Detection of gravitational wave polarization with space-borne detectors
DDetection of gravitational wave polarization with space-bornedetectors
Chao Zhang, ∗ Yungui Gong, † Dicong Liang, ‡ and Chunyu Zhang § School of Physics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
General Relativity predicts only two tensor polarization modes for gravitationalwaves while at most six possible polarization modes of gravitational waves are allowedin general metric theory of gravity. The number of polarization modes is totallydetermined by the specific modified theory of gravity. Therefore, the determinationof polarization modes can be used to test gravitational theory. We introduce aconcrete data analysis pipeline for a single space-based detector such as LISA todetect the polarization modes of gravitational waves. Apart from being able to detectmixtures of tensor and extra polarization modes, this method also has the addedadvantage that no waveform model is needed and monochromatic gravitational wavesemitted from any compact binary system with known sky position and frequency canbe used. We apply the data analysis pipeline to the reference source J0806.3+1527 ofTianQin with 90-days’ simulation data, and we show that α viewed as an indicativeof the intrinsic strength of the extra polarization component relative to the tensormodes can be limited below 0.5 for LISA and below 0.2 for Taiji. We investigate thepossibility to detect the nontensorial polarization modes with the combined networkof LISA, TianQin and Taiji and find that α can be limited below 0.2. I. INTRODUCTION
So far there have been tens of confirmed gravitational wave (GW) detections [1–13] sincethe first GW event GW150914 observed by the Laser Interferometer Gravitational-WaveObservatory (LIGO) Scientific Collaboration and the Virgo Collaboration [1, 2]. The tran-sient GWs detected by ground-based detectors are the merging signals with the duration of ∗ chao [email protected] † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ g r- q c ] F e b seconds to minutes in the frequency band around several-hundreds hertz, so it is impossibleto measure the signals’ polarization content with advanced LIGO because the two detectorsare co-oriented [14, 15] and the observed signal is so short that we can ignore the motion ofthe detector around the Sun. However, some preliminary results on the signals’ polarizationcontent were obtained with the LIGO-Virgo network [5, 8, 13, 16]. In general metric theoryof gravity, GWs can have up to six polarization modes [17, 18]: two transverse-tracelesstensor modes (+ and × ), two vector modes ( x and y ), a scalar breathing mode ( b ) and ascalar longitudinal mode ( l ). The specific modified theory of gravity uniquely determines thepolarization modes. For example, in Brans-Dicke theory [19] there exists one extra breath-ing mode beyond the two transverse-traceless tensor modes of General Relativity (GR). Thescalar polarization mode is a mixture of breathing mode and longitudinal mode if the scalarfield is massive in generic theory of gravity [20–23]. Einstein-Æther theory [24] predictsthe existence of scalar and vector polarization modes [23, 25, 26] while generalized tensor-vector-scalar theories, such as TeVeS theory [27], predict the existence of all 6 polarizationmodes [23]. Therefore, the detection of extra polarization modes allows us to falsify GR. Toseparate the polarization modes of GWs, in principle the number of ground-based detectorsoriented differently should be equal to or larger than the number of the polarization modes.The network of ground-based detectors including advanced LIGO [14, 15], advanced Virgo[28], KAGRA [29, 30] and LIGO India has the ability of probing extra polarization modes[31–33]. In the past years, different methods were developed to probe nontensorial polariza-tions in stochastic GW backgrounds [34–36], continuous GWs [37–40], GW bursts [41, 42]and GWs from compact binary coalescences [43, 44]. In particular, the Fisher informationmatrix approximation was usually used to estimate the parameters of the source and todiscuss the measurement of polarization modes [44–58].For stellar or intermediate black hole binaries with the mass range 100 − M (cid:12) , in theearly inspiral phase the GW frequency is in the mHz range and its evolution can be neglectedduring the mission of the space-based detector. The proposed space-based observatoriesincluding LISA [59, 60], TianQin [61] and Taiji [62] can detect the monochromatic GWsignals emitted by these wealthy sources. Due to the orbital motion of the detector inspace, along its trajectory the single detector can be effectively regarded as a set of virtualdetectors and therefore forms a virtual network to measure the polarization content of themonochromatic GW signals. By taking a specific linear combination of the outputs inthe network of detectors it is possible to remove any tensor signal present in the data[63, 64]. A particular χ distribution is followed by the null energy constructed with thismethod [64] when the null energy is calculated at the true sky position. If nontensorialpolarizations present in the data, then the null energy evaluated at the true sky positionno longer follows the particular χ distribution [33]. Based on these results, we introduceone concrete data analysis pipeline to check the existence of extra polarizations for a singlespace-borne detector. Apart from being able to detect mixtures of polarization modes andalternative polarization modes, this method has the added advantage that no waveformmodel is needed, and monochromatic GWs from any kind of compact binary systems withknown sky positions and frequencies can be used.The paper is organized as follows. In Section II, we describe the basics of the GW signalregistered in the space-borne GW detector. In Section III, we present general monochro-matic waveforms including extra polarization modes and construct the method to discoveralternative polarization modes in Section III A. We then apply the method on the referencesource J0806.3+1527 of TianQin with 90-days simulation data for both LISA and Taiji inSection III B. In Section IV, we explore the possibility of detecting extra polarization modesfrom the network consisting of LISA, TianQin and Taiji. We present our conclusion anddiscussion in Section V. II. GRAVITATIONAL WAVE SIGNAL
It is convenient to describe GWs and the motion of space-based GW detectors like LISA,TianQin and Taiji in the heliocentric coordinate system. For GWs propagating in thedirection ˆ w , we introduce a set of unit vectors { ˆ w, ˆ θ, ˆ φ } which are perpendicular to eachother to form an orthonormal coordinate system,ˆ θ = cos( θ ) cos( φ )ˆ e x + cos( θ ) sin( φ )ˆ e y − sin( θ )ˆ e z , ˆ φ = − sin( φ )ˆ e x + cos( φ )ˆ e y , ˆ w = − sin( θ ) cos( φ )ˆ e x − sin( θ ) sin( φ )ˆ e y − cos( θ )ˆ e z . (1)To describe the six possible polarization modes of GWs in general metric theory of gravity,the polarization angle ψ is introduced to form polarization axes of the gravitational radiation,ˆ p = cos ψ ˆ θ + sin ψ ˆ φ, ˆ q = − sin ψ ˆ θ + cos ψ ˆ φ. (2)The polarization tensors are e + ij = ˆ p i ˆ p j − ˆ q i ˆ q j , e × ij = ˆ p i ˆ q j + ˆ q i ˆ p j ,e xij = ˆ p i ˆ w j + ˆ w i ˆ p j , e yij = ˆ q i ˆ w j + ˆ w i ˆ q j ,e lij = ˆ w i ˆ w j , e bij = ˆ p i ˆ p j + ˆ q i ˆ q j . (3)For this particular choice of { ˆ w, ˆ p, ˆ q } , GWs in general metric theory of gravity have the form h ij ( t ) = (cid:88) A e Aij h A ( t ) , (4)where A = + , × , x, y, l, b stands for the six polarization states: two transverse-tracelesstensor modes (+ and × ), two vector modes ( x and y ), a breathing mode ( b ) and a longitudinalmode ( l ).For a monochromatic GW with the frequency f propagating along the direction ˆ ω , theoutput in an equal-arm space-based interferometric detector such as LISA, TianQin andTaiji with a single round trip of light travel is s ( t ) = (cid:88) A F A h A ( t ) e iφ D ( t ) , (5) φ D ( t ) = 2 πf Rc sin θ cos (cid:18) πtP − φ − φ α (cid:19) , (6) F A = (cid:88) i,j D ij e Aij , (7)where φ D ( t ) is the Doppler phase, φ α is the ecliptic longitude of the detector α at t = 0, therotational period P is 1 year and the radius R of the orbit is 1 AU. The detector tensor D ij is D ij = 12 [ˆ u i ˆ u j T ( f, ˆ u · ˆ ω ) − ˆ v i ˆ v j T ( f, ˆ v · ˆ ω )] , (8)where ˆ u and ˆ v are the unit vectors along the arms of the detector and T ( f, ˆ u · ˆ ω ) is [65, 66] T ( f, ˆ u · ˆ w ) = 12 { sinc[ f f ∗ (1 − ˆ u · ˆ ω )] exp[ − i f f ∗ (3 + ˆ u · ˆ ω )]+ sinc[ f f ∗ (1 + ˆ u · ˆ ω )] exp[ − i f f ∗ (1 + ˆ u · ˆ ω )] } , (9)sinc( x ) = sin x/x , f ∗ = c/ (2 πL ) is the transfer frequency of the detector, c is the speed oflight and L is the arm length of the detector. Note that in the long wavelength approximation f (cid:28) f ∗ , we have T ( f, ˆ u · ˆ w ) → | F A | wasdiscussed in [67–73] III. METHODOLOGY
Now we consider the strain output d ( t ) produced by a monochromatic GW for the space-based GW detector in heliocentric coordinate system. A monochromatic GW assumed tobe emitted from a source with the sky location ˆΩ( θ, φ ), arrives at the center of the detectorat time t . If only the tensor polarization modes are present, one has d w ( t ) = F + w ( ˆΩ , f, t ) h + ( t ) e iφ D ( t ) + F × w ( ˆΩ , f, t ) h × ( t ) e iφ D ( t ) + n w ( t ) , (10)where F + w and F × w are the noise-weighted beam pattern functions and n w ( t ) is the whitenednoise. The noise-weighted beam pattern functions and noise-weighted data are [74] F w = F (cid:112) S n ( f ) , d w = d (cid:112) S n ( f ) . (11)For space-based interferometers, the noise power spectral density S n ( f ) is [60, 61, 75] S n ( f ) = S x L + 4 S a (2 πf ) L (1 + 10 − Hz f ) . (12)For LISA, the acceleration noise is √ S a = 3 × − m s − / Hz / , the displacement noiseis √ S x = 15 pm/Hz / , the arm length is L = 2 . × km , and its transfer frequency is f ∗ = 0 .
02 Hz [60]. Similarly, for TianQin the acceleration noise is √ S a = 10 − m s − / Hz / ,the displacement noise is √ S x = 1 pm/Hz / , the arm length is L = √ × km, andits transfer frequency is f ∗ = 0 .
28 Hz [61]. For Taiji the acceleration noise is √ S a =3 × − m s − / Hz / , the displacement noise is √ S x = 8 pm/Hz / , the arm length is L = 3 × km, and its transfer frequency is f ∗ = 0 .
016 Hz [76].In the following we always use the noise-weighted beam pattern functions and noise-weighted data, so we ignore the label w for simplicity. The signal in Eq. (10) can berewritten in another form d ( t ) = ¯ h + F + ( ˆΩ , f, t ) e πift + iφ D ( t ) + ¯ h × F × ( ˆΩ , f, t ) e πift + iφ D ( t ) + n ( t ) , (13)where ¯ h + = A (cid:2) ( ι ) (cid:3) exp( iφ ) , ¯ h × =2 i A cos( ι ) exp( iφ ) , A is the amplitude and we assume the quadrupole formula for the GW waveform of a binarysystem with the lowest-order post-Newtonian approximation. A. The Method for a Single Detector
We denote the observational data d i = d ( t i ) at discrete times in a more compact matrixform d = Fh + n , (14)where d = d ... d D , h = ¯ h + ¯ h × , n = n ... n D , (15) F = (cid:16) F + F × (cid:17) = F + ( t ) e πift + iφ D ( t ) F × ( t ) e πift + iφ D ( t ) ... ... F + ( t D ) e πift D + iφ D ( t D ) F × ( t D ) e πift D + iφ D ( t D ) , (16) D = 0 , , , , ..., i, ... labels the data observed by the detector at the time t i = i ∗ ∆ t and1 / ∆ t is the sampling rate. The GW signal s = Fh spanned by F + and F × can be viewedas being in a subspace of the space of detector outputs. We can construct the null projector P null ( ˆΩ) [77] to project away the signal if the projector is constructed with the source’s skylocation and frequency [33]. The null projector is given by P null = I − F ( F † F ) − F † , (17)where † denotes Hermitian conjugation. Applying the null projector on the strain data d inEq. (14), we obtain z ( ˆΩ , f ) = P null ( ˆΩ , f ) d = P null ( ˆΩ , f ) F ( ˆΩ , f ) h + P null ( ˆΩ , f ) n = P null ( ˆΩ , f ) n , (18)where z ( ˆΩ , f ) is the null stream which only consists of the noise living in a subspace that isorthogonal to the one spanned by F + and F × .To consider the effect of the polarization modes other than the tensor modes, we param-eterize the other polarization modes as [78] h x = α A sin ι cos ι exp(2 πif t + iφ ) ,h y = α A sin ι exp(2 πif t + iφ ) ,h l = α A sin ι exp(2 πif t + iφ ) ,h b = α A sin ι exp(2 πif t + iφ ) , (19)where α can be viewed as an indicative of the intrinsic strength of the other polarizationcomponents relative to the tensor modes. Including the polarization content beyond thetensor polarizations, the data matrix becomes d = F te ( ˆΩ , f ) h te + F e ( ˆΩ , f ) h e + n , (20)where the superscript te means summing over + and × , while the superscript e meanssumming over whatever additional polarizations present. The null stream obtained from thenull projector with pure-tensor beam pattern matrix is given by z ( ˆΩ , f ) = P null d = P null n + P null F e h e . (21)The last term signifies the presence of extra polarizations other than the tensor modes.If there are additional polarization modes in the GW, then the data ˜ z ( ˆΩ , f ) which is thediscrete Fourier transform of z ( ˆΩ , f ) [77], acted by the null projector with the known skyposition ˆΩ and frequency f has a discrete component at f in the frequency domain. B. Simulation Result
Taking TianQin’s reference source J0806.3+1527 located at ( θ = 94 . ◦ , φ = 120 . ◦ ) asan example, we simulate the strain output in the spaced-based GW detector. In GR thequadrupole formula provides the lowest-order post-Newtonian GW waveform for a binarysystem as h + = A (cid:2) ( ι ) (cid:3) exp(2 πif t + iφ ) ,h × =2 i A cos( ι ) exp(2 πif t + iφ ) , (22)where A = 2( G M /c ) / ( πf /c ) / /D L is the GW overall amplitude, M = 0 . M (cid:12) (wetake the component masses 0 . M (cid:12) and 0 . M (cid:12) ) is the chirp mass, D L = 0 . ι = π/ φ is the initial GW phase at the start of observation, f = 6 .
22 mHz is theemitted GW frequency of the reference source [79–83].For the reference source J0806.3+1527 and the total observation time of 90 days, wechoose the sampling rate as 0.02 Hz. We inject a set of mock waveforms (19) in additionto simulated signals from GR. Fig. 1 shows that extra polarization modes with α ≥ . α ≥ . α takes the values (0.5, 0.8, 1.0). The SNR for Taiji is about 368 if only the tensormodes exist and the SNR is about (390, 415, 436) when α takes the values (0.5, 0.8, 1.0).The SNR for TiaQin is about 255 whatever the value α takes.
1. The influence of observation time
We analyze the effect of the observation time on the amplitude of | ˜ z | using the referencesource J0806.3+1527 and the result is shown in Fig. 2. In Fig. 2, we take α = 10 and f is theGW frequency emitted by the reference source. From Fig. 2 we find that as the observationtime increases, the amplitude of | ˜ z ( f ) | increases. So we can increase the observation timeto improve the possibility of the detection of extra polarization modes.
2. The influence of virtual detector number
It is well known that three detectors with different orientations are enough to discriminateextra polarization mode from the tensor modes. For the reference source J0806.3+1527 andthe total observation time of 90 days, we split the data into three identical lengthy segmentswith one-month data each and regard them as three independent detectors’ data. Thismethod decreases the number of virtual detectors but increases the effective observationtime for each detector. The observation time T for each virtual detector becomes 30 days, Frequency [Hz]
LISA | z | = 0.0 Frequency [Hz]
LISA | z | = 0.5 Frequency [Hz] TianQin | z | = 0.5 Frequency [Hz] Taiji | z | = 0.2 FIG. 1. The results of the null stream | ˜ z | , α = 0 in the top left panel and α = 0 . α = 0 . α = 0 . Observation time (days) | z ( f )| FIG. 2. The dependence of the amplitude of | ˜ z | on the observation time for LISA. We choose α = 10. s ( t ) = (cid:0) F + ( t ) h + ( t ) + F × ( t ) h × ( t ) (cid:1) e iφ D ( t ) , (23) s ( t ) = (cid:0) F + ( t + T ) h + ( t + T ) + F × ( t + T ) h × ( t + T ) (cid:1) e iφ D ( t + T ) , (24) s ( t ) = (cid:0) F + ( t + 2 T ) h + ( t + 2 T ) + F × ( t + 2 T ) h × ( t + 2 T ) (cid:1) e iφ D ( t +2 T ) . (25)We rewrite the three detectors’ observation data in the matrix form d ( t ) = F ( t ) h + n ( t ) , (26)where d ( t ) = d ( t ) d ( t ) d ( t ) , h = ¯ h + ¯ h × , n ( t ) = n ( t ) n ( t ) n ( t ) = n ( t ) n ( t + T ) n ( t + 2 T ) , (27)and F ( t ) = F + ( t ) e πift + iφ D ( t ) F × ( t ) e πift + iφ D ( t ) F + ( t + T ) e πif ( t + T )+ iφ D ( t + T ) F × ( t + T ) e πif ( t + T )+ iφ D ( t + T ) F + ( t + 2 T ) e πif ( t +2 T )+ iφ D ( t +2 T ) F × ( t + 2 T ) e πif ( t +2 T )+ iφ D ( t +2 T ) . (28)The signal can be seen as the data observed at a given time t by three different detectorsat the same time. Within the observation period of one month, there are many observationpoints. For any given time, we get z ( ˆΩ true , f, t ) = P null ( t ) d ( t )= P null n + P null F e ( t ) h e . (29)Simulating the data in LISA with the waveforms (19), we then apply the method (29) tothe data and the result is shown in Fig. 3. From Fig. 3, we see that this method candetect extra polarization modes with α ≥ .
8. Therefore, null result of extra polarizationmodes limits α to be below 0.8 with this method. The method with fewer virtual detectorsperforms worse than the method in the previous section with more virtual detectors. IV. THE NETWORK
LISA, TianQin and Taiji have different orientations in the orbit around the Sun, so wecan combine these three detectors to form a network to measure the polarization of GWs1
Frequency [Hz] | z | = 0.0 Frequency [Hz] | z | = 0.5 Frequency [Hz] | z | = 0.8 Frequency [Hz] | z | = 1.0 FIG. 3. LISA’s results of the null stream for three virtual detectors with one-month data each. if they all will be in operation around the same time. Combining the data in the threedetectors, we get d ( t ) = F ( t ) h + n ( t ) , (30)where d ( t ) = d ( t ) d ( t ) d ( t ) , h = ¯ h + ¯ h × , n ( t ) = n ( t ) n ( t ) n ( t ) , (31)and F ( t ) = F + LISA ( t ) e πift + iφ D ( t ) F × LISA ( t ) e πift + iφ D ( t ) F + T Q ( t ) e πif ( t )+ iφ D ( t ) F × T Q ( t ) e πif ( t )+ iφ D ( t ) F + T J ( t ) e πif ( t )+ iφ D ( t ) F × T J ( t ) e πif ( t )+ iφ D ( t ) . (32)Simulating the data in each detector with the waveforms (19) for the observation time of90 days, we then apply the method (29) to the combined data and the results are shown inFig. 4. From Fig. 4, we see that null stream limits α to be below 0.2 and extra polarizationmodes with α ≥ . Frequency [Hz] | z | = 0.0 Frequency [Hz] | z | = 0.1 Frequency [Hz] | z | = 0.2 Frequency [Hz] | z | = 0.5 FIG. 4. The results for the combined network of LISA, Taiji and TianQin. sensitivity of the detection of polarization by about 2.5 times compared with LISA alone.
V. CONCLUSION
We introduce one concrete data analysis pipeline to test extra polarization modes ofGWs for a single space-borne detector. This method is valid for LISA and Taiji due totheir changing orientation of the detector plane. For the reference source J0806.3+1527of TianQin, we show that α can be limited below 0.5 for LISA and below 0.2 for Taiji.We also analyzed the influence of observation time and virtual number of detectors on thesensitivity of the detection of extra polarization modes and we find that the more virtualdetector number and the more observation time, the better sensitivity of the detectionof extra polarization modes. However the large number of virtual detectors costs morecomputational memory and time because we need to handle larger matrix. We concludethat a single space-borne detector can detect non-tensorial polarizations. For the networkconsisting of LISA, Taiji and TianQin, it improves the limit on the parameter α by about32.5 times for GWs with the frequency f = 6 .
22 mHz compared with LISA alone.
ACKNOWLEDGMENTS
This work is supported by the National Key Research and Development Program of Chinaunder Grant No. 2020YFC2201504, the National Natural Science Foundation of China underGrant No. 11875136 and the Major Program of the National Natural Science Foundation ofChina under Grant No. 11690021 [1] B. P. Abbott et al. (LIGO Scientific, Virgo), Observation of Gravitational Waves from a BinaryBlack Hole Merger, Phys. Rev. Lett. , 061102 (2016), arXiv:1602.03837.[2] B. P. Abbott et al. (LIGO Scientific, Virgo), GW150914: The Advanced LIGO Detectors inthe Era of First Discoveries, Phys. Rev. Lett. , 131103 (2016), arXiv:1602.03838.[3] B. P. Abbott et al. (LIGO Scientific, Virgo), GW151226: Observation of Gravitational Wavesfrom a 22-Solar-Mass Binary Black Hole Coalescence, Phys. Rev. Lett. , 241103 (2016),arXiv:1606.04855.[4] B. P. Abbott et al. (LIGO Scientific, VIRGO), GW170104: Observation of a 50-Solar-MassBinary Black Hole Coalescence at Redshift 0.2, Phys. Rev. Lett. , 221101 (2017), [Erratum:Phys.Rev.Lett. 121, 129901 (2018)], arXiv:1706.01812.[5] B. P. Abbott et al. (LIGO Scientific, Virgo), GW170814: A Three-Detector Observation ofGravitational Waves from a Binary Black Hole Coalescence, Phys. Rev. Lett. , 141101(2017), arXiv:1709.09660.[6] B. P. Abbott et al. (LIGO Scientific, Virgo), GW170817: Observation of Gravitational Wavesfrom a Binary Neutron Star Inspiral, Phys. Rev. Lett. , 161101 (2017), arXiv:1710.05832.[7] B. . P. . Abbott et al. (LIGO Scientific, Virgo), GW170608: Observation of a 19-solar-massBinary Black Hole Coalescence, Astrophys. J. , L35 (2017), arXiv:1711.05578.[8] B. P. Abbott et al. (LIGO Scientific, Virgo), GWTC-1: A Gravitational-Wave TransientCatalog of Compact Binary Mergers Observed by LIGO and Virgo during the First andSecond Observing Runs, Phys. Rev. X , 031040 (2019), arXiv:1811.12907. [9] B. P. Abbott et al. (LIGO Scientific, Virgo), GW190425: Observation of a Compact BinaryCoalescence with Total Mass ∼ . M (cid:12) , Astrophys. J. Lett. , L3 (2020), arXiv:2001.01761.[10] R. Abbott et al. (LIGO Scientific, Virgo), GW190412: Observation of a Binary-Black-HoleCoalescence with Asymmetric Masses, Phys. Rev. D , 043015 (2020), arXiv:2004.08342.[11] R. Abbott et al. (LIGO Scientific, Virgo), GW190814: Gravitational Waves from the Coales-cence of a 23 Solar Mass Black Hole with a 2.6 Solar Mass Compact Object, Astrophys. J.Lett. , L44 (2020), arXiv:2006.12611.[12] R. Abbott et al. (LIGO Scientific, Virgo), GW190521: A Binary Black Hole Merger with aTotal Mass of 150 M (cid:12) , Phys. Rev. Lett. , 101102 (2020), arXiv:2009.01075.[13] R. Abbott et al. (LIGO Scientific, Virgo), GWTC-2: Compact Binary Coalescences Observedby LIGO and Virgo During the First Half of the Third Observing Run, arXiv:2010.14527.[14] G. M. Harry (LIGO Scientific), Advanced LIGO: The next generation of gravitational wavedetectors, Class. Quant. Grav. , 084006 (2010).[15] J. Aasi et al. (LIGO Scientific), Advanced LIGO, Class. Quant. Grav. , 074001 (2015),arXiv:1411.4547.[16] B. P. Abbott et al. (LIGO Scientific, Virgo), Tests of General Relativity with GW170817,Phys. Rev. Lett. , 011102 (2019), arXiv:1811.00364.[17] D. M. Eardley, D. L. Lee, A. P. Lightman, R. V. Wagoner, and C. M. Will, Gravitational-waveobservations as a tool for testing relativistic gravity, Phys. Rev. Lett. , 884 (1973).[18] D. M. Eardley, D. L. Lee, and A. P. Lightman, Gravitational-wave observations as a tool fortesting relativistic gravity, Phys. Rev. D , 3308 (1973).[19] C. Brans and R. H. Dicke, Mach’s principle and a relativistic theory of gravitation, Phys. Rev. , 925 (1961).[20] D. Liang, Y. Gong, S. Hou, and Y. Liu, Polarizations of gravitational waves in f ( R ) gravity,Phys. Rev. D , 104034 (2017), arXiv:1701.05998.[21] S. Hou, Y. Gong, and Y. Liu, Polarizations of Gravitational Waves in Horndeski Theory, Eur.Phys. J. C , 378 (2018), arXiv:1704.01899.[22] Y. Gong, S. Hou, E. Papantonopoulos, and D. Tzortzis, Gravitational waves and the polariza-tions in Hoˇrava gravity after GW170817, Phys. Rev. D , 104017 (2018), arXiv:1808.00632.[23] Y. Gong, S. Hou, D. Liang, and E. Papantonopoulos, Gravitational waves in Einstein-æther and generalized TeVeS theory after GW170817, Phys. Rev. D , 084040 (2018), arXiv:1801.03382.[24] T. Jacobson and D. Mattingly, Einstein-Aether waves, Phys. Rev. D , 024003 (2004),arXiv:gr-qc/0402005.[25] K. Lin, X. Zhao, C. Zhang, T. Liu, B. Wang, S. Zhang, X. Zhang, W. Zhao, T. Zhu, andA. Wang, Gravitational waveforms, polarizations, response functions, and energy losses oftriple systems in Einstein-aether theory, Phys. Rev. D , 023010 (2019), arXiv:1810.07707.[26] C. Zhang, X. Zhao, A. Wang, B. Wang, K. Yagi, N. Yunes, W. Zhao, and T. Zhu, Gravitationalwaves from the quasicircular inspiral of compact binaries in Einstein-aether theory, Phys. Rev.D , 044002 (2020), arXiv:1911.10278.[27] J. D. Bekenstein, Relativistic gravitation theory for the MOND paradigm, Phys. Rev. D ,083509 (2004), [Erratum: Phys.Rev.D 71, 069901 (2005)], arXiv:astro-ph/0403694.[28] F. Acernese et al. (VIRGO), Advanced Virgo: a second-generation interferometric gravita-tional wave detector, Class. Quant. Grav. , 024001 (2015), arXiv:1408.3978.[29] K. Somiya (KAGRA), Detector configuration of KAGRA: The Japanese cryogenicgravitational-wave detector, Class. Quant. Grav. , 124007 (2012), arXiv:1111.7185.[30] Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. Sekiguchi, D. Tatsumi, andH. Yamamoto (KAGRA), Interferometer design of the KAGRA gravitational wave detector,Phys. Rev. D , 043007 (2013), arXiv:1306.6747.[31] Y. Hagihara, N. Era, D. Iikawa, and H. Asada, Probing gravitational wave polarizationswith Advanced LIGO, Advanced Virgo and KAGRA, Phys. Rev. D , 064035 (2018),arXiv:1807.07234.[32] Y. Hagihara, N. Era, D. Iikawa, A. Nishizawa, and H. Asada, Constraining extra gravitationalwave polarizations with Advanced LIGO, Advanced Virgo and KAGRA and upper boundsfrom GW170817, Phys. Rev. D , 064010 (2019), arXiv:1904.02300.[33] P. T. H. Pang, R. K. L. Lo, I. C. F. Wong, T. G. F. Li, and C. Van Den Broeck, Generic searchesfor alternative gravitational wave polarizations with networks of interferometric detectors,Phys. Rev. D , 104055 (2020), arXiv:2003.07375.[34] A. Nishizawa, A. Taruya, K. Hayama, S. Kawamura, and M.-a. Sakagami, Probing non-tensorial polarizations of stochastic gravitational-wave backgrounds with ground-based laserinterferometers, Phys. Rev. D , 082002 (2009), arXiv:0903.0528. [35] T. Callister, A. S. Biscoveanu, N. Christensen, M. Isi, A. Matas, O. Minazzoli, T. Regimbau,M. Sakellariadou, J. Tasson, and E. Thrane, Polarization-based Tests of Gravity with theStochastic Gravitational-Wave Background, Phys. Rev. X , 041058 (2017), arXiv:1704.08373.[36] B. P. Abbott et al. (LIGO Scientific, Virgo), Search for Tensor, Vector, and Scalar Polariza-tions in the Stochastic Gravitational-Wave Background, Phys. Rev. Lett. , 201102 (2018),arXiv:1802.10194.[37] M. Isi, A. J. Weinstein, C. Mead, and M. Pitkin, Detecting Beyond-Einstein Polarizations ofContinuous Gravitational Waves, Phys. Rev. D , 082002 (2015), arXiv:1502.00333.[38] M. Isi, M. Pitkin, and A. J. Weinstein, Probing Dynamical Gravity with the Polarization ofContinuous Gravitational Waves, Phys. Rev. D , 042001 (2017), arXiv:1703.07530.[39] B. P. Abbott et al. (LIGO Scientific, Virgo), First search for nontensorial gravitational wavesfrom known pulsars, Phys. Rev. Lett. , 031104 (2018), arXiv:1709.09203.[40] L. O’Beirne, N. J. Cornish, S. J. Vigeland, and S. R. Taylor, Constraining alternative polar-ization states of gravitational waves from individual black hole binaries using pulsar timingarrays, Phys. Rev. D , 124039 (2019), arXiv:1904.02744.[41] K. Hayama and A. Nishizawa, Model-independent test of gravity with a network of ground-based gravitational-wave detectors, Phys. Rev. D , 062003 (2013), arXiv:1208.4596.[42] I. Di Palma and M. Drago, Estimation of the gravitational wave polarizations from a nontem-plate search, Phys. Rev. D , 023011 (2018), arXiv:1712.05580.[43] H. Takeda, A. Nishizawa, Y. Michimura, K. Nagano, K. Komori, M. Ando, and K. Hayama,Polarization test of gravitational waves from compact binary coalescences, Phys. Rev. D ,022008 (2018), arXiv:1806.02182.[44] H. Takeda, A. Nishizawa, K. Nagano, Y. Michimura, K. Komori, M. Ando, and K. Hayama,Prospects for gravitational-wave polarization tests from compact binary mergers with futureground-based detectors, Phys. Rev. D , 042001 (2019), arXiv:1904.09989.[45] M. Vallisneri, Use and abuse of the Fisher information matrix in the assessment ofgravitational-wave parameter-estimation prospects, Phys. Rev. D , 042001 (2008), arXiv:gr-qc/0703086.[46] L. Wen and Y. Chen, Geometrical Expression for the Angular Resolution of a Network ofGravitational-Wave Detectors, Phys. Rev. D , 082001 (2010), arXiv:1003.2504. [47] B. P. Abbott et al. (KAGRA, LIGO Scientific, VIRGO), Prospects for Observing and Lo-calizing Gravitational-Wave Transients with Advanced LIGO, Advanced Virgo and KAGRA,Living Rev. Rel. , 3 (2018), arXiv:1304.0670.[48] K. Grover, S. Fairhurst, B. F. Farr, I. Mandel, C. Rodriguez, T. Sidery, and A. Vecchio,Comparison of Gravitational Wave Detector Network Sky Localization Approximations, Phys.Rev. D , 042004 (2014), arXiv:1310.7454.[49] C. P. L. Berry et al. , Parameter estimation for binary neutron-star coalescences with realisticnoise during the Advanced LIGO era, Astrophys. J. , 114 (2015), arXiv:1411.6934.[50] L. P. Singer and L. R. Price, Rapid Bayesian position reconstruction for gravitational-wavetransients, Phys. Rev. D , 024013 (2016), arXiv:1508.03634.[51] B. B´ecsy, P. Raffai, N. J. Cornish, R. Essick, J. Kanner, E. Katsavounidis, T. B. Littenberg,M. Millhouse, and S. Vitale, Parameter estimation for gravitational-wave bursts with theBayesWave pipeline, Astrophys. J. , 15 (2017), arXiv:1612.02003.[52] W. Zhao and L. Wen, Localization accuracy of compact binary coalescences detected by thethird-generation gravitational-wave detectors and implication for cosmology, Phys. Rev. D , 064031 (2018), arXiv:1710.05325.[53] C. Mills, V. Tiwari, and S. Fairhurst, Localization of binary neutron star mergers with sec-ond and third generation gravitational-wave detectors, Phys. Rev. D , 104064 (2018),arXiv:1708.00806.[54] S. Fairhurst, Localization of transient gravitational wave sources: beyond triangulation, Class.Quant. Grav. , 105002 (2018), arXiv:1712.04724.[55] Y. Fujii, T. Adams, F. Marion, and R. Flaminio, Fast localization of coalescing binaries witha heterogeneous network of advanced gravitational wave detectors, Astropart. Phys. , 1(2019), arXiv:1905.02362.[56] C. Liu, W.-H. Ruan, and Z.-K. Guo, Constraining gravitational-wave polarizations with Taiji,Phys. Rev. D , 124050 (2020), arXiv:2006.04413.[57] C. Zhang, Y. Gong, H. Liu, B. Wang, and C. Zhang, Sky localization of space-based gravita-tional wave detectors, arXiv:2009.03476.[58] C. Zhang, Y. Gong, B. Wang, and C. Zhang, Accuracy of estimation of parameters withspace-borne gravitational wave observatory, arXiv:2012.01043. [59] K. Danzmann, LISA: An ESA cornerstone mission for a gravitational wave observatory, Class.Quant. Grav. , 1399 (1997).[60] P. Amaro-Seoane et al. (LISA), Laser Interferometer Space Antenna, arXiv:1702.00786.[61] J. Luo et al. (TianQin), TianQin: a space-borne gravitational wave detector, Class. Quant.Grav. , 035010 (2016), arXiv:1512.02076.[62] W.-R. Hu and Y.-L. Wu, The Taiji Program in Space for gravitational wave physics and thenature of gravity, Natl. Sci. Rev. , 685 (2017).[63] Y. Guersel and M. Tinto, Near optimal solution to the inverse problem for gravitational wavebursts, Phys. Rev. D , 3884 (1989).[64] S. Chatterji, A. Lazzarini, L. Stein, P. J. Sutton, A. Searle, and M. Tinto, Coherent networkanalysis technique for discriminating gravitational-wave bursts from instrumental noise, Phys.Rev. D , 082005 (2006), arXiv:gr-qc/0605002.[65] F. B. Estabrook and H. D. Wahlquist, Response of Doppler spacecraft tracking to gravitationalradiation, Gen. Relat. Gravit. , 439 (1975).[66] N. J. Cornish and S. L. Larson, Space missions to detect the cosmic gravitational wave back-ground, Class. Quant. Grav. , 3473 (2001), arXiv:gr-qc/0103075.[67] S. L. Larson, W. A. Hiscock, and R. W. Hellings, Sensitivity curves for spaceborne gravita-tional wave interferometers, Phys. Rev. D , 062001 (2000), arXiv:gr-qc/9909080.[68] S. L. Larson, R. W. Hellings, and W. A. Hiscock, Unequal arm space borne gravitational wavedetectors, Phys. Rev. D , 062001 (2002), arXiv:gr-qc/0206081.[69] M. Tinto and M. E. da Silva Alves, LISA Sensitivities to Gravitational Waves from RelativisticMetric Theories of Gravity, Phys. Rev. D , 122003 (2010), arXiv:1010.1302.[70] A. Blaut, Angular and frequency response of the gravitational wave interferometers in themetric theories of gravity, Phys. Rev. D , 043005 (2012), arXiv:1901.11268.[71] D. Liang, Y. Gong, A. J. Weinstein, C. Zhang, and C. Zhang, Frequency response ofspace-based interferometric gravitational-wave detectors, Phys. Rev. D , 104027 (2019),arXiv:1901.09624.[72] C. Zhang, Q. Gao, Y. Gong, D. Liang, A. J. Weinstein, and C. Zhang, Frequency responseof time-delay interferometry for space-based gravitational wave antenna, Phys. Rev. D ,064033 (2019), arXiv:1906.10901. [73] C. Zhang, Q. Gao, Y. Gong, B. Wang, A. J. Weinstein, and C. Zhang, Full analytical formulasfor frequency response of space-based gravitational wave detectors, Phys. Rev. D , 124027(2020), arXiv:2003.01441.[74] B. Allen, W. G. Anderson, P. R. Brady, D. A. Brown, and J. D. E. Creighton, FINDCHIRP:An Algorithm for detection of gravitational waves from inspiraling compact binaries, Phys.Rev. D , 122006 (2012), arXiv:gr-qc/0509116.[75] X.-C. Hu, X.-H. Li, Y. Wang, W.-F. Feng, M.-Y. Zhou, Y.-M. Hu, S.-C. Hu, J.-W. Mei, andC.-G. Shao, Fundamentals of the orbit and response for TianQin, Class. Quant. Grav. ,095008 (2018), arXiv:1803.03368.[76] W.-H. Ruan, Z.-K. Guo, R.-G. Cai, and Y.-Z. Zhang, Taiji program: Gravitational-wavesources, Int. J. Mod. Phys. A , 2050075 (2020), arXiv:1807.09495.[77] P. J. Sutton et al. , X-Pipeline: An Analysis package for autonomous gravitational-wave burstsearches, New J. Phys. , 053034 (2010), arXiv:0908.3665.[78] K. Chatziioannou, N. Yunes, and N. Cornish, Model-Independent Test of General Relativity:An Extended post-Einsteinian Framework with Complete Polarization Content, Phys. Rev. D , 022004 (2012), [Erratum: Phys.Rev.D 95, 129901 (2017)], arXiv:1204.2585.[79] G. L. Israel et al. , Rxj0806.3+1527: a double degenerate binary with the shortest knownorbital period (321s), Astron. Astrophys. , L13 (2002), arXiv:astro-ph/0203043.[80] S. C. C. Barros, T. R. Marsh, P. Groot, G. Nelemans, G. Ramsay, G. Roelofs, D. Steeghs,and J. Wilms, Geometrical constraints upon the unipolar model of V407 Vul and RXJ0806.3+1527, Mon. Not. Roy. Astron. Soc. , 1306 (2005), arXiv:astro-ph/0412368.[81] G. H. A. Roelofs, A. Rau, T. R. Marsh, D. Steeghs, P. J. Groot, and G. Nelemans, Spectro-scopic Evidence for a 5.4-Minute Orbital Period in HM Cancri, Astrophys. J. Lett. , L138(2010), arXiv:1003.0658.[82] P. Esposito, G. L. Israel, S. Dall’Osso, and S. Covino, Swift X-ray and ultraviolet observationsof the shortest orbital period double-degenerate system RX J0806.3+1527 (HM Cnc), Astron.Astrophys. , A117 (2014), arXiv:1311.6973.[83] T. Kupfer, V. Korol, S. Shah, G. Nelemans, T. R. Marsh, G. Ramsay, P. J. Groot, D. T. H.Steeghs, and E. M. Rossi, LISA verification binaries with updated distances from Gaia DataRelease 2, Mon. Not. Roy. Astron. Soc.480