Bayesian analysis of the stochastic gravitational-wave background with alternative polarizations for space-borne detectors

Future space gravitational-wave detectors will detect gravitational waves with high sensitivity in the mHz frequency band. One possible source is the stochastic gravitational-wave background (SGWB), possibly from astronomy and cosmology. Detecting SGWB could provide an opportunity to directly examine the polarization of gravitational waves. While general relativity predicts only two tensor modes for gravitational-wave polarization, general metric theories of gravity allow up to four additional modes, including two vector and two scalar modes. Observing other polarization modes of gravitational waves would directly indicate that general relativity needs to be modified. However, the application of polarization identification methods developed for ground-borne detectors to space-borne detectors will require improvement. In this paper, we design a new statistic for the characteristics of space-borne detectors and perform Bayesian analysis. We analyze the performance of the new statistics, including the signal-to-noise ratio, ability to identify SGWB, and parameter estimation. The results show that the Bayesian method with new statistics is good enough to meet the needs of space detectors to identify polarized SGWB. In particular, space-borne gravitational-wave detectors will have the ability to distinguish the scalar-breathing mode and the scalar-longitudinal mode with this Bayesian method, which ground-based detectors cannot.

Figure 1

The comparison of the ORF of LISA-TianQin network for tensor-polarized SGWB calculated by different methods. The above subfigure is the real part of the ORF; the lower one is the imaginary part of the ORF. The ORF of the TDI-X channel at a certain point in the orbit is calculated, adopting numerical integration (black solid), small antenna approximation (blue dashed), and our method (red dotted), respectively.

Figure 2

The average ORF of TDI-A channel for different networks. The red solid line represents the LISA-TianQin network; the blue dashed line represents the LISA-Taiji network; and the green dot-dashed line represents the TQ $\mathrm{I}\text{−}\mathrm{II}$.

Figure 3

The ratio of arithmetic mean and root mean square of ORF of TDI-A channel for different networks. Different networks are represented by different colors: LISA-TianQin (red), LISA-Taiji (blue), and TQ $\mathrm{I}\text{−}\mathrm{II}$ (green). And the polarizations are distinguished by line type: tensor (solid), vector (dashed), scalar-breaking (dotted), and scalar-longitudinal (dot-dashed).

Figure 4

The mean and root mean square of the ORFs for the LISA-TianQin.

Figure 5

The simulated measurements $\widehat{C}(f)$ (upper) and ${\widehat{C}}^T(f)$ (lower) for a background with mixture of tensor and vector polarizations, recovered after one year of observation with LISA-TianQin network. The parameters of spectra of background for the simulation are ${\mathrm{\Omega }}^T(f)=8.59\times {10}^{-12}(f/1\mathrm{mHz}{)}^{2/3}$ and ${\mathrm{\Omega }}^V(f)=7.58\times {10}^{-12}(f/1\mathrm{mHz}{)}^{2/3}$, with optimal SNR ${\widetilde{\mathrm{SNR}}}_{\mathrm{opt}}=14.87$.

Figure 6

The distribution of odds ratios ${\mathcal{O}}_N^{\mathrm{SIG}}$ (left) and ${\mathcal{O}}_{\mathrm{GR}}^{\mathrm{SIG}}$ (right) for different simulated data. For the Gaussian noise (gray), the distribution of ${\mathcal{O}}_N^{\mathrm{SIG}}$ and ${\mathcal{O}}_{\mathrm{GR}}^{\mathrm{SIG}}$ is concentrated and less than 0. And for the pure tensor polarized signal with index $2/3$ (blue), the amplitude is set reasonably so that the SNR is 10. Finally, for a background with mixture of tensor and scalar-breathing polarization (light brown), the indices for two polarizations are both $2/3$, and the amplitude of each polarization component is set reasonably so that the SNR of the individual components is 10 and the total SNR is 15.

Figure 7

Odds ratios ${\mathcal{O}}_N^{\mathrm{SIG}}$ (left) and ${\mathcal{O}}_{\mathrm{GR}}^{\mathrm{SIG}}$ (right) for different simulated data. Each point in the graph represents a simulation result. The top row is the change in odds ratios as the increase of amplitude for pure tensor background with index $2/3$. The lower row shows the increase in the odds ratio as the amplitude of the scalar-breathing polarization in background increases, mixed with a fixed tensor component.

Figure 8

Odds ratios ${\mathcal{O}}_N^{\mathrm{SIG}}$ (left) and ${\mathcal{O}}_{\mathrm{GR}}^{\mathrm{SIG}}$ (right) for simulated data with a background with mixture of tensor and scalar-breathing polarizations. The tensor and scalar-breathing components have the same index $2/3$. The horizontal and vertical coordinates of the left and right images are the same, and they are the amplitudes of the tensor and scalar-breathing components, respectively.

Figure 9

The posteriors for a simulation of a background with pure tensor polarizations (case 2). The colored histograms along the diagonal show the marginalized one-dimensional posteriors for the amplitudes and slopes of the tensor, vector, scalar-breathingm and scalar-longitudinal polarizations (blue, green, orange, and red, respectively). The rest of the subplots show the joint two-dimensional posteriors between each pair of parameters. The parameters of tensor polarizations are peak around their true values.

Figure 10

The posteriors for a simulation of a background with mixed tensor and vector polarizations (case 3). The colored histograms along the diagonal show the marginalized one-dimensional posteriors for the amplitudes and slopes of the tensor, vector, scalar-breathing, and scalar-longitudinal polarizations (blue, green, orange, and red, respectively). The rest of the subplots show the joint two-dimensional posteriors between each pair of parameters. The parameters of tensor and vector polarizations are peak around their true values.

Figure 11

The posteriors for a simulation of a background with mixed tensor and scalar-breathing polarizations (case 4). The parameters of the tensor polarization agree with the true value within the error range. However, the injected scalar-breathing component was incorrectly identified as a mixture of scalar-breathing and scalar-longitudinal modes.

Figure 12

The posteriors for a simulation of a background with mixed tensor and two scalar polarizations (case 5). The parameters of the tensor and the two scalar polarizations are all consistent with the true values within the error range. This is because the spectra of the two scalars are very different: 3 for the scalar-breathing and $2/3$ for the scalar-longitudinal mode.