# Constraining the Extra Polarization Modes of Gravitational Waves with Double
White Dwarfs

Ning Xie  MOE Key Laboratory of TianQin Mission, TianQin Research Center for
Gravitational Physics & School of Physics and Astronomy, Frontiers Science
Center for TianQin, Gravitational Wave Research Center of CNSA, Sun Yat-sen
University (Zhuhai Campus), Zhuhai 519082, China.  Jian-dong Zhang
<EMAIL_ADDRESS>MOE Key Laboratory of TianQin Mission, TianQin
Research Center for Gravitational Physics & School of Physics and Astronomy,
Frontiers Science Center for TianQin, Gravitational Wave Research Center of
CNSA, Sun Yat-sen University (Zhuhai Campus), Zhuhai 519082, China.  Shun-Jia
Huang  MOE Key Laboratory of TianQin Mission, TianQin Research Center for
Gravitational Physics & School of Physics and Astronomy, Frontiers Science
Center for TianQin, Gravitational Wave Research Center of CNSA, Sun Yat-sen
University (Zhuhai Campus), Zhuhai 519082, China.  Yi-Ming Hu  MOE Key
Laboratory of TianQin Mission, TianQin Research Center for Gravitational
Physics & School of Physics and Astronomy, Frontiers Science Center for
TianQin, Gravitational Wave Research Center of CNSA, Sun Yat-sen University
(Zhuhai Campus), Zhuhai 519082, China.  Jianwei Mei  MOE Key Laboratory of
TianQin Mission, TianQin Research Center for Gravitational Physics & School of
Physics and Astronomy, Frontiers Science Center for TianQin, Gravitational
Wave Research Center of CNSA, Sun Yat-sen University (Zhuhai Campus), Zhuhai
519082, China.

###### Abstract

The detection of gravitational waves has opened a new window to test the
theory of gravity in the strong field regime. In general relativity (GR), GW
can only possess two tensor polarization modes, which are known as the $+$ and
$\times$ modes. However, vector and scalar modes can exist in some modified
theories of gravity, and we can test the gravitational theories by probing
these extra polarization modes. As a space-borne GW detector, TianQin will be
launched in the 2030s, and it is expected to observe plenty of GW signals,
including those from nearly ten thousand double white dwarfs in our Galaxy.
This offers an excellent chance for testing the existence of extra
polarization modes. In this article, we analyze the capability of detecting
the extra polarization modes with DWDs. For the extra modes, we consider both
the leading order dipole radiation and the sub-leading quadrupole radiation.
We find that the capability of TianQin has very strong dependency on the
source orientation. We also analyze all the verification binaries with
determined position and frequency, and find that ZTF J1539 can give the best
constraint on the extra polarization modes. We have also considered the case
of TianQin twin constellation, and the joint observation with LISA.

##  I Introduction

As a prediction of general relativity (GR), gravitational waves (GWs) were
first observed by aLIGO on September 14, 2015 Abbott  _et al._ (2016a), and 90
GW events have been reported by the LVK Collaboration up until the present day
Abbott  _et al._ (2016b, 2019a, 2021a, 2021b, 2021c). The analysis of GW150914
and all the GWTC events Abbott  _et al._ (2016c, 2019b, 2019c, 2021d, 2021e)
shows that no deviation of GR has been detected with current observations.
However, more sensitive observatories such as ET Punturo  _et al._ (2010) and
CE Abbott  _et al._ (2017a) on the Earth, or TianQin Luo  _et al._ (2016) and
LISA Amaro-Seoane  _et al._ (2017) in space, will be built in the future. So
the deviation of GR may be observed, or be constrained to a higher level with
the data which is more accurate. Most of the constraint is obtained by model
dependent method, which means that we need to calculate the waveform of the
sources in GR or in the alternative theory, thus the inaccuracy of the
waveform model may cause misinterpretation of the observed data. However, the
polarization mode, an intrinsic feature of GW, can be used to test GR in a
model independent way.

In GR, GWs have only two tensor polarization modes. But in some modified
theories of gravity, vector and scalar polarization modes can also exist, and
there could be at most six polarization modes in the most general cases
Eardley  _et al._ (1973a, b); Gong and Hou (2018a). For example, in the
massless Scalar-Tensor theory, GWs have an additional scalar breathing mode.
In a general Scalar-Tensor theory Will (2014); Maggiore and Nicolis (2000);
Capozziello and Corda (2006) and the $f(R)$ theory Rizwana Kausar  _et al._
(2016); Gong and Hou (2018b); Katsuragawa  _et al._ (2019); Moretti  _et al._
(2019), GWs both have two scalar mode: the breathing mode and the longitude
mode. Moreover, in the Einstein-Æther theory Gong  _et al._ (2018), TeVeS
theory Sagi (2010), and bimetric theory de Paula  _et al._ (2004), all the six
polarization modes could exist. Thus by testing whether these extra modes
exist or not in the detected GW signal, we can test GR.

For the compact binary coalescence (CBC) signals detected by ground-based
detectors like LIGO and Virgo, a Bayesian model selection method has been used
by comparing the assumptions of the GW is constituted of pure vector or scalar
mode against pure tensor mode, and the obtained results still prefer the pure
tensor mode assumption Abbott  _et al._ (2017b, 2019c). Among all the events,
GW170817 gives the best constraint, with the Bayes factor larger than
$10^{20}$ Abbott  _et al._ (2019b). For the sources detected by at least three
detectors, the null-stream method Guersel and Tinto (1989); Wen and Schutz
(2005); Wen (2008); Chatterji  _et al._ (2006); Hagihara  _et al._ (2018,
2019, 2020) can be used to test if there exist non-tensor modes. This method
has been used for the O2, O3a and O3b GW events Pang  _et al._ (2020); Wong
_et al._ (2021), but the data is still consistent with the pure tensor mode
hypothesis and other polarization hypotheses beyond GR are relatively
disfavoured. Moreover, using the position information of GW170817 from
electromagnetic (EM) observation, Ref. Hagihara  _et al._ (2019) puts the
upper limit on the amplitude of vector modes, and Ref. Takeda  _et al._ (2022)
constrains the relative amplitude of scalar modes in a specific scalar-tensor
mixed polarization model. However, in order to distinguish all the extra
modes, we need a network with more detectors to get enough degree of freedom
Chatziioannou  _et al._ (2012); Takeda  _et al._ (2018).

For the CBC signals shown in ground-based detectors the duration will be very
short, and the motion of the Earth can be neglected. Nevertheless, for
continuous wave (CW) generated by fast spinning neutron star (NS), the
monochromatic signal can last all over the observation period, which means the
rotation of the Earth and the movement of the Earth around the Sun need to be
considered in the signal modelling. Then different polarization modes can be
distinguished by the observational data of single detector Isi  _et al._
(2015, 2017); Abbott  _et al._ (2018); Isi  _et al._ (2020). However, CW has
not been observed up to date, but this method is also applicable for space-
borne detectors, since the signals in mHz band also last for a long time.

On the other hand, the extra polarization modes of GWs can also be scouted
using pulsar timing arrays (PTAs) da Silva Alves and Tinto (2011); Lee  _et
al._ (2008); Niu and Zhao (2019); O’Beirne  _et al._ (2019); Boîtier  _et al._
(2020). North American Nanohertz Observatory for Gravitational Waves
(NANOGrav) has searched for evidence of stochastic gravitational wave
background (SGWB) in the 12.5 yr pulsar-timing data in Arzoumanian  _et al._
(2020), while it is found that the result shows very strong prefer of a pure
breathing mode instead of tensor mode Chen  _et al._ (2021). The result from
Chen  _et al._ (2021) is proven by the NANOGrav group, but the conclusion
seems to be strongly related with single pulsar in catalogue Arzoumanian  _et
al._ (2021). They figure out that if one of the pulsar is removed from the
data set, the tensor mode is preferred instead of the transverse scalar mode,
and also the subject is still under debate.

For a space-borne GW observatory, the extra mode can be detected by
considering the motion of detector, with similar method used for CW by ground-
based detector. The space-borne detector, such as TianQin Luo  _et al._
(2016), is expected to be sufficient to detect GW from binary of supermassive
black hole Klein  _et al._ (2016); Feng  _et al._ (2019); Wang  _et al._
(2019), extreme mass ratio inspiral (EMRI) Babak  _et al._ (2017); Fan  _et
al._ (2020), galactic compact binary (CB) Robson  _et al._ (2018); Lau  _et
al._ (2020); Liu  _et al._ (2020); Huang  _et al._ (2020) and cosmic
background Romano and Cornish (2017); Liang  _et al._ (2022). During the
operation period, the detector can discover many non-transient signals, which
can be used to distinguish different polarizations. It has been studied that
there exist $10^{8}$ CBs in our Galaxy, and most of them are double white
dwarfs (DWDs). According to previous calculation, TianQin and LISA could
detect about $10^{4}$ pairs of DWDs Lau  _et al._ (2020); Huang  _et al._
(2020). Moreover, we have already observed some verification binarys which
could be definitely detected by TianQin and LISA. On the other hand, the GW
generated by these DWDs can be regarded as quasi-monochromatic signals Burdge
_et al._ (2019), since the evolution of their frequencies is quite small. So
the waveform model for these DWDs will be very simple, and the consideration
of the extra modes for them is also very easy. So DWDs will be very ideal
source to test GR by testing the existence of the extra polarization modes.

In this work, we focus on the capability of constraining the extra
polarization modes of GW with DWDs. The signals are assumed to be
monochromatic, and the Doppler effect caused by the motion of the detector is
considered in the response of the detectors. For the extra modes, we consider
both the leading order dipole radiation, and the sub-leading order quadrupole
radiation, separately. For the dipole radiation, its amplitude is stronger,
but its frequency is half of the tensor mode, so it causes some difficulty to
identify the common origin with the tensor mode. However, for the quadrupole
radiation, although its amplitude is weaker, its frequency is the same as the
tensor mode. We calculate the capability of constraining the extra modes for
DWDs at different location, since the location governs the result. We consider
both TianQin and its twin constellation, we also consider the joint detection
with LISA. Furthermore, we also perform the analysis for VBs with certain
positions Huang  _et al._ (2020).

This paper is organized as following. In Sec. II, we introduce some basic
methods, including the waveform of extra polarization mode, the response of
the GW detector, and the parameter estimation method. In Sec. III, we focus on
the source location dependency of the parameter estimation (PE) accuracy for
different detector configurations, and we also discuss the effect of the
source inclination angle on the PE accuracy of extra polarization amplitude.
In Sec. IV, we analyze the capability of extra mode with the verification
binaries. Finally, we give a brief summary and discussion in Sec. V. In this
paper, we adopt $G=c=1$.

##  II Model and Method

For a general metric theory of gravity, the metric tensor has 6 propagation
d.o.f., and so there exist 6 polarization modes. More explicitly, there are
two tensor modes, plus ($+$) and cross ($\times$), two vector modes, x (x) and
y (y), and two scalar modes, breathing (b) and longitudinal (l).

In general relativity, only $+$ and $\times$ modes survive, and all the other
modes vanish. For this reason, if the existence of the extra polarization
modes are confirmed in the detected GW signal, it means that GR definitely
needs to be modified.

Although there already exist several waveforms including the extra
polarizations for some specific theories Will (1994); Chatziioannou  _et al._
(2012); Sennett  _et al._ (2016), we keep our attention to the parameterized
model in this work. Due to the radiation of the extra modes, more energy is
carried out compared to GR, so the evolution of the binary system is faster.
However, for most of the DWDs in our Galaxy, the evolution of their
frequencies is very slow. Therefore they can be regarded as monochromatic
source, and the energy loss caused by extra modes can also be neglected. Then
the correction due to the existence of the extra modes can be characterised by
their amplitude.

In general, the detected signal can be written as a linear combination of the
responses of all polarizations:

$h(t)=\sum_{P}F_{P}h_{P}(t),$ (1)

where $P$ stands for different polarizations. $h_{P}(t)$ are the waveform for
each polarization, and $F_{P}$ are the antenna pattern function Poisson and
Will (2014), which described the response of the detector. In the following
part of this section, a more detailed introduction of the waveform and the
response will be presented.

###  II.1 Waveform

For a DWD system, its GW can be regarded as a nearly monochromatic wave, since
the evolution of its orbit is extremely small Burdge  _et al._ (2019). For
example, the GW frequency $f$ of J0806 is about $6$mHz, while the changing
rate of the frequency $\dot{f}$ is about $10^{-16}$Hz/s Roelofs  _et al._
(2010). Thus for a 5 year observation, the variation of $f$ is about
$10^{-8}$Hz, about the same order as the PE accuracy of $f$. Note that for DWD
with eccentricity, the GW frequency can change with a faster rate due to the
fact that the orbital energy of DWD brought by the GW radiation increases
Peters and Mathews (1963). However, most of the DWDs circularize during the
common envelope phase, and we expect the binary system would have a low
eccentricity Nelemans  _et al._ (2001). In this situation, the eccentricity
effect is insignificant and the monochromaticity approximation is still
applicable. Therefore we will not consider the $\dot{f}$ term in the waveform
in this work. The eccentricity will also introduce higher order harmonics in
the GW waveform, but these will not change our method.

In GR, the leading order of GW comes from the quadrupole radiation, and the
corresponding waveform for the $+$ and $\times$ modes in the source frame is:

$\displaystyle h_{+}(t)$
$\displaystyle=\mathcal{A}[(1+\cos^{2}{\iota})/2]\cos(\Phi),$ (2)
$\displaystyle h_{\times}(t)$ $\displaystyle=\mathcal{A}\cos\iota\sin(\Phi).$
(3)

The amplitude $\mathcal{A}$ is given by

$\displaystyle\mathcal{A}=\frac{4\mathcal{M}}{D_{L}}(\pi\mathcal{M}f)^{2/3},$
(4)

where $\mathcal{M}=(m_{1}m_{2})^{3/5}/(m_{2}+m_{2})^{1/5}$ is the chirp mass
of the DWD system constitute by two white dwarfs with mass $m_{1}$ and
$m_{2}$, and $D_{L}$ is the luminosity distance of the system. The inclination
angle $\iota$ is the angle between the direction of angular momentum of the
source $\hat{L}$ and the direction of GW propagation $\hat{k}$, i.e.,
$\cos\iota=\hat{k}\cdot\hat{L}$, where $\hat{k}$ is the unit vector along the
direction of GW propagation. Since we focus on the detectability of space-
borne gravitational wave observatory, it is convenient to discuss in the
ecliptic coordinate system. For a GW located at $(\theta_{s},\phi_{s})$ in the
heliocentric ecliptic coordinate, the GW phase $\Phi$ is given by

$\Phi=2\pi ft+2\pi fR\sin\theta_{s}\cos(2\pi
f_{m}t-\phi_{s}+\phi_{m})+\phi_{0},$ (5)

where $f$ is the frequency of tensor modes, which is twice of the DWD’s
orbital frequency. The initial GW phase $\phi_{0}$ is chosen to be 0 in this
work, and we will not consider it in the Fisher information matrix analysis
(see Sec. II.3). The second term arises from the Doppler effect since the
space-borne detector is moving around the Sun following the Earth. Thus
$\phi_{m}$ represents the initial position of detector, $R=1~{}\text{AU}$ is
the radius of the Earth orbit, and $f_{m}=1~{}\text{yr}^{-1}$ is the frequency
of the Earth motion around the Sun.

To the leading order, the extra polarization modes are dipole radiation, and
the waveform can be written as Will (2014)

$\displaystyle h_{x}(t)$ $\displaystyle=\mathcal{A}_{v}\sin(\Phi/2),$ (6)
$\displaystyle h_{y}(t)$ $\displaystyle=\mathcal{A}_{v}\cos\iota\cos(\Phi/2),$
(7) $\displaystyle h_{b}(t)$
$\displaystyle=\mathcal{A}_{b}\sin\iota\cos(\Phi/2),$ (8) $\displaystyle
h_{l}(t)$ $\displaystyle=\mathcal{A}_{l}\sin\iota\cos(\Phi/2).$ (9)

Note that the frequency is half that of the tensor modes. The amplitudes of
the two vector modes $x$ and $y$ have the same value, since they can be
transformed between each other by a rotation.

However, since we can detect about ten-thousands of DWDs, it is difficult to
distinguish whether it is the dipole radiation of an extra polarization mode,
or it is the tensor mode of another DWD which coincidentally has the half
frequency of the previous one. Also, the misinterpretation of polarization
mode of a GW signal can lead to a incorrect source location, owing to the fact
that each polarization has an unique response function. The wrong sky location
information brings more difficulty to pinpoint the common origin of DWDs.

For the sub-leading quadrupole radiation, the waveform can be written as:

$\displaystyle h_{x}(t)$
$\displaystyle=\mathcal{A}^{\prime}_{v}\sin\iota\sin(\Phi),$ (10)
$\displaystyle h_{y}(t)$
$\displaystyle=\mathcal{A}^{\prime}_{v}\sin\iota\cos\iota\cos(\Phi),$ (11)
$\displaystyle h_{b}(t)$
$\displaystyle=\mathcal{A}^{\prime}_{b}\sin^{2}\iota\cos(\Phi),$ (12)
$\displaystyle h_{l}(t)$
$\displaystyle=\mathcal{A}^{\prime}_{l}\sin^{2}\iota\cos(\Phi).$ (13)

Here, the amplitudes are smaller than that for the leading dipole modes.

We have assumed that all modes are massless and propagate at the speed of
light. However, the non-transverse mode are in fact massive. But due to the
monochromaticity, the dispersion only causes a constant phase shift. Thus the
approximation will not influence the final result. We will take the dispersion
effect into consideration in our future work.

###  II.2 Detector Response

Space-borne GW detectors such as TianQin and LISA have equilateral triangle
configuration, from which we can construct two independent orthogonal
effective Michelson interferometers. We will not consider the time delay
interferometry (TDI) in this work, since it will not change the result
significantly Zhang  _et al._ (2019).

The antenna pattern function $F_{P}$ in Eq. (1) for the two Michelson
interferometers $I$ and $II$ are

$F_{{P}}^{I,II}(f)=D_{ij}^{I,II}e_{P,ij}.$ (14)

In the low frequency limit, the detector tensor $D_{ij}$ for an interferometer
with equal arms is independent of frequency, and only depends on the position
of three spacecrafts, thus it can be expressed as

$\displaystyle D^{I}$
$\displaystyle=\dfrac{1}{2}(\hat{r}_{12}\otimes\hat{r}_{12}-\hat{r}_{13}\otimes\hat{r}_{13}),$
(15) $\displaystyle D^{II}$
$\displaystyle=\frac{1}{2\sqrt{3}}\left(\hat{r}_{12}\otimes\hat{r}_{12}+\hat{r}_{13}\otimes\hat{r}_{13}-2\hat{r}_{23}\otimes\hat{r}_{23}\right)$
(16)

where $\hat{r}_{ij}$ is the unit vector from the $i$-th spacecraft to the
$j$-th spacecraft. Once the design scheme of space-based observatory is given,
the ecliptic coordinates of each spacecraft can be derived from the motion of
constellation. The analytical formulae of spacecraft coordinates can be found
in Hu  _et al._ (2018); Zhang  _et al._ (2020) for TianQin and Rubbo  _et al._
(2004) for LISA.

In order to express the six polarizations mathematically, we define a source
coordinate $(\hat{p},\,\hat{q},\,\hat{k})$. The basis vectors $\hat{p}$ and
$\hat{q}$ are defined as

$\displaystyle\hat{p}$
$\displaystyle=\dfrac{\hat{L}\times\hat{k}}{|{\hat{L}}\times{\hat{k}}|},$ (17)
$\displaystyle\hat{q}$ $\displaystyle={\hat{k}}\times{\hat{p}},$ (18)

Therefore, the basis tensors can be written as

$\displaystyle\textbf{e}_{+}$
$\displaystyle=\hat{p}\otimes\hat{p}-\hat{q}\otimes\hat{q},$ (19)
$\displaystyle\textbf{e}_{\times}$
$\displaystyle=\hat{p}\otimes\hat{q}+\hat{q}\otimes\hat{p},$
$\displaystyle\textbf{e}_{\textit{x}\,}$
$\displaystyle=\hat{p}\otimes\hat{k}+\hat{k}\otimes\hat{p},$
$\displaystyle\textbf{e}_{\textit{y}\,}$
$\displaystyle=\hat{q}\otimes\hat{k}+\hat{k}\otimes\hat{q},$
$\displaystyle\textbf{e}_{\textit{b}\,}$
$\displaystyle=\hat{p}\otimes\hat{p}+\hat{q}\otimes\hat{q},$
$\displaystyle\textbf{e}_{\textit{l}\ }$
$\displaystyle=\hat{k}\otimes\hat{k}.$

Here $\otimes$ denotes tensor product of two vectors.

Moreover, in another frame $(\hat{x},\hat{y},\hat{z})$, the polarization angle
$\psi$ is determined as

$\psi=-\arctan\Big{[}\frac{\hat{L}\cdot\hat{z}-(\hat{k}\cdot\hat{L})(\hat{k}\cdot\hat{z})}{\hat{k}\cdot(\hat{L}\times\hat{z})}\Big{]},$
(20)

Different from the inclination angle, the value of the polarization angle
dependents on the choice of coordinate. In the detector frame, $\hat{z}$ is
along the direction of the detector’ plane, and the polarization angle is
noted as $\bar{\psi}$. In the ecliptic frame, $\hat{z}$ is along the direction
the ecliptic pole, and the polarization angle is noted as $\psi_{s}$. It’s
convenient to use the detector frame to describe the response, but the
polarization angle will change due to the motion of the detector. However, in
the ecliptic frame, the polarization angle can be treated as a constant, since
the precession of the DWDs is negligible. So we will define the source
parameter in the ecliptic frame, and transform to the detector frame to
calculate the response.

In the detector frame, the antenna pattern function for each polarization is:

$\displaystyle F_{+}$
$\displaystyle=\frac{\sqrt{3}}{2}\left(\frac{1+\cos^{2}\bar{\theta}}{2}\cos
2\bar{\phi}\cos 2\bar{\psi}-\cos\bar{\theta}\sin 2\bar{\phi}\sin
2\bar{\psi}\right),$ (21) $\displaystyle F_{\times}$
$\displaystyle=\frac{\sqrt{3}}{2}\left(\frac{1+\cos^{2}\bar{\theta}}{2}\cos
2\bar{\phi}\sin 2\bar{\psi}+\cos\bar{\theta}\sin 2\bar{\phi}\cos
2\bar{\psi}\right),$ $\displaystyle F_{x}$
$\displaystyle=-\frac{\sqrt{3}}{2}\sin\bar{\theta}\left(\cos\bar{\theta}\cos
2\bar{\phi}\cos\bar{\psi}-\sin 2\bar{\phi}\sin\bar{\psi}\right),$
$\displaystyle F_{y}$
$\displaystyle=-\frac{\sqrt{3}}{2}\sin\bar{\theta}\left(\cos\bar{\theta}\cos
2\bar{\phi}\sin\bar{\psi}+\sin 2\bar{\phi}\cos\bar{\psi}\right),$
$\displaystyle F_{b}$
$\displaystyle=-F_{l}=-\frac{\sqrt{3}}{4}\sin^{2}\bar{\theta}\cos
2\bar{\phi}.$

The last equality means that the response of the breathing mode and the
longitude mode are degenerate. The degeneracy can be broken if we consider the
differences in their waveform, such as the propagation speed due to their
mass. But we will not consider this feature in this work, and thus we can’t
distinguish the two scalar modes in this waveform model. So we have

$F_{b}h_{b}(t)+F_{l}h_{l}(t)=F_{b}(h_{b}(t)-h_{l}(t))=F_{b}h_{s}(t),$ (22)

the we can define the scalar mode amplitude as

$\mathcal{A}_{s}=\mathcal{A}_{b}-\mathcal{A}_{l},~{}~{}~{}\mathcal{A}^{\prime}_{s}=\mathcal{A}^{\prime}_{b}-\mathcal{A}^{\prime}_{l}.$
(23)

If $\mathcal{A}_{s},\mathcal{A}^{\prime}_{s}\neq 0$, it can be inferred that
there must exist at least one kind of scalar mode.

###  II.3 Parameterization and Fisher Information Matrix

To evaluate the PE accuracy, we adopt Fisher information matrix (FIM)
Vallisneri (2008) to deduce the uncertainty of the involved parameters. The
definition of FIM is

$\Gamma_{ij}=\left(\frac{\partial
h}{\partial\lambda_{i}}\bigg{|}\frac{\partial h}{\partial\lambda_{j}}\right),$
(24)

where $\lambda_{i}$ is the $i$-th parameter and $(~{}|~{})$ denotes the noise-
weighted inner product Finn (1992); Cutler and Flanagan (1994), which is
defined as

$\left(h_{1}|h_{2}\right)=2\int_{0}^{\infty}\mathrm{d}f\dfrac{\tilde{h}_{1}(f)\tilde{h}_{2}^{*}(f)+\tilde{h}_{1}^{*}(f)\tilde{h}_{2}(f)}{S_{n}(f)},$
(25)

Here $\tilde{h}_{i}$ is the Fourier transformations of $h_{i}$, and $S_{N}(f)$
is the one-sided power spectral density (PSD) for the instrument noise of the
detector, whose explicit formula for TianQin and LISA can be found in Hu  _et
al._ (2018) and Robson  _et al._ (2019), respectively. For the joint detection
with multiple detectors, the total FIM is the sum of each one. The PE accuracy
for each parameter is then given by

$\Delta\lambda_{i}=\sqrt{\Gamma^{-1}_{ii}}.$ (26)

In this work, in order to characterise the magnitude of the extra polarization
modes, we define

$\alpha_{v,s}=\mathcal{A}_{v,s}/\mathcal{A},~{}~{}~{}\alpha^{\prime}_{v,s}=\mathcal{A}^{\prime}_{v,s}/\mathcal{A}$
(27)

as the amplitudes of the extra polarization modes relative to the tensor mode.
The parameters we choose are

$\bm{\lambda}=(\theta_{s},\phi_{s},\psi_{s},\iota,f,\mathcal{A},\alpha_{v},\alpha_{s}),$
(28)

or

$\bm{\lambda}^{\prime}=(\theta_{s},\phi_{s},\psi_{s},\iota,f,\mathcal{A},\alpha^{\prime}_{v},\alpha^{\prime}_{s}).$
(29)

For verification binaries, the position of the source $(\theta_{s},\phi_{s})$
is fixed. The frequency can also be determined by electromagnetic observation,
but since its accuracy is worse than GW observation, we will not fix the
frequencies.

In this work, the capabilities of detectors to probe extra polarizations are
described by $\Delta\alpha_{v}$ and $\Delta\alpha_{s}$ (or
$\Delta\alpha^{\prime}_{v}$ and $\Delta\alpha^{\prime}_{s}$). The value for
the amplitude of the extra modes is chosen as $0$ in the following FIM
analysis, since we assume GR as the standard theory. Thus the value of the PE
accuracy means that we can only probe the extra modes beyond these result,
otherwise it will be confused with the statistic error caused by the noise.

##  III Constraint with general DWDs

For the capability towards extra polarizations, the influence of the amplitude
and frequency is almost trivial. On the other hand, the response of different
polarization modes and the correction due to the Doppler effect is
significantly different for the sources at different positions. So the most
non-trivial parameters are the position of source, and thus the result will be
influenced by the configuration of the detector. TianQin is a geocentric orbit
detector with fixed orientation, and LISA is a heliocentric orbit detector
with its orientation moving around the ecliptic pole. So their capability for
the sources at different positions could be totally different.

Figure 1:  The distribution of SNR on the celestial sphere in ecliptic
coordinate for TianQin and LISA.

In the following calculation, the position of the source will vary all over
the celestial sphere, and the other parameters of the source are chosen as:
$f=0.02\text{Hz},~{}\mathcal{A}=10^{-22},~{}\iota=\pi/4,~{}\psi_{s}=\pi/4$.
The observation time is 1 year, and the results for $n$ years are just divided
by $\sqrt{n}$ due to the monochromaticity of the source and the periodicity of
the detector’s orbit. The SNR for the source at different position is given in
Fig. 1, with upper panel for TianQin and bottom panel for LISA. We can see
that for the chosen parameters, the SNR for the sources all over the sky is
far beyond the threshold for detection, and thus satisfy the high SNR
condition in FIM analysis.

TianQin will point to the DWD J0806 (the coordinate in ecliptic frame is
$\theta_{s}=-4.7^{\circ}$ and $\phi_{s}=120.4^{\circ}$) and operate in the so-
called “3+3” mode Luo  _et al._ (2016), which means that the detector will
follow the “three months on and three months off” working scheme, and thus the
effective observation time ratio is $50\%$. So, we will also consider the twin
constellation, which has another three crafts move along the orbit orthogonal
to the primary TianQin constellation, so it will work during the primary
constellation is off. We denote the two constellations as “TQ” and “TQ II”,
respectively. The TianQin twin constellation will be denoted as “TQ I+II”. We
will also calculate the capability of LISA and its joint detection with
TianQin and the twin constellation.

###  III.1 Constraint on the Dipole Radiation

In this subsection, we will only consider the leading order dipole radiation
of the extra modes.

For TianQin, the result is shown in Fig. 2. The result shows that the PE
accuracy of $\alpha_{v}$ and $\alpha_{s}$ are strongly dependent on the
location of the source. Especially, the result diverges for the direction of
J0806 and its antipodal point. This feature is caused by the fact that the
antenna pattern $F_{P}$ will be 0 for the sources at $\bar{\theta}=0,\pi$, and
thus in the response signal, the components for the extra modes will be 0.
However, for the other regions, the accuracy can be a few percent for vector
and scalar mode, and the capability for vector mode is better than scalar
mode.

Figure 2:  The distribution of $\Delta\alpha_{v}$ and $\Delta\alpha_{s}$ on
the celestial sphere in ecliptic coordinate for TianQin. The red dot is J0806,
and the white dot is the galactic center.

Figure 3: The distribution of $\Delta\alpha_{v}$ and $\Delta\alpha_{s}$ on the
celestial sphere in ecliptic coordinate for the TianQin twin constellation.

The TianQin twin constellation is useful to avoid the divergent. Thus the
results are a few percent for both extra modes all over the sky, as plotted in
Fig. 3. However, we can see that the contour plot will be different for vector
and scalar modes. For vector modes, the region near the orientations of the
two constellations will be worse. But for scalar modes, the region between the
orientations will be worse. This caused by the difference on the dependence of
$\bar{\theta}$ for the response of the extra modes. For vector modes, as
$\bar{\theta}$ approaches to 0, the response approaches to 0 as
$\sin\bar{\theta}$. But the response for scalar modes will approaches to 0 as
$\sin^{2}\bar{\theta}$. So although the result for both modes will divergent
at $\bar{\theta}=0,\pi$, but the region near these points will be larger for
scalar modes, since the response will grow slower as the sources move away
from the divergent points.

Figure 4: The distribution of $\Delta\alpha_{v}$ and $\Delta\alpha_{s}$ along
the ecliptic plane for TQ (red line), TQ II (orange line) and TQ I+II (green
line). Note that for TQ I+II, the better regions for the vector modes and the
scalar modes located at different longitude.

In Fig. 4, we plot the accuracy of each mode for different detectors for the
sources at the ecliptic plane. We should notice that the divergent point for
TQ II located just on the ecliptic plane, and for TQ the result will be worst
for the points has the same longitude as the divergent point. At the worst
point for one detector, the accuracy will be the best for the other detector,
and the result will be almost exact the value for the best one. But for the
intermediate region between the divergent points, the accuracy will be almost
the same for both detectors, and the joint detection will have an improvement
for about $\sqrt{2}$ times. So for the vector modes, the PE result will
decrease faster, and although the accuracy for the intermediate points will be
worse than the best point for each detector, an improvement of $\sqrt{2}$
times will make it better than the best value. But for the scalar modes, the
PE result will decrease slower, and an improvement of $\sqrt{2}$ times is
still worse than the best value. Although the shape of the contours will be
different, the results are still at the same order for the twin constellation
all over the sky.

Figure 5: The distribution of $\Delta\alpha_{v}$ and $\Delta\alpha_{s}$ on the
celestial sphere in ecliptic coordinate for LISA.

For LISA, as showed in Fig. 5, we can find that the accuracy is mainly
determined by the latitude due to the periodic motion of LISA, and the result
is about a few percent all over the sky. Additionally, we find that the
distribution is different for vector and scalar modes. For the vector modes,
the regions with better accuracy are located near the north and south pole.
For the scalar modes, the accuracy will be worse for the region at the
ecliptic plane and the poles. For the joint detection for LISA with TianQin or
TianQin twin constellation, the accuracy will be improved, and the results are
shown in Fig. 15 and Fig. 17 at Appendix A.

Figure 6: Top panel: the area ratio of sky region where the $\Delta\alpha_{v}$ always takes on a value less than a certain quantity. Bottom panel: same as the former one but for $\Delta\alpha_{s}$. The configurations considered include TQ, LISA, TQ I+II, TQ + LISA and TQ I+II + LISA. The dashed lines corresponding to the 90% threshold.  Table 1: The representative detecting accuracy for the 90% level and best of $\alpha_{v}$ and $\alpha_{s}$ for each configuration.  | TQ | TQ I+II | LISA | TQ + LISA | TQ I+II + LISA
---|---|---|---|---|---
$\Delta\alpha_{v}(90\%)$ | 0.040 | 0.020 | 0.016 | 0.015 | 0.013
$\Delta\alpha_{v}(\text{best})$ | 0.022 | 0.015 | 0.015 | 0.013 | 0.010
$\Delta\alpha_{s}(90\%)$ | 0.304 | 0.069 | 0.052 | 0.050 | 0.041
$\Delta\alpha_{s}(\text{best})$ | 0.054 | 0.040 | 0.048 | 0.036 | 0.030

For a more intuitive exhibition, we also plot the cumulative distribution
function (CDF) of the accuracy distribution on the sky for each detector
configuration in Fig. 6. The dashed line corresponds to the accuracy which can
be achieved by 90% area in celestial sphere. The detailed results
corresponding to this 90% threshold and the best accuracy for each
configuration are shown in Table 1. From these figures, we can find that for
TianQin, the distribution of the accuracy will have a very wide range due to
the existence of the divergent point, although the best value can be
comparable with other configurations. For all the other configurations, the
distribution of the accuracy will be narrow, and Moreover, the accuracy for
vector modes will be a few times better than the scalar modes in general.

Figure 7:  The distribution of $\Delta\alpha_{v}$ and $\Delta\alpha_{s}$ with
different inclination angle for TianQin and LISA.

Despite the position of the source, the inclination angle will also have
significant influence on the PE accuracy for the amplitude of extra
polarizations. So here we consider the DWD source with different inclination
angle, while the source position is chosen as $\theta_{s}=30^{\circ}$ and
$\phi_{s}=60^{\circ}$. The dependence on inclination angle $\iota$ for
$\Delta\alpha_{v}$ and $\Delta\alpha_{s}$ are shown in Fig. 7. If we choose
another pair of position parameter, the result of $\Delta\alpha_{v}$ and
$\Delta\alpha_{s}$ might have a moderate difference quantitatively, but the
tendency of their dependence on $\iota$ will be similar. We find that
$\Delta\alpha_{s}$ diverges when $\iota$ approaches to zero (face-on case)
since the amplitude of scalar modes vanish for this case (cf. Eqs. (8) and
(9)). Moreover, due to the different on the factor of $\iota$, the PE of the
amplitude for extra modes will reach its best value at different $\iota$,
which is $\pi/2$ for scalar modes, and $0$ for vector modes.

###  III.2 Constraint on the Quadrupole Radiation

For the sub-leading quadrupole radiation, the calculations are almost the
same, except that we only consider these sub-leading term, and ignore the
leading order dipole radiation.

The results for TianQin are shown in Fig. 8. We can find that the capability
will be a few times worse than the dipole situation, and it will also have
strong position dependence due to the vanish of the response. Due to the fact
that the frequency of the extra modes at quadrupole order will be the same as
the tensor mode, the degeneracy will happen between the angles and the
relative amplitude of the extra modes. Thus except the divergent at the
orientation of the detector, the result will also be worse on the plane of the
detector for scalar modes, and on the direction perpendicular to the
orientation and located on the ecliptic plane. The second additional
degeneracy is caused by the degeneracy between $\alpha^{\prime}_{v}$ and
$\theta_{s}$. As we plotted in Fig. 9, if we exclude the parameter
$\theta_{s}$, then the region on the ecliptic plane and perpendicular to the
orientation will disappear.

Figure 8:  The distribution of $\Delta\alpha^{\prime}_{v}$ and
$\Delta\alpha^{\prime}_{s}$ on the celestial sphere in ecliptic coordinate for
TianQin. Figure 9:  The distribution of $\Delta\alpha^{\prime}_{v}$ on the
celestial sphere for TianQin, with $\theta_{s}$ excluded in the PE.

Figure 10: The distribution of $\Delta\alpha^{\prime}_{v}$ and
$\Delta\alpha^{\prime}_{s}$ on the celestial sphere in ecliptic coordinate for
the TianQin twin constellation.  Figure 11:  The distribution of
$\Delta\alpha^{\prime}_{v}$ on the celestial sphere for TianQin twin
constellation, with $\iota$ excluded in the PE.

The results for the TianQin twin constellation are displayed in Fig. 10. The
position dependence will also be deduced as for the dipole case, but the shape
will be different for the vector mode. This is caused by the degeneracy
between $\alpha^{\prime}_{v}$ and $\iota$. After removing the inclination
angle $\iota$ in the FIM analysis, we can find that the result looks like the
plot for dipole case (see upper panel of Fig. 3). As a comparison, we show it
in Fig. 11. The detailed shape is caused by the choice of the value for
$\iota$, and it will also change if we change $\iota$.

Figure 12: The distribution of $\Delta\alpha^{\prime}_{v}$ and
$\Delta\alpha^{\prime}_{s}$ on the celestial sphere in ecliptic coordinate for
LISA.

For LISA, we can find that the detecting accuracy is mainly determined by the
latitude of the source location, same as the dipole case. But this time, the
worse part will be the pole region for both vector and scalar modes. The
result is shown in Fig. 12. For the joint detection for LISA with TianQin or
TianQin twin constellation, the accuracy will also be improved, and the
results are shown in Fig. 16 and Fig. 18 at Appendix A.

Again, we plot the CDF of the accuracy distribution on the sky for each
detector configuration in Fig. 13, and list the result for the best value and
$90\%$ threshold in Table 2. Except for TianQin, LISA will also have a very
wide distribution due to the degeneracy between the relative amplitude and the
angles. But the result for joint detection will still be narrow due to the
complementary of different detectors. Overall, the accuracy for dipole
radiation will be a few times better than that for the quadrupole radiation.
Since the amplitude of the leading dipole radiation would also be stronger
than the sub-leading quadrepole radiation, this result means that it will be
better to constrain the extra modes with dipole radiation, if we ignore the
problem of identification of the common origin.

Figure 13: Upper panel: the area ratio of sky region where the $\Delta\alpha^{\prime}_{v}$ always takes on a value less than a certain quantity. Bottom panel: same as the former one but for $\Delta\alpha^{\prime}_{s}$. The configurations considered include TQ, LISA, TQ I+II, TQ + LISA and TQ I+II + LISA. The dashed lines corresponding to the 90% threshold.  Table 2: The representative detecting accuracy for the 90% level and best of $\alpha^{\prime}_{v}$ and $\alpha^{\prime}_{s}$ for each configuration.  | TQ | TQ I+II | LISA | TQ + LISA | TQ I+II + LISA
---|---|---|---|---|---
$\Delta\alpha^{\prime}_{v}(90\%)$ | 0.076 | 0.030 | 0.049 | 0.024 | 0.019
$\Delta\alpha^{\prime}_{v}(\text{best})$ | 0.034 | 0.019 | 0.023 | 0.018 | 0.014
$\Delta\alpha^{\prime}_{s}(90\%)$ | 0.958 | 0.139 | 0.284 | 0.117 | 0.083
$\Delta\alpha^{\prime}_{s}(\text{best})$ | 0.156 | 0.073 | 0.088 | 0.069 | 0.056
Figure 14:  The distribution of $\Delta\alpha^{\prime}_{v}$ and
$\Delta\alpha^{\prime}_{s}$ with different inclination angle for TianQin and
LISA.

Same as the dipole case, the PE accuracy for the sub-leading quadrupole
radiation will also be sensitive to inclination angle of the source. Here we
also consider the DWD source with different inclination angle by fixing the
source position to $\theta_{s}=30^{\circ}$ and $\phi_{s}=60^{\circ}$. The
dependence on inclination angle $\iota$ for $\Delta\alpha^{\prime}_{v}$ and
$\Delta\alpha^{\prime}_{s}$ are shown in Fig. 14. We find that both
$\Delta\alpha^{\prime}_{v}$ and $\Delta\alpha^{\prime}_{s}$ diverge when
$\iota$ becomes zero since all the extra modes both vanish for the face-on DWD
(cf. Eqs. (10-13)). Moreover, we can see that only for the TianQin
consternation, $\iota\to\pi/2$ can bring the degeneracy between some angle
parameters and the extra mode amplitudes (especially $\iota$ and
$\alpha^{\prime}_{v}$, $\psi$ and $\alpha^{\prime}_{s}$). When we exclude
$\iota$ and $\psi$ respectively in the PE, the degeneracy disappears and
$\Delta\alpha^{\prime}_{v,s}$ has the smallest value at $\iota=\pi/2$.

##  IV Constraint with verification binaries

As mentioned in Huang  _et al._ (2020), despite the thousands of DWDs which
could be detected in the future observation, we can also detect several VBs
which have already been detected by EM observations. For VBs, the position and
frequency of the VBs can be determined by EM observation, and the accuracy of
the position is much better than the GW observations, which means that we
don’t need to consider the PE of the two position angles in the FIM analysis.
However, since the accuracy of frequency with EM observation is worse than the
GW observation, the frequency will still be considered in the FIM analysis.
Thus the parameters will be reduced to only six while $\theta_{s}$ and
$\phi_{s}$ will be known parameters. As we mentioned above, there will be
degeneracy between the relative amplitudes and the positions of the source, so
if the position is determined through the EM observation, the constraint of
the extra modes will be better. Moreover, for part of VBs, their inclination
angle could also be determined by EM observations such as intensity
interferometry Baumgartner  _et al._ (2020) with better precision comparing to
the GW observation. Thus the parameter $\iota$ can be excluded in the FIM
analysis. The parameters of the VBs are listed in Table 3. For another part of
VBs there is no direct measurement on the inclination angle through EM
observation and the angle is assigned based on other information of the binary
system Huang  _et al._ (2020). We label the estimated values of $\iota$ with
the asterisk. Note that we must consider the PE of the inclination angle for
these special VBs in FIM analysis.

Table 3:  The parameters of VBs which can be detected by TianQin within 5 years. The asterisk after the values means the parameter has no precise EM measurement.  source | $\theta$ | $\phi$ | $\iota$ |  $f$/mHz | $\mathcal{A}$
---|---|---|---|---|---
J0806 | 94.7 | 120.4 | 38 | 6.22 | $1.28\times 10^{-22}$
ZTF J1539 | 23.8 | 205.0 | 84 | 4.82 | $3.68\times 10^{-22}$
V407 Vul | 43.2 | 295.0 | 60∗ | 3.51 | $2.20\times 10^{-22}$
ES Cet | 110.3 | 24.6 | 60∗ | 3.22 | $2.14\times 10^{-22}$
SDSS J0651 | 94.2 | 101.3 | 87 | 2.61 | $3.24\times 10^{-22}$
SDSS J1351 | 85.5 | 208.4 | 60∗ | 2.12 | $1.24\times 10^{-22}$
AM CVn | 52.6 | 170.4 | 43 | 1.94 | $5.66\times 10^{-22}$
SDSS J1908 | 28.5 | 298.2 | 15 | 1.84 | $1.22\times 10^{-22}$
HP Lib | 85.0 | 235.1 | 30∗ | 1.81 | $3.14\times 10^{-22}$
SDSS J0935 | 61.9 | 131.0 | 60∗ | 1.68 | $5.98\times 10^{-22}$
SDSS J2322 | 81.5 | 353.4 | 27 | 1.66 | $1.74\times 10^{-22}$
PTF J0533 | 111.1 | 82.9 | 73 | 1.62 | $1.52\times 10^{-22}$
CR Boo | 72.1 | 202.3 | 30∗ | 1.36 | $2.58\times 10^{-22}$
V803 Cen | 120.3 | 216.2 | 14 | 1.25 | $3.20\times 10^{-22}$

Here we will consider TQ, LISA, TQ I+II, TQ + LISA and TQ I+II + LISA. The
mission lifetime is 4 yrs for LISA, and 5 yrs for TianQin with 4 years overlap
with LISA.The results are shown in Table 4 for dipole case, and in Table 5 for
quadrupole case. The results of J0806 in the first two column are invalid
since the estimated errors diverge in the sky location of J0806 for TQ due to
the zero response of the extra modes. However, due to the suitable frequency
and the strong amplitude, the capability for other configurations can still
reach to a few percent or less. In Tables 4 and 5, for J0806, ZTF J1539, SDSS
J0651, AM CVn, SDSS J1908, SDSS J2322, PTF J0533 and V803 Cen, the FIMs are
calculated with fixed $\iota$.

Table 4:  The PE accuracy of dipole amplitude $\alpha_{v}$ and $\alpha_{s}$ for the VBs.  | TQ | LISA | TQ I+II | TQ + LISA | TQ I+II + LISA
---|---|---|---|---|---
| $\Delta\alpha_{v}$ | $\Delta\alpha_{s}$ | $\Delta\alpha_{v}$ | $\Delta\alpha_{s}$ | $\Delta\alpha_{v}$ | $\Delta\alpha_{s}$ | $\Delta\alpha_{v}$ | $\Delta\alpha_{s}$ | $\Delta\alpha_{v}$ | $\Delta\alpha_{s}$
J0806 | \ | \ | 0.008 | 0.031 | 0.035 | 0.103 | 0.008 | 0.031 | 0.008 | 0.029
ZTF J1539 | 0.026 | 0.037 | 0.005 | 0.009 | 0.018 | 0.028 | 0.005 | 0.009 | 0.005 | 0.008
V407 Vul | 0.090 | 0.305 | 0.022 | 0.046 | 0.056 | 0.123 | 0.021 | 0.046 | 0.020 | 0.043
ES Cet | 0.089 | 0.164 | 0.030 | 0.068 | 0.080 | 0.162 | 0.029 | 0.063 | 0.028 | 0.063
SDSS J0651 | 0.225 | 1.342 | 0.041 | 0.077 | 0.093 | 0.159 | 0.041 | 0.076 | 0.038 | 0.069
SDSS J1351 | 0.360 | 0.658 | 0.158 | 0.377 | 0.358 | 0.658 | 0.145 | 0.327 | 0.145 | 0.327
AM CVn | 0.090 | 0.285 | 0.038 | 0.113 | 0.063 | 0.217 | 0.035 | 0.105 | 0.032 | 0.100
SDSS J1908 | 0.433 | 4.538 | 0.167 | 1.526 | 0.286 | 2.477 | 0.155 | 1.445 | 0.144 | 1.299
HP Lib | 0.173 | 0.784 | 0.072 | 0.350 | 0.150 | 0.767 | 0.066 | 0.318 | 0.064 | 0.317
SDSS J0935 | 0.172 | 0.712 | 0.051 | 0.113 | 0.097 | 0.214 | 0.049 | 0.111 | 0.045 | 0.100
SDSS J2322 | 0.400 | 2.485 | 0.149 | 0.804 | 0.291 | 2.007 | 0.138 | 0.760 | 0.132 | 0.746
PTF J0533 | 0.686 | 2.065 | 0.230 | 0.439 | 0.445 | 1.070 | 0.218 | 0.429 | 0.204 | 0.406
CR Boo | 0.362 | 1.363 | 0.142 | 0.664 | 0.327 | 1.353 | 0.132 | 0.597 | 0.130 | 0.596
V803 Cen | 0.326 | 2.787 | 0.124 | 1.252 | 0.270 | 2.709 | 0.116 | 1.142 | 0.113 | 1.136
Table 5:  The PE accuracy of quadrupole amplitude $\alpha^{\prime}_{v}$ and $\alpha^{\prime}_{s}$ for the VBs.  | TQ | LISA | TQ I+II | TQ + LISA | TQ I+II + LISA
---|---|---|---|---|---
| $\Delta\alpha^{\prime}_{v}$ | $\Delta\alpha^{\prime}_{s}$ | $\Delta\alpha^{\prime}_{v}$ | $\Delta\alpha^{\prime}_{s}$ | $\Delta\alpha^{\prime}_{v}$ | $\Delta\alpha^{\prime}_{s}$ | $\Delta\alpha^{\prime}_{v}$ | $\Delta\alpha^{\prime}_{s}$ | $\Delta\alpha^{\prime}_{v}$ | $\Delta\alpha^{\prime}_{s}$
J0806 | \ | \ | 0.010 | 0.049 | 0.018 | 0.054 | 0.010 | 0.044 | 0.009 | 0.036
ZTF J1539 | 0.007 | 0.651 | 0.003 | 0.010 | 0.006 | 0.043 | 0.003 | 0.009 | 0.003 | 0.009
V407 Vul | 0.035 | 0.141 | 0.006 | 0.016 | 0.019 | 0.053 | 0.006 | 0.016 | 0.006 | 0.015
ES Cet | 0.029 | 0.453 | 0.006 | 0.018 | 0.024 | 0.054 | 0.006 | 0.018 | 0.006 | 0.017
SDSS J0651 | 0.056 | 3.114 | 0.005 | 0.017 | 0.025 | 0.049 | 0.005 | 0.016 | 0.005 | 0.016
SDSS J1351 | 0.114 | 4.142 | 0.024 | 0.076 | 0.102 | 0.201 | 0.023 | 0.073 | 0.023 | 0.071
AM CVn | 0.033 | 0.172 | 0.008 | 0.030 | 0.023 | 0.123 | 0.008 | 0.029 | 0.008 | 0.029
SDSS J1908 | 0.395 | 6.216 | 0.107 | 1.293 | 0.290 | 4.088 | 0.104 | 1.277 | 0.100 | 1.218
HP Lib | 0.115 | 0.635 | 0.023 | 0.145 | 0.099 | 0.482 | 0.022 | 0.140 | 0.022 | 0.137
SDSS J0935 | 0.065 | 0.317 | 0.011 | 0.031 | 0.028 | 0.078 | 0.011 | 0.030 | 0.010 | 0.029
SDSS J2322 | 0.206 | 1.878 | 0.061 | 0.421 | 0.158 | 1.503 | 0.059 | 0.411 | 0.057 | 0.404
PTF J0533 | 0.230 | 1.132 | 0.047 | 0.127 | 0.220 | 0.792 | 0.046 | 0.125 | 0.044 | 0.123
CR Boo | 0.236 | 3.229 | 0.071 | 0.415 | 0.164 | 0.735 | 0.066 | 0.377 | 0.065 | 0.360
V803 Cen | 0.342 | 31.95 | 0.146 | 1.802 | 0.286 | 3.360 | 0.135 | 1.633 | 0.130 | 1.584

We can find that, among all the VBs, ZTF J1539 can constrain the amplitude
$\alpha_{v}$ and $\alpha_{s}$ to 0.005 and 0.008 respectively with TQ I+II +
LISA, which is the best result. That is the combined effect of the higher
amplitude of ZTF J1539 with suitable frequency, and the advantage sky location
which at the best region as plotted in the previous figures. However, the
accuracy of quadrupole mode constrained with TianQin will be worse for ZTF
J1539, and the reason is that its position located near the detector’s plane
of TianQin, and we plotted in the bottom panel in Fig. 8, those sources will
have strong degeneracy with angular parameters.

##  V Conclusion and Discussion

We investigated the capability of TianQin to probe the extra polarizations in
GWs. We considered not only the standard TianQin constellation , but also the
twin constellations and their joint detection with LISA. We construct the
responded signal with all the six polarizations, and assume that the GW
waveform is monochromatic for DWDs. Fisher analysis is adopted to estimate the
accuracy to measure the parameters. We defined two parameters, $\alpha_{v}$
and $\alpha_{s}$, to describe the dipole amplitudes of the extra modes
relative to the tensor modes, and $\alpha^{\prime}_{v}$ and
$\alpha^{\prime}_{s}$ for quadrupole amplitudes. Doppler effect due to the
periodical motion of the detector around the Sun is also considered in the
signal construction.

We first analyze the behaviour of the PE accuracy for the relative amplitudes
as the function of the source position, and listed the best accuracy and the
accuracy which can be attained by more than 90% sky zone to describe the
overall capability in the polarization detection. We find that, for TQ, the PE
accuracy will divergent at the orientation due to the zero response, and thus
have very strong dependency on the sky location of the source, ranging across
about 2 orders. However, the capability of TianQin can still reach to a few
percent for most of the regions. The application of the twin constellation can
significantly improve this strong position dependent feature. A joint
detection with LISA will also improve the result for a few times.

Generally, for all the detector configurations and both dipole and quadrupole
situations, the capability for vector modes will be better than that for
scalar modes. On the other hand, the capability for dipole radiation will also
be better than that for quadrupole radiation. For the quadrupole case, the
amplitude of the extra modes will have strong degeneracy with the position and
the direction of the angular momentum. However, due to the difficulty in the
identification of the common origin, it’s still very useful to consider the
quadrupole radiation.

Finally, we also considered the capabilities with the observation of the VBs.
For most of the VBs, the capabilities are about a few percents. However, due
to the zero response of the extra modes, TianQin could not detect the extra
modes with J0806, although its expected to be one of the most loudest DWDs. We
also find that, for the best source ZTF J1539, the capability will be even
less than one percent.

However, our analysis only focus on the monochromatic DWD signals. But the
space borne GW detectors can also detect many other chirping sources, such as
binary black holes or EMRIs. Some modelling works for these sources which
focused on the extra polarization modes have already been done in recent years
for some specific gravitational theories. But for a more general analysis, we
still need a parametric model. In the future, we will consider the non-
monochromatic sources and including the propagation effect and the
modification of the phase.

###### Acknowledgements.

We thank Changfu Shi for many helpful discussions. This work is supported by
the Guangdong Major Project of Basic and Applied Basic Research (Grant No.
2019B030302001). Y. H. is supported by the Natural Science Foundation of China
(Grants No. 12173104).

##  Appendix A Results for the joint detection

To reduce the length of the main text, we plot the results for the joint
detection of TianQin and LISA in this appendix. The analysis of these results
is presented in Sec. III, and we will not repeat it here.

Figure 15:  The distribution of $\Delta\alpha_{v}$ and $\Delta\alpha_{s}$ on
the celestial sphere in ecliptic coordinate for the joint detection of TianQin
and LISA.

Figure 16:  The distribution of $\Delta\alpha^{\prime}_{v}$ and
$\Delta\alpha^{\prime}_{s}$ on the celestial sphere in ecliptic coordinate for
the joint detection of TianQin and LISA.

Figure 17:  The distribution of $\Delta\alpha_{v}$ and $\Delta\alpha_{s}$ on
the celestial sphere in ecliptic coordinate for the joint detection of TianQin
twin constellation and LISA.

Figure 18: The distribution of $\Delta\alpha^{\prime}_{v}$ and
$\Delta\alpha^{\prime}_{s}$ on the celestial sphere in ecliptic coordinate for
the joint detection of TianQin twin constellation and LISA.

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  *[GR]: general relativity
  *[GW]: gravitational wave
  *[DWDs]: double white dwarf
  *[GWs]: gravitational wave
  *[CBC]: compact binary coalescence
  *[EM]: electromagnetic
  *[CW]: continuous wave
  *[NS]: neutron star
  *[SGWB]: stochastic gravitational wave background
  *[EMRI]: extreme mass ratio inspiral
  *[CB]: compact binary
  *[CBs]: compact binary
  *[VBs]: verification binary
  *[PE]: parameter estimation
  *[DWD]: double white dwarf
  *[TDI]: time delay interferometry
  *[FIM]: Fisher information matrix
  *[PSD]: power spectral density
  *[SNR]: signal-to-noise ratio
  *[CDF]: cumulative distribution function
  *[FIMs]: Fisher information matrix
  *[EMRIs]: extreme mass ratio inspiral