Constraint on gravitational-wave polarizations for space-based detectors with time-delay interferometry

Probing extra polarizations in gravitational waves (GWs) with space-based detectors is the most direct method for testing theories of gravity. In this paper, by employing the second-generation time-delay interferometry (TDI) to cancel out the laser frequency noise in a rotating and flexing configuration with arm lengths varying linearly in time, we study the detectors' constraint ability on extra polarizations, and explore the impacts of TDI on the constraint of polarizations. Working in the parametrized post-Einstein (ppE) waveform framework, we find that the constraints on extra polarizations are significantly weaker than those for the tensor mode. For the tensor mode, the constraint ability for the detectors scales with signal-to-noise ratio (SNR). At low frequency, due to signal cancellation effects, the SNR registered is lower for the detectors with TDI method than that with the simplified equal-arm Michelson interferometer method. Therefore, tensor-mode constraints are degraded when TDI is applied. Although the direct detection of extra polarizations remains challenging, the constraint ability of the space-based detectors on the vector mode is better than the scalar modes. Besides, the detectability on extra polarizations will be enhanced with TDI, and the improvement of the constraint on the vector mode is less than scalar modes due to the inclination-dependent waveforms.