Testing the Polarization of Gravitational-wave Background with the LISA-TianQin Network

While general relativity predicts only two tensor modes for gravitational-wave (GW) polarization, general metric theories of gravity allow for up to four additional modes, including two vector and two scalar modes. Observing the polarization modes of GWs could provide a direct test of the modified gravity. The stochastic GW background (SGWB), which can be detected by space-based laser-interferometric detectors at design sensitivity, will provide an opportunity to directly measure alternative polarization. In this paper, we investigate the performance of the LISA-TianQin network for detecting alternative polarizations of stochastic backgrounds, and propose a method to separate different polarization modes. First, we generalize the small antenna approximation to compute the overlap reduction functions for the SGWB with arbitrary polarization, which is suitable for any time-delay interferometry combination. Then we analyze the detection capability of LISA-TianQin for the SGWB with different polarizations. Based on the orbital characteristics of LISA-TianQin, we propose a method to distinguish different polarization modes from their mixed data. Finally, simulation tests are performed to verify the effectiveness of the method. The results of the simulations demonstrate that LISA-TianQin, when employing our proposed method, has the ability to differentiate between various polarization modes, with a specific emphasis on the ability to distinguish between the breathing and longitudinal modes.


Introduction
Since the first gravitational-wave (GW) signal GW150914 was detected by Advanced LIGO in 2015 (Abbott et al. 2016a), Advanced LIGO and later Advanced Virgo have detected an increasing number of GWs sourced from compact binary mergers (Abbott et al. 2016b(Abbott et al. , 2017(Abbott et al. , 2019a(Abbott et al. , 2021a(Abbott et al. , 2021b)).This also implies that there should be many weak GWs that cannot be recognized by detectors.The combined weak signal from the population of compact binary constitutes the stochastic GW background (SGWB).In addition to the astrophysical sources, there are many ways to generate the SGWB in the early Universe, such as cosmological phase transitions (Hogan 1986;Caprini et al. 2010), inflation (Mukhanov et al. 1992;Turner 1997;Vagnozzi 2021;Benetti et al. 2022), cosmic strings (Damour & Vilenkin 2005;Siemens et al. 2007), etc.The detection of the GWB can provide us with information on the astronomical distribution (Mazumder et al. 2014;Callister et al. 2016) and cosmology (Ungarelli & Vecchio 2001;Bartolo et al. 2016;Caprini et al. 2016;Auclair et al. 2020;Caprini et al. 2020), and also provide an opportunity to test the theory of gravity (Maselli et al. 2016;Callister et al. 2017).
Pulsar timed arrays (PTAs) have been accumulating data for more than a decade, and the common-spectrum signal (Arzoumanian et al. 2020;Chen et al. 2021;Goncharov et al. 2021;Antoniadis et al. 2022) discovered in previous years has recently been confirmed to be from SGWBs (Agazie et al. 2023;Antoniadis et al. 2023;Reardon et al. 2023;Xu et al. 2023).In the hertz band, ground-based laser interferometers provide upper limits on the fractional energy density of the SGWB Ω GW 5.8 × 10 −9 (Abbott et al. 2021c).Future spacebased interferometers, such as LISA (Amaro-Seoane et al. 2017), TianQin (Luo et al. 2016), Taiji (Hu & Wu 2017), and DECIGO (Kawamura et al. 2011), will be able to detect the SGWB with high sensitivity in the megahertz band (Seto 2020;Wang & Han 2021;Liang et al. 2022).Once the SGWB is detected, the next step is to investigate what information can be obtained from it.An important role of the SGWB is to test the theory of gravity.General relativity (GR) predicts that only two tensor polarizations exist for the GW, the plus and cross modes.However, there are additional four polarizations allowed by the generic metric theories of gravity, including two vector modes and two scalar modes (Brans & Dicke 1961;Eardley et al. 1973;Jacobson & Mattingly 2004;Sagi 2010;Liang et al. 2017a;Moretti et al. 2019Moretti et al. , 2020Moretti et al. , 2021)).The observation of vector or scalar modes would cast doubt on general relatively, while their absence could be used to constrain modified gravity (Will 2014).If parity is considered, there is circular polarization in the SGWB, describing the asymmetry of the two tensor polarizations (Seto & Taruya 2007, 2008;Kato & Soda 2016;Chu et al. 2021).
Research on the detection of the polarization modes of the SGWB has become a hot topic, as more and more modified theories of gravity are proposed.For PTA, the detectability of non-GR polarizations of the SGWB was first investigated in Lee et al. (2008).Later findings provide a more detailed extension (Chamberlin & Siemens 2012;Gair et al. 2015).No evidence of non-GR polarizations of the SGWB has been found in more than a decade of PTA data.The constraints of the amplitude or energy density for non-GR modes in the SGWB at a frequency of 1 yr −1 can be found in Cornish et al. (2018), Arzoumanian et al. (2021), and Wu et al. (2022).The upgraded data will place tighter restrictions on non-GR polarizations.
Meanwhile, the ground-based GW detection network is gradually forming, as more and more detectors join it.It has the potential to detect the SGWB produced by compact binary mergers.Data from Advanced LIGO's and Advanced Virgo's three observing runs can constrain the fractional energy density of different polarization modes at the hertz band (Abbott et al. 2018(Abbott et al. , 2019b(Abbott et al. , 2021d)).However, the ability of the ground-based laser interferometer network to study the polarizations of the SGWB is limited.The two kinds of scalar modes are completely different, one is transverse and the other is longitudinal.Therefore, they cannot be converted into one another via a rotational coordinate transformation along the direction of propagation (whereas two tensor modes or two vector modes can).In the scalar-tensor theory of gravity, there is only one extra breathing mode if the scalar field is massless, or a mixture of breathing mode and longitudinal mode if the scalar field is massive (Liang et al. 2017b).It is therefore necessary to distinguish between the two scalar modes in the detectors.Unfortunately, the responses of the ground-based laser interferometer to scalar-breathing and scalar-longitudinal modes are completely degenerate, which means that the two modes cannot be distinguished, no matter how sensitive the detectors are (Nishizawa et al. 2009).
For the future space-based GW detectors, the situation is different.The degeneracy of the response between the breathing and longitudinal modes is broken at relatively high frequencies (Liang et al. 2019;Liu et al. 2020), which implies that space-based detectors have the potential to distinguish the breathing and longitudinal modes of GWs.Besides, space-based detectors can detect signals from a much larger range of frequencies, compared to ground-based detectors.The abundance of sources in the sensitive band and the sufficiently high sensitivity of the detectors ensure that a polarization analysis can be performed.What is more, the relative orientation of two space-based detectors may change as they are in different positions in their respective orbits.For example, for the LISA-TianQin network, the normal vector of the constellation plane of LISA varies with time and that of TianQin points to a fixed direction (Luo et al. 2016).This means that the overlap reduction function (ORF), the transfer function between the spectrum of the SGWB and the power spectrum of the cross-correlation signal, will vary accordingly.This feature will facilitate the differentiation of the polarization modes, which can be inferred from the viewpoint that the detectors can be regarded as different detectors at different positions.
For the space-based detectors, the laser phase noise is usually orders of magnitude higher than other noises due to the mismatch of arm lengths, and significantly exceeds the magnitude of the GW signal.Fortunately, time-delay interferometry (TDI) can be used to suppress the laser phase noise (Tinto et al. 2000;Prince et al. 2002;Tinto et al. 2002;Tinto & Dhurandhar 2020).It is worth mentioning that three noise quadrature channels can be constructed, called A, E, and T. The T channel is relatively insensitive to the GW compared with other channels.Therefore, the readings of the T channel can be used to identify noise, which can be combined with the A and E channels to extract the SGWB.This is called the null channel method (Hogan & Bender 2001;Adams & Cornish 2010;Romano & Cornish et al. 2017).The other more conventional approach is cross-correlation analysis (Flanagan 1993;Allen & Romano 1999), which is typically employed to detect the SGWB.When there are multiple spacebased detectors, the cross-correlation analysis between different detectors based on the TDI combination can be adopted to search for the SGWB.In recent years, there have been several studies on the performance of space-based detectors for GWs with different polarizations (Liu et al. 2020;Orlando et al. 2021;Wang & Han 2021;Liang et al. 2022).However, for the space-based detectors, the study of the data analysis methods for the separation and identification of different polarization SGWBs is lacking, as well as the treatment of the time-varying effect.Among the promising space-based detectors, the difference in the orbital configuration of LISA and TianQin is the most significant, so the time-varying effect of the LISA-TianQin network is the most pronounced.In this paper, we investigate the detection capability of the LISA-TianQin network for the SGWB with different polarizations, and propose a method inspired by the time-varying effect to distinguish different polarization modes from their mixed data.
The outline of this paper is as follows.In Section 2, we review the SGWB in general metric theories of gravity and introduce the LISA-TianQin network.In Section 3, we review the correlation analysis for detecting the SGWB and calculate the ORF for the LISA-TianQin network.Then, in Section 4, we describe the method to separate the polarizations for the LISA-TianQin network.The simulations are performed to verify the separation ability in Section 5. Finally, a discussion is presented in Section 6.

Stochastic Background of Non-GR Polarizations
The metric perturbations corresponding to the SGWB can be expressed as a superposition of plane waves of different frequencies from different directions (Romano & Cornish et al. 2017): The Fourier coefficients h f n , ab ( ˆ) are random variables, whose statistical properties contain information on the SGWB.In generic metric theory, the coefficients can be expanded in terms of the six spin-two polarization tensors: where A = { + , × , X, Y, B, L} represents different polarization modes; where +, × represent tensor modes predicted by GR; and X, Y and B, L represent vector and scalar modes allowed by the general metric theory of gravity.Explicitly, the six spin-two polarization tensors are q ˆ, f ˆare the standard angular unit vectors tangent to the sphere: n sin cos , sin sin , cos , cos cos , cos sin , sin , sin , cos , 0 .3 The statistical properties of the SGWB are described by the probability distribution of the metric perturbations.In this work, we assume that the SGWB is Gaussian, stationary, and isotropic.And without loss of generality, we can assume that the background has zero mean á . So all the information is encoded in the quadratic expectation where S f h A ( ) can be regarded as the component corresponding to the A polarization of the one-sided GW strain power spectral density (PSD) function.We further assume that both the tensor and vector modes are unpolarized, which implies that However, the two scalar modes should be considered as two independent polarization modes, since one is the longitudinal and the other is transverse.It is worth noting that some models predict the presence of parity-violating SGWB sources (Alexander et al. 2006;Sorbo 2011), which can be recognized with interferometers (Seto 2006a;Domcke et al. 2020;Orlando et al. 2021).
The function S f h A ( ), which characterizes the spectral shape of the SGWB within each polarization sector, can be detected directly without assuming a model.However, the amplitude of the SGWB for each polarization is characterized by the fractional energy density (Allen & Romano 1999), .Here, G is the gravitational constant, and H 0 = 67.4km s −1 Mpc −1 is the Hubble constant (Planck Collaboration et al. 2020).In GR, the relation between S f h A ( ) and In alternative theories of gravity, Equation (7) may not hold unless the stress energy of GWs also obeys Isaacson's formula (Isaacson 1968): In this case, W f A gw ( ) can be understood as a function of the observable S f h A ( ) rather than the fractional energy density.Many theoretical models of the SGWB predict that the shape of W f A gw ( ) can be modeled as power laws (Romano & Cornish et al. 2017), such that Here, W a 0 A is the amplitude of polarization A at a reference frequency f 0 and α A is the corresponding spectral index.For instance, the tensor polarization background from compact binary coalescences is modeled by a power law with an index of α T = 2/3 (Abbott et al. 2019b), and the inflationary cosmic background is α T = 0 (Cornish & Larson 2001).

LISA-TianQin Network
LISA and TianQin are proposed space GW missions targeting the detection of the GW in the frequency band of 0.1 mHz-1 Hz.The difference is that LISA is a heliocentric orbit and TianQin is a geocentric orbit.In addition, the relative angle between their detector plane will vary with time.TianQin has a geocentric orbit and consists of three satellite formations in a nearly equilateral triangle.Accurate to the first order, the coordinates of the three satellites of TianQin are (Hu et  ) are the coordinates of the satellites in the geocentric coordinate system, Here, R = 1 au, e TQ = 0.0167, α TQ = 2πf m t − α 0 , f m = 1/yr, α 0 = 102°.9,α n = 2πf sc t + κ n , κ n = 2/3(n − 1)π, R 1 = 1 × 10 8 m, θ s = − 4°.7, f s = 120°.5,f sc = 1/(3.64days),and n = 1, 2, 3, respectively, denotes one of the three satellites.The detector plane orientation is fixed as q f q f q cos cos , cos sin , sin 1 2 .LISA has a heliocentric orbit at 20°behind Earth.The satellite formation consists of three satellites to form an approximate equilateral triangle (Dhurandhar et al. 2005), and the coordinates of the three LISA satellites are = r x y z , ,  1 2 .

Noise and Response for TDI
There are six laser links between the three satellites.Laser noise can be effectively reduced by constructing TDI combinations.Any TDI combination can be expressed in terms of a polynomial of the delay operator acting on the six received signals (Tinto & Dhurandhar 2020) Here, Pi is the polynomial of the delay operators and s i represents the measured signal of the reception of spacecraft i, and whether the subscript is primed or unprimed, depends on whether it is clockwise or counterclockwise.For example, s 1 is the time series signal measured at reception at spacecraft 1 with transmission from spacecraft 2 (along L 3 ), while ¢ s 1 is from spacecraft 3 (along ¢ L 3 ).The detector configuration is illustrated in Figure 1.A TDI combination can be expressed by coefficients Pi .For example, the coefficients of the firstgeneration TDI Michelson combination X are given by which represents a laser interferometry link: The time-delay operator is defined as D i s j (t) = s j (t − L i /c), and can be converted to the frequency domain, namely, The equivalent expressions for Michelson channel Y and Z can be obtained by permuting the labels {1, 2, 3}.Three noise-orthogonal channels can be constrained by the linear combinations of X, Y, and Z, After removing the phase noise, there are residual acceleration noise and displacement measurement noise in the TDI combination.The PSDs of the remaining noises are (Flauger et al. 2021;Wang et al. 2021) b p where β = 2πfL/c and the coefficients For example, the PSD of the X channel is b p b The PSDs of the X and A channels for LISA and TianQin are shown in Figure 2. Since GWs are weak, it is accurate enough to calculate the detector response to the linear order.In the frequency domain, the GW's signal can be expressed as (Romano & Cornish et al. 2017) is the impulse response of a single arm, u i ˆis the direction unit vector of the arm, and r i is the midpoint of the arm.Here, the choice of r i is different from that in the literature for the convenience of calculation, such that the impulse response becomes (Estabrook & Wahlquist 1975  where is the transfer function.

Cross-correlation Signal
Usually, the SGWB is very weak, which is masked by the noise of the detector, and the characteristics are close to the noise.So it is difficult to distinguish the noise and the GWB signal in a single detector.The correlation analysis is a powerful method to detect the SGWB (Allen & Romano 1999).We review this method and apply it to the LISA-TianQin network in this section.
We start with the output signals of the two detectors, where I and J represent TianQin and LISA in this paper.And the correlation signal is or in the frequency domain where ) .Assume that the two detector noises are uncorrelated, such that the mean of the correlation signal is ò ò Combining Equations ( 4) and (20), , 28 where the ORFs are will calculate the ORFs in next subsection.So the mean of the signal is Assume that the signal is much smaller than the noise, such that The variance is A prerequisite for the cross-correlation analysis is that the ORFs remain constant for the duration of observation.LISA-TianQin does not meet this condition, so some improvements need to be made.The total observation time T is divided into N segments, with each segment ΔT. ΔT needs to be greater than the light travel time between the two detectors and less than the timescale over which the ORF will vary.Therefore, ORFs in each segment can be treated as constants, labeled as G f IJ k A , ( ), and the cross-correlation analysis can be performed.The S/N for each segment is and the total S/N for observation time T reads as

ORF
The ORF can be interpreted as the response of the correlation of two detectors to the isotropic SGWB (Romano & Cornish et al. 2017).The ORF mainly depends on three factors: detector similarity, separation, and orientation relative to one another.In the past literature, one usually considers two identical detectors placed in different locations.In fact, for detectors like LISA-TianQin, the different arm lengths result in slightly different frequency bands for their respective responses, which is an important factor leading to the reduction of ORF.On the other hand, changes in orbits cause ORFs to change over time.In addition, the small antenna limit that applies to ground-based detectors is no longer always applicable to space-based detectors in the detection band.For example, the characteristic frequencies f = c/(2L) for LISA and TianQin are 0.06 Hz and 0.86 Hz, respectively.the small antenna limit required that the frequency be much smaller than the characteristic frequencies.However, the most sensitive frequency band of LISA-TianQin is 10 −3 Hz-10 −1 Hz, so the small antenna limit is not always satisfied.Based on the above considerations, the ORF of the LISA-TianQin network deserves a careful discussion.
) L for A = L.In this way, the ORF of any TDI channel can be disassembled and calculated between two separate arms.In general, the integral in Equation (37) cannot be calculated analytically.In the short antenna limit b b¢  , 1, it can be calculated analytically (Flanagan 1993;Allen & Romano 1999;Nishizawa et al. 2009) where the coefficients are Here, 9 6 6 4 1  6 24 4 16 2  6 4 8 8 2  4 16 8 24 4  1 2 2 4 1 , 40 So the ORF of any TDI channel in the small antenna limit is Notice that there is an extra phase factor b b ) , due to the difference in the arm lengths of LISA and TianQin.For the LISA-TianQin network, the phase factor e i2 π f/65mHz cannot be ignored in the detection band.
Since the small antenna limit may not work well, we extend it slightly.We expand Equation (37) as a Taylor series of the frequency f.To the zeroth order, the term is the expression for the small antenna approximation above.Expanding to the next order, we obtain that where Following the same method, we construct it with δ ab , s a, and u a under the premise of ensuring its symmetry.Then solving the linear equations for the coefficients, we can get the ORF of the second order.The details of the calculation are provided in Appendix A.
To quantify the accuracy of the expanded ORF, we compared the zero order (small antenna approximation) and second order with numerical integration, as shown in Figure 3, taking the X channel as an example.For frequencies below the characteristic frequency of f < c/(2L LS ) = 0.06 Hz, the secondorder expression is more accurate than the zero-order expression, which agrees well with the numerical integration.If the phase factor b b ) is ignored, the zero-order accuracy will be even worse.For frequencies above the characteristic frequency f > c/(2L LS ) = 0.06 Hz, the accuracy of the second order is worse compared with that of the zero order since the error of the second order grows faster than that of the zero order.To improve the accuracy, we choose to splice the two at the characteristic frequency of They are smooth at the connection point, both equal to 0. This concatenated expression has high enough precision for SGWB data analysis.The detector is insensitive to frequency bands above the characteristic frequency due to loud noise.Conventionally, ORFs are normalized so that the normalized ORFs only contain the effects of two detector separation and relative orientation.For the Michelson interferometer, the ORFs for low frequencies can be regarded as constant.Therefore, a constant normalization factor κ A can be selected so that g = 0 1 IJ A ( ) for two co-located and co-aligned detectors, where normalized ORFs are . However, the ORFs of TDI tend to 0 at low frequencies due to the extra factor p p fL c fL c sin 2 sin 2 et al. 2004;Seto 2006b).Normalization for ORFs of TDI may not be convenient, and the normalization factor will be proportional to f 2 .For the TDI-X combination, the normalization factors will be k for the tensor and vector modes, and κ B = κ L /2 = κ T /3 for the breathing and longitudinal modes.The normalization factor for TDI-A or TDI-E is multiplied by 3/2 compared to that of TDI-X.Furthermore, the normalization for other non-Michelsonian-type TDI combinations is even more complicated.
In Figure 4, we show the normalized ORF of the A channel for different polarizations.The ORFs of different polarizations share some common characteristics.Their zero crossings are similar: f ≈ nc/(2|Δx|), f = nc/(2L I ), and f = nc/(2L J ).The normalized ORFs decay rapidly to 0 beyond f = c/(2L LS ) ≈ 60mHz.
What is more, we plot the ORFs for different polarizations at different times in Figure 5.The time-varying property makes data analysis more difficult, but also increases the chance of resolving different polarizations.Correlations at different locations can be used as multi-correlation channels to distinguish between the four polarizations.Based on this idea, an algorithm can be developed to achieve the separation of different polarizations.And the accuracy of a polarization identification depends on how different the rate of change in time of its ORF is from that of other modes.Details are provided in Section 5.
It is well acknowledged that the angular antenna response of the Michelson interferometer-type detector to breathing and longitudinal modes is degenerate at low-frequency approximations, namely, This results in the degeneracy of the ORFs for the two scalar polarizations in the low-frequency range, with a quantitative relationship of γ L = γ B .However, as the frequency increases, the degeneracy between them is removed as shown in Figure 5.This is reflected in the fact that the curves of the two modes coincide at low frequencies, exhibiting differentiation as frequency increases.This implies that it is possible to resolve two scalar modes through the LISA-TianQin network.The key point is that the degeneracy is broken, not how different the ORFs of the two scalar modes are, as distinguishing between the two modes requires different correlation channels to provide more information.As the quantitative relationship of γ L = γ B always remains in the low-frequency range, it does not provide any information to distinguish between the two scalar modes, regardless of the number of correlated channels added.In the high-frequency range, the relationship of γ L = γ B is no longer maintained, which means that the ratio of γ L and γ B for different correlation channels may vary.This provides the possibility of distinguishing between breathing and longitudinal modes.

Sensitivity for the Background of Alternative Polarizations
In addition to the naive method of directly correlating the output of two detectors, the idea of a matched filter is often used to improve the S/N.Matched filtering starts by multiplying the correlation signal (Equation ( 26)) by a filter function, And the optimal filter is proved to be and the resulting optimal S/N is given by The premise of the above expression is that the noise amplitude is much larger than the signal.In general, it will be modified to (Cornish 2001) Once again, considering the time-varying effect of LISA-TianQin, the total S/N after segmented processing is where the average ORFs is defined as Assuming that the spectrum of the SGWB is a power law, the optimal S/N calculated by Equation (51) can be used to evaluate the detection capability of the LISA-TianQin network for the SGWB.We show the S/N of the SGWB with different polarizations in Figure 6, where the observation time is chosen as T = 1 yr, and the TDI channel is selected as channel A here and in the rest of the article.The result shows that the LISA-TianQin network is more sensitive to tensor and vector polarizations than the two scalar polarizations.For example, for the same spectral index of α A = 2/3, in order to achieve an S/N of 10, the amplitudes at the reference frequency for different polarizations are required to be W = ´-8.59 10 ´-7.58 10 ´-1.71 10 B 0 11 , and W = ´-8.61 10 L 0 12 .The contribution of high-frequency bands to the S/N is negligible.In fact, for an SGWB with a spectral index of α = 2/3, the contributions over 0.06 Hz to the S/N for each polarization are 8.5 × 10 −8 , 6.5 × 10 −8 , 7.7 × 10 −8, and 1.1 × 10 −7 , respectively.This reflects the fact that strong noise prevents high-frequency bands from being accurately detected at present.The error in the ORF in the high-frequency region does not matter.Of course, as detector technology advances, noise levels decrease, and the development of high-precision ORFs in the high-frequency region is necessary to detect potential high-frequency sources.

Polarization Separation
If the alternative polarization really exists, different polarization modes will be mixed in the cross-correlation signal.We need suitable methods to distinguish difference modes in the cross-correlation signal.One possible approach is the Bayesian model selection proposed in Callister et al. (2017), which is adopted to detect alternative polarization backgrounds in the cross-correlation data of ground-based detectors (Abbott et al. 2018(Abbott et al. , 2019b(Abbott et al. , 2021d)).This approach is efficient and easy to implement.However, it may be prone to bias if the model does not fit well with the true background.
If one wants to get away from being bound by model assumptions, there is an intuitive method to separate polarizations using multiple detector pairs (Nishizawa et al. 2009).Different detector pairs have different ORFs, which breaks the degeneracy between different polarizations and provides enough degrees of freedom to separate different polarizations.Notably, this approach can be implemented independently for each frequency bin.The accuracy of the results for each frequency bin depends on the corresponding signal and noise strength.However, this method requires a higher S/N than the previous method.Another disadvantage is that multiple detector pairs naturally mean that more than two detectors are required.
Since the LISA-TianQin network varies with orbits, their ORFs for difference polarizations vary accordingly as can be seen in Section 3.So their correlation signals at different positions respond differently to different polarizations, which means that they can be equivalently regarded as different detector pairs at different times.There is an opportunity to extract different polarization patterns from the data from the two detectors, using an approach similar to the multi-detectorbased approach (Nishizawa et al. 2009).
The key point is to break the coupling of different polarizations in the data through multiple correlation channels.If data for n detectors is available, a total of N = n(n − 1)/2 correlation channels can be constructed as where 1 i < j n.These statistics satisfy is the detector correlation matrix.If the rank of the matrix is greater than or equal to 4, rank (M) 4, the polarization components can be separated by inverting Equation (57), namely, Here, M(f) −1 is the pseudo-inverse of the non-square matrix M.
In this way, the spectrum of each polarization component can be reconstructed from the detector data by Equation (59).However, the degeneracy of ORFs of two scalar polarizations for ground-based causes the rank of the detector correlation matrix to be at most 3.So the SGWB detected by the groundbased detector network can only be separated into tensor, vector, and scalar components.Two space-based detectors may achieve the capabilities of the abovementioned ground-based detector network to distinguish polarizations in the SGWB.It is only necessary to replace the different correlation channels with the correlation of the two space-based detectors at different positions.Below we use another description of the maximum likelihood estimation mentioned in Romano & Cornish et al. (2017) to obtain the same result.First, the data is divided into N segments (indexed by i), and the duration of each segment of data ΔT satisfies the condition that it is greater than the light travel time between the two detectors and less than the timescale of the ORF changes.A cross-correlation statistic can be constructed as Figure 6.The S/N of a power-law SGWB with different amplitude Ω A and index α A for different polarizations.The reference frequency is chosen to be f 0 = 1 mHz.The A channel is chosen for calculation here and in the rest of the article.
normalized such that the statistic's mean is acts as the role of different correlation channels Z( f ).And the variance is where Δf is the frequency bin width, P 1,i ( f ) and P 2,i ( f ) are the noise PSD of the detectors.
The likelihood function for represents the fractional energy density of different polarizations.Equivalently, we can write it as where the detector correlation matrix becomes And the noise covariance matrix is ).The maximum likelihood estimator for the fractional energy density of different polarizations is (Romano & Cornish et al. 2017 are the Fisher matrices and dirty map for this analysis.The element of the matrix The maximum likelihood estimator  ˆis equivalent to Equation (59), which is easier to calculate in practice and avoids finding the pseudo-inverse of a huge matrix.After the data is available, the reconstructed spectrum of each polarization component can be obtained by calculating Equation (66) for each frequency.The error will be determined by the detector noise at that frequency and the degeneracy of ORFs of different polarizations.Specifically, the marginalized uncertainties of the maximum likelihood estimates are given by the inverse of the Fisher matrices, In order to easily evaluate the detection capability, we assume that the noise power spectrum for different time segments is equal.And the effective overlap function can be defined by which can reflect the intensity of the response to different polarization components.The effective overlap functions for the LISA-TianQin network are shown in the top panel of Figure 7.The effective overlap functions of the two scalar components are significantly lower than that of tensors and vectors, due to the similar ORFs of the scalar-breathing and scalar-longitudinal modes.G f A eff ( ) decreases to 0 at the characteristic frequency of f LS = 60 mHz because Γ A ( f LS ) = 0.The peaks in the high-frequency region (around 0.1 Hz) seem strange, but are actually due to the fact that Γ A ( f ) for each polarization is less degenerate in this region.In order to quantify the sensitivity of this method to different polarization components, detector noise should also be included.This is the role of uncertainties s W A , which are shown in the bottom panel of Figure 7.For a certain frequency f, the intensity of the A component must meet ) peaks in the high-frequency region, the recognition ability for this band is very poor due to the influence of noise.
To visually measure the method's ability to distinguish between different polarizations, an overall S/N can be defined as By convention, an S/N of 10 is considered to be a signal extract from the data.For an astronomical background spectrum with an index of α = 2/3, the amplitudes at the reference frequency, f 0 = 1 mHz, for different polarizations are required to be W = ´-1.27 10 ´-9.55 10 ´-1.43 10 B 0 10 , and W = ´-7.23 10 L 0 11 .Comparing with the results of optimal S/N calculated by Equation (54), we find that the ability of the method to resolve vector polarization is the best, followed by tensor.The ability to distinguish two scalars is the worst, and the S/N is equivalent to 12% of the optimal S/N.For comparison, for the flat spectral index α = 0, the amplitudes for different polarizations need to exceed W = ´-2.90 10 ´-2.59 10

Simulation
To verify the ability of the above method to resolve polarization modes, we apply it to the simulated data of different situations.Notably, we simulated a mixture of tensor and breathing modes to examine whether it can distinguish between breathing and longitudinal modes.We simulated a year of data and divided it into 3600 segments, and the noise was generated according to Equation (62).The spectrum injected into each piece of data are Ω T = 8.59 × 10 −12 ( f/1 mHz) 2/3 , and Ω B = 1.71 × 10 −11 ( f/1 mHz) 2/3 .The parameters are chosen such that individual tensor and scalarbreathing components both correspond to an optimal S/N of 10, mixed together with an optimal S/N of 15.The spectrum for each polarization reconstructed with Equation (66) is shown in Figure 8.The spectral accuracy of the reconstructed spectrum in the frequency domain of 10 −3 Hz-10 −2 Hz is relatively high.It can be concluded that in the most sensitive frequency band, the different modes can indeed be distinguished.Even two scalar modes are capable of distinguishing modes, but the accuracy is worse than the tensor or vector modes.
As discussed above, the separate breathing mode of the two scalar modes will not be mistaken for the longitudinal mode.Furthermore, some alternative theories of gravity would predict more complex mixed polarizations.On the other hand, the spectral index for different polarizations may be different.Therefore, we simulate a more complex case with all four polarizations at the same time, in which the spectral index of the vector mode is set to distinguish it from the other three.The special spectrum injected into the data are Ω T = 1.42 × 10 −11 ( f/1 mHz) 2/3 , Ω V = 1.62 × 10 −11 , Ω B = 1.71 × 10 −11 ( f/ 1 mHz) 2/3 , and Ω L = 8.61 × 10 −12 ( f/1 mHz) 2/3 .The indices for tensor, breathing, and longitudinal modes are set to 2/3, while the spectra of vector mode are set to be flat, namely, α V = 0.As shown in Figure 9, the spectra of each polarization are correctly reconstructed.The noise of the breathing and longitudinal modes is significantly greater than that of the other two modes, but each can be identified independently.The reconstructed polarization spectrum falls exactly near the injected true value, and the   difference between the polarizations is only the strength of the noise.
Figure 10 shows the result of increasing the observation time for the same signal, where the parameters of the signal are exactly the same as in Figure 9, and the observation time is increased to 5 yr.The S/N increases with observation time, specifically / µ T S N .Equivalently, five times the observation time would reduce the uncertainty by a factor of 2.2.This behavior is clearly presented in Figure 10.The noise of the reconstruction spectrum observed over 5 yr is significantly reduced for the same signal.
The simulation results verify the ability of the equivalent multi-detector-based method to separate different polarizations.The reconstructed spectrum for each polarization obtained by this method falls near the true value, and the uncertainty depends on the noise of the detector and the correlation between one component and the other polarization component in the correlation matrix in Equation (65).It may be misunderstood that this correlation between different polarized components refers to their ORF similarity.In fact, it does not matter how many times the ORF for one mode is that for the other, what matters is how different the values of each of them change at different positions in orbit.For example, the degeneracy of the breathing and longitudinal modes in the low-frequency band results in the ORF for the longitudinal mode being twice that of the breathing mode.But the key point is that this double relationship is always maintained in the lowfrequency band, regardless of how the detectors move.For high-frequency bands, the degenerate relation is broken, and the ORFs of the breathing and longitudinal modes have the opportunity to vary with time differently.The time-varying effect has been encapsulated in the effective ORFs G f A eff ( ) constructed by the inverse of the correlation matrix, which can quantify the ability to separate polarization.

Discussion
In this paper, we studied the detectability of alternative polarizations of the SGWB with the LISA-TianQin network.The different orbits of LISA and TianQin make them very different from ground-based detectors.The relative orientation of their orbital planes changes with time, which means that the ORF will vary accordingly.In other words, the SGWB signal in their cross-correlation signal varies over time.This will pose certain challenges for data analysis.On the other hand, LISA-TianQin has advantages for resolving polarization modes in the SGWB in principle.
TDI technology is applied to space GW detectors to suppress laser phase noise.Based on the small antenna approximation, we obtain the ORF of the second-order expansion of the frequency for any TDI channel.This method can be applied to any laser interferometric detector, even with different configurations and different arm lengths.For the LISA-TianQin network, the accuracy can be effectively improved in its most sensitive frequency band.Then we study the detectability of alternative polarizations in the SGWB with the LISA-TianQin network.We calculated the S/N for a power-law SGWB with different polarization modes.
Once the SGWB is detected, it is necessary to distinguish different polarization modes from it.An equivalent multidetector-based approach can be applied to the LISA-TianQin network, thanks to its special orbital variation.The ORFs of the LISA-TianQin network are different at different times, so they can be equivalently regarded as different detectors.In fact, the cross-correlated signals at different times form a system of linear equations for different polarization modes.In turn, the system of equations can be solved to obtain different polarization modes, including the influence of noise of course.Although this method does not rely on model assumptions, it requires a high S/N.Its resolution at a specific frequency bin is influenced by the degeneracy of the ORFs for different polarizations, as well as the noise present at that frequency bin.What is more, it has more advantages in distinguishing scalar-breathing and scalar-longitudinal modes.
Our proposed method for resolving polarization relies on the change in the relative orientation of the two detectors over time, with LISA-TianQin being the most pronounced, due to differences in orbital configuration.However, our analysis can be extended to include any space-based detector.We will compare the resolution capabilities of different detector pairs and evaluate the ability to improve multi-detector network cooperation in future work.

Appendix A Second-order ORF of Any TDI Channel
The ORF frequency second-order expansion of any TDI channel requires the calculation of the coefficients  ( ), the linear system of equations for the coefficients A A(2) , B A(2) , ..., E A 4 2 ( ) is obtained, which can be expressed as matrix equations as Here, where h = u s The coefficients Y A(2) for the tensor, α LS = α TQ −20°and e LS = 0.0048.The detector plane is inclined to the orbit plane by 60°and the arm length is L LS = 2.5 × 10 9 m.The displacement measurement noise =

Figure 1 .
Figure 1.The configuration of the space-based laser interferometry consists of laser sources and links.

Figure 2 .
Figure 2. The PSD P n ( f ) for different channels.The solid and dashed lines denote the X and A channels, respectively.We use the red/blue lines to label the LISA and TianQin.
For any TDI channel, the ORF for A polarization is To keep the definition consistent, the sum means +

Figure 3 .
Figure 3.The tensor ORF of the X channel for the LISA-TianQin network.The real and imaginary parts of the ORF are shown the upper and lower panels, respectively, and the time is t = 0.

Figure 4 .
Figure 4.The different polarizations of the ORF of the A channel for the LISA-TianQin network.The normalized ORF curve is plotted at t = 0.And the real and imaginary parts are represented by different types of curves.

Figure 5 .
Figure 5.The normalized ORF at different times.The left and right sides from top to bottom are the real and imaginary parts of the ORF at t = 0, t = 1 day, and t = yr −2 , respectively.

Figure 7 .
Figure 7.The effective overlap functions (above) and effective sensitivity curves (below) for different polarization.

Figure 10 .
Figure 10.The comparison of the reconstructed spectrum with observation time T = 1 yr and T = 5 yr.The injected spectra are the same as in Figure 9.
Contracting the expression with δ ab δ cd , . The result is Âccordingto the symmetry of the index, we construct it with δ ab , s a, and u a :