displays that $\delta\varphi_{GW}(t_{\rm end})$
increases with the increase of central BH’s spin. The blue dotted line in the
left bottom panel of Fig. 4 labels the EMRI with mass ratio $10^{-5}$ and
dimensionless spin $a=0.9$, which was discussed in [58]. With the specific
choice of parameters, it was argued that the presence of a scalar charge as
small as $d\sim 0.005-0.01$ could be detected in space. We see that as the
primary body loses mass, the dephasing increases for a fixed scalar charge so
that the accumulative effect increases as the mass ratio increases. It is
possible to measure a smaller scalar charge in EMRIs with a larger mass ratio.

To show the dependence of dephasing $\delta\varphi_{GW}(t_{\rm end})$ on
scalar charge $d$ and central BH’s spin $a$, in Fig. 6 we set $M=5\times
10^{5}M_{\odot}$, $m_{p}=10M_{\odot}$, and plot $\delta\varphi_{GW}(t_{\rm
end})$ in the parameter space $a$ and $d$. The red isoline corresponds to the
conservative value for a detectable dephasing, i.e., one radian, while the
blue and green curves are the constant value of two and three radians,
respectively. We see that fast rotating BH with larger $a$ can allow to detect
smaller scalar charge $d$, e.g. $d\simeq 0.0048$ for $a=0.1$, $d\simeq 0.004$
for $a=0.99$. This scenario indicates that dephasing is more sensitive to the
scalar charge in EMRIs with a larger $a$.

By investigating the dephasing, we estimated the effects of the scalar field
on EMRIs. For a more quantitive and accurate assessment, we proceed to analyze
the measurability by the future space-borne detector LISA. A relevant quantity
is faithfulness, first introduced in Ref. [58] based on the inner product
weighted by the power spectral density of LISA’s floor noise [67]. By
definition, it provides an estimation of how the two signals are distinct from
each other and potentially lead to the detection of the scalar charge. A
comprehensive study has been performed recently in Ref. [58], and the authors
demonstrated that LISA could measure up to a few percent of scalar charges.
For the current study, we calculate the faithfulness $\mathcal{F}_{n}$ between
the GW signals in GR and the present model. To be specific, we consider two
values of the spin parameter $a=0$ and $a=0.9M$, while assuming
$M=10^{6}M_{\odot}$ , $q=2\times 10^{-5}$, $\varphi_{0}=0$,
$\theta_{S}=\phi_{S}=0.2$, and $\theta_{K}=\phi_{K}=0.8$. For a given SNR
$\rho=30$, one can identify the presence of the scalar charge through the GW
signals when $\mathcal{F}_{n}\leq 0.988$. We show the results in Fig. 7, where
the faithfulness is evaluated as a function of the scalar charge $d$. Also,
the details regarding the calculations are relegated to the appendix. The
results presented in Fig. 7 largely agree with those found in [58]. It is
observed that the calculated faithfulness for $a=0.9M$ always stays below
those for $a=0$. The lower bound for the detection is found to be as small as
$d\approx 0.005$. These results support the above results on the dephase of GW
owing to a nonvanishing scalar charge.

It is also worth mentioning that the above calculations have been mainly
focused on the role of the scalar charge. In other words, the preceding
analysis has not taken into account possible degeneracy among different model
parameters. The remaining model parameters may also play a substantial role,
leading to a possible degeneracy in the theory’s parameter space. To this end,
one may carry out an analysis in terms of the Fisher information matrix [68,
26]. Indeed, this aspect has been convincingly investigated recently in Ref.
[63]. It was shown that a more sophisticated survey of the parameter space
does not undermine the feasibility of measuring the scalar charge. In this
regard, as an educated guess, we understand that it is plausible that the
above conclusion will continue to be valid under a joint investigation of the
entire parameter space. A more detailed study is left to future work.

Figure 7: Faithfulness between the GW signals with and without of the scalar
charge $d$ as the function of the latter. The calculations are carried out for
$a=0$ and $a=0.9$. The horizontal solid line represents the detection limit
$\mathcal{F}_{n}\leq 0.988$ by assuming a SNR of $30$.

Before closing this section, we comment on how the observations might give
rise to a more stringent bound on the theories of gravity. It is well known
that the experimental data from the solar system brings relevant constraints
on the parameters of the theories of gravity [69], and in particular, in terms
of those derived by the parameterized post-Newtonian (PPN) formalism [70]. In
this context, the Brans-Dicke theories have been largely ruled out by the
solar system observations [71, 72]. On the other hand, the solar system might
not be suitable to test the strong, dynamical, and non-linear features of the
gravitational interaction [31] owing to the strength of the gravity and
magnitude of the planet’s velocities. Therefore, it is not very surprising
that calculations carried out for theories with high-curvature corrections,
such as Einstein-dilaton Gauss-Bonnet theory [73] and 4D Einstein-Gauss-Bonnet
[74] models, have all succeeded in passing the solar test. In this regard,
EMRIs are expected to provide further constraints on the validity of candidate
theories of modified gravity, especially for theories featured by strong-
curvature modifications. As a result, joint observations that unite solar
observations, future EMRI, and binary system [31] are expected to provide a
more stringent constraint on the scalar charge. In practice, for a specific
modified gravity, the scalar charge $d$ is often related to the parameter of
the theory. By the skeletonization procedure, in Refs. [46, 75, 49, 50, 76],
for Einstein-dilaton Gauss-Bonnet theory with $f(\phi)\propto e^{\phi}$, it
was found that

$d=2\beta+\frac{73}{30}\beta^{2}+\frac{15577}{2520}\beta^{3}+\mathcal{O}(\beta^{4}).$
(49)

While for shift-symmetric scalar-Gauss-Bonnet theory with
$f(\phi)\propto\phi$, one has

$d=2\beta+\frac{73}{60}\beta^{3}+\mathcal{O}(\beta^{5}),$ (50)

where the dimensionless constant $\beta$ is defined as
$\beta=q^{-2}\zeta=\alpha/m^{2}_{p}$ [46, 75, 76], where $\alpha$ is the
coupling between the curvature and the scalar field and $m_{p}$ is the mass of
the small object. Subsequently, a measurement of the charge $d$ could imply a
bound on the underlying modified gravity. On the other hand, the model
investigated in our study is rather generic and does not necessarily satisfy
these relations. Nonetheless, our simulations indicate for the spin parameter
$a\leq 0.99M$, the least detectable scalar charge is approximataly $d\gtrsim
0.004$. If one further assumes the above relations, it implies that the
coupling $\alpha\gtrsim 0.001995m_{p}^{2}$ and $\alpha\gtrsim 0.002m_{p}^{2}$,
respectively.

We conclude this section by reiterating that dephasing due to the scalar wave
emission has been found to be more significant for EMRIs with either a larger
mass ratio or a faster rotating central black hole. The calculations indicate
that EMRI can serve as a sensitive probe to the scalar field in modified
gravity.

##  V Further discussions and concluding remarks

We studied the EMRIs in an alternative theory of gravity with an additional
scalar field and investigated the imprint of the scalar field on the
gravitational waveform. Besides GWs of tensor modes, scalar radiation also
plays a crucial role in the present model. The equation of metric perturbation
is numerically solved in the Teukolsky formalism, while the equation for
scalar perturbation is computed in Green’s method. We simulated the EMRIs up
to four years before the merger and explored its dependence on the parameters
such as the mass ratio $q$, scalar charge $d$, and central BH’s spin $a$.

We computed the total energy flux radiated to the spatial infinity. It was
found that increasing either the scalar charge or the central BH spin enhances
the strengths of both gravitational and scalar energy emissions. The flux of
the GW emission increases more rapidly than that of the scalar wave radiation
in the high-frequency region, and the correction
$\delta\dot{E}_{d}/\dot{E}_{GR}$ decreases when a smaller object spirals into
ISCO.

The dephasing generated by the scalar field emission was also investigated.
The additional scalar emission increases the total energy flux, resulting in
the orbit shrinking faster than that observed in GR. Compared with the case in
GR, a smaller object experiences fewer orbital cycles before the merger for
identical initial conditions. Hence, the orbital phase difference
$\delta\varphi_{\rm orb}(t)$ is always positive. Through a scan in the
parameter space, we found that the dephasing changes monotonously with the
parameters. As the mass ratio increases, the dephasing
$\delta\varphi_{GW}(t_{\rm end})$ increases. The dephasing also increases with
the central BH’s spin and the secondary object’s scalar charge. Thus we
concluded that the dephasing in EMRI due to the scalar radiation is sensitive
to the scalar charge when the system possesses a significant mass ratio and
fast rotating central BH. Since EMRI is more likely to be observable in space-
based gravitational observation, the signature of the scalar field can be
potentially probed in a BH binary system containing the above characteristics.

It is interesting to further generalize the present study to the case of
generic orbits and examine the effect of the small compact object’s spin.
Besides, it is worth exploring more realistic scenarios in astrophysics,
inclusively when the central massive BH is surrounded by an accretion disk or
immersed in a superlight dark matter cloud. The specific waveform in such a
case is a pertinent topic that is crucial for extracting novel physics from
empirical data. These open questions call for more studies.

##  Appendix A Calculation of the faithfulness

The GW strain in the space-based detector under the long-wavelength
approximation reads

$s(t)=\frac{\sqrt{3}}{2}\left[h_{+}(t)F^{+}(t)+h_{\times}(t)F^{\times}(t)\right],$
(51)

where $h_{+}(t)=\mathcal{A}\cos\left[2\varphi_{\rm
orb}+2\varphi_{0}\right]\left(1+\cos^{2}\iota\right)$,
$h_{\times}(t)=-2\mathcal{A}\sin\left[2\varphi_{\rm
orb}+2\varphi_{0}\right]\cos\iota$, $\iota$ is the inclination angle between
the binary orbital angular momentum and the line of sight, and the GW
amplitude $\mathcal{A}=2m_{\rm p}\left[M\Omega(t)\right]^{2/3}/d_{L}$, $d_{L}$
is the luminosity distance. The interferometer pattern functions
$F^{+,\times}(t)$ and $\iota$ can be expressed in terms of four angles which
specify the source orientation, $(\theta_{S},\phi_{S})$, and the orbital
angular direction $(\theta_{K},\phi_{K})$ [77, 78]. Further, we take the
modulation of Doppler phase due to the LISA orbital motion into the GW signal
(51) [79]

$\varphi_{\rm orb}(t)\rightarrow\varphi_{\rm orb}(t)+\frac{d\varphi_{\rm
orb}(t)}{dt}R_{\text{AU}}\sin\theta_{S}\cos\left(2\pi
t/(1~{}\text{year})-\phi_{S}\right).$ (52)

To estimate the scalar charge distinguishability, one introduces the noise-
weighted inner product to define the the faithfulness between two templates

$\langle s_{1}|s_{2}\rangle=2\int_{f_{\rm min}}^{f_{\rm
max}}\frac{\tilde{s_{1}}(f)\tilde{s_{2}}^{*}(f)+\tilde{s_{1}}^{*}(f)\tilde{s_{2}}(f)}{S_{n}(f)}df,$
(53)

where $S_{n}(f)$ is the noise power spectral density of LISA [80] without
Galactic binary foreground noise for convenience, $f_{\text{min}}$ and
$f_{\text{max}}$ are defined by Eq.(44). Faithfulness between two signals can
also be determined

$\mathcal{F}_{n}[s_{1},s_{2}]=\max_{\\{t_{c},\phi_{c}\\}}\frac{\langle
s_{1}|s_{2}\rangle}{\sqrt{\langle s_{1}|s_{1}\rangle\langle
s_{2}|s_{2}\rangle}}\ ,$ (54)

where $(t_{c},\phi_{c})$ are time and phase offsets [67]. The two resulting
signals can be distinguished by LISA if $\mathcal{F}_{n}\leq 0.988$ [81].

###### Acknowledgements.

The authors thank the helpful discussions with Prof. Bin Wang. This research
is supported by the National Key Research and Development Program of China
under Grant No.2020YFC2201400. YG acknowledges the support by the National
Natural Science Foundation of China under Grant No. 11875136. We also
acknowledge the financial support from Brazilian agencies Fundação de Amparo à
Pesquisa do Estado de São Paulo (FAPESP), Fundação de Amparo à Pesquisa do
Estado do Rio de Janeiro (FAPERJ), Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq), Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior (CAPES).

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# Square-Triangle Tilings:
Lift & Flip to Sample?

Thomas Fernique HSE
Moscow, Russia<EMAIL_ADDRESS>Semenov Institute of Chemical Physics
Moscow, Russia Olga Mikhailovna Sizova Semenov Institute of Chemical Physics
Moscow, Russia<EMAIL_ADDRESS>
###### Abstract

We introduce an elementary transformation called flips on tilings by squares
and triangles and conjecture that it connects any two tilings of the same
region of the Euclidean plane.

##  1 Introduction

We consider the tilings of a simply-connected bounded region of the Euclidean
plane by two tiles: a unit square and a regular unit triangle (Fig. 1).

Figure 1: Two square-triangle tilings of the same region.

The goal is to introduce an elementary local transformation that allows to
travel the space of all the possible tilings, in order to sample them using
mixing times techniques as in, e.g., [3]. To do this, we shall embed these
tilings in a larger set obtained by adding a rhombus tile, and show how these
tilings can be naturally seen as discrete surfaces in 4-dimensional Euclidean
space. This is a work-in-progress.

##  2 Lift

Let $j=\exp\frac{2i\pi}{3}$ and $a=\exp\frac{2i\pi}{12}$ be the third and
twelfth roots of the unity, respectively. For $k=0,1,2$, define the complex
numbers $u_{\\_}k:=j^{k}$ and $v_{\\_}k=a\cdot j^{k}$, seen as vectors in the
Euclidean plane. Without loss of generality, the edges of the square and
triangles tiles are directed by the $u_{\\_}i$’s and $v_{\\_}i$’s.

We want to associate with every edge $w_{\\_}k\in\\{u_{\\_}k,v_{\\_}k\\}$ a
vector $\hat{w}_{\\_}k$ in some higher dimensional space $\mathbb{R}^{n}$ so
that, when travelling around a tile, the sum of the vectors associated with
the traversed edges (with negative sign if travelled in backward direction) is
zero. This does not yield any restriction for the squares since each pair of
parallel edges are travelled in opposite directions, hence their sum is always
zero. The same holds for any parallelogram (this will be used later). For the
triangles, this yields two conditions (Fig. 2):

$\hat{u}_{\\_}0+\hat{u}_{\\_}1+\hat{u}_{\\_}2=0\quad\textrm{and}\qquad\hat{v}_{\\_}0+\hat{v}_{\\_}1+\hat{v}_{\\_}2=0.$

Figure 2: The vectors which define tiles (left) and the conditions on their
lifts.

Since we have $2$ constraints on $6$ vectors, that is, a linear system of rank
$4$, it is natural to take vectors in $\mathbb{R}^{4}$ (higher dimension is
useless, lower dimension will loose generality). We will define them as
vectors in $\mathbb{C}^{2}$. For example, take

$\forall
i\in\\{0,1,2\\},\qquad\hat{u}_{\\_}k:=(j^{k},0)\qquad\textrm{and}\qquad\hat{v}_{\\_}k:=(0,j^{k}).$

The condition on the sum of vectors around each tile ensures (by induction)
that the sum of the vectors associated with the traversed edges along every
cycle is zero. In particular, if we map an arbitrary vertex $s_{\\_}0$ of the
tiling to $0\in\mathbb{C}^{2}$, then the sum $h(s)\in\mathbb{C}^{2}$ of
vectors associated with the edges along a path from $v_{\\_}0$ to a vertex $s$
of the tiling does not depend on the path. The vector $h(s)$ is called the
height of $s$. This allows to see any square-triangle tiling as a sort of
“stepped” two-dimensional surface embedded in $\mathbb{C}^{2}$. Figure 3
illustrates this.

Figure 3: Square-triangle tilings with the height of every vertex indicated.

Every triangle has edges directed either by three $u_{\\_}k$’s or by three
$v_{\\_}k$’s. We call it a $u$-triangle in the former case, a $v$-triangle in
the latter case. The way the $\hat{u}_{\\_}k$’s have been chosen ensure that,
in a lifted tiling, the vertices of every $u$-triangle have all the same
second coordinate (in $\mathbb{C}$); this common coordinate defines the
elevation of the triangle. These three vertices differ by their first
coordinate (in $\mathbb{C}$). The barycenter of these coordinates is called
the position of the triangle. The situation is similar for $v$-triangles, with
first and second coordinates exchanged.

##  3 Flip

In the context of tilings, it is common to introduce elementary
transformations on tilings generally called flips. For example, in the case of
tilings by rhombi, a flip denotes the half-turn rotation of a hexagon formed
by three rhombi. Square-triangle tilings lack of such a “natural” flip.
However, such a local transformation has been defined in [4], which features a
third tile, namely a rhombus. This flip is depicted in Figure 4.

Figure 4: A flip.

It may seem strange to involve a new tile that takes us out of the space of
square-triangle tilings. But if we look on this flip in the lifted tiling111As
for squares, pairs of parallel edges in rhombi do not yield any restriction
for the lift., it can be seen as an exchange between the faces of a triangular
prism: the upper triangle and one “square” lateral face are replaced by the
lower triangle and the other two “square” lateral faces (Figure 5). We are
arguing here that squares and rhombi play the same role, differing only in the
way they are represented in the Euclidean plane. To emphasize this, we use the
same color for both tiles. We will also still use the term square-triangle
tilings even when rhombi are present, with the special cases of lack of rhombi
being called pure square-triangle tilings. The position and elevation of a
triangle are affected as follows by a flip:

  * •

the position does not change;

  * •

the elevation changes by $\pm j^{k}$ for some $k\in\\{0,1,2\\}$.

Figure 5: A flip seen in the lift.

##  4 Flip-connectivity

In [4] is defined the zipper move, which is a sequence of flips between two
pure square-triangle tilings (while intermediary tilings may not be pure).

Here, we would like to prove that, any two square-triangle tilings of the same
region (not necessarily pure, that is, with possibly rhombus tiles) can be
connected by a sequence of flips. This would yield the above claim as a
corollary.

Recall that the position of a triangle is unchanged by a flip, only its
elevation. Hence, if flip-connectivity holds, then for any pair $T$ and
$T^{\prime}$ of tilings of the same region, to each triangle of $T$
corresponds a triangle of $T^{\prime}$ with the same position. Transforming
$T$ into $T^{\prime}$ by flips thus amounts to equal the elevations of pairs
of triangles with identical position. In other words, we can “spot” a triangle
in every possible tilings. Figure 6 illustrates this.

Figure 6: Every pair of triangles with the same position have the same label.

Hence, the wanted result will follows if we can prove the following result (a
similar result have been proven for rhombus tilings in [2]):

###### Conjecture 1

If $T\neq T^{\prime}$ are two tilings of the same region, then there is a
triangle in $T$ that can be moved by a flip towards its elevation in
$T^{\prime}$, so that no triangle which has already at the same elevation in
$T$ and $T^{\prime}$ is moved.

## References

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