Prospects of constraining on the polarizations of gravitational waves from binary black holes using space- and ground-based detectors

The theory of general relativity (GR) predicts the existence of gravitational waves (GWs) with two tensor modes, while alternative theories propose up to six polarization modes. In this study, we investigate constraints on GW polarization using a model-independent parametrized post-Einsteinian framework and consider both space- and ground-based detectors. By evaluating the capabilities and network performance of LISA, Taiji, TianQin, LIGO, Virgo, KAGRA, and the Einstein Telescope (ET), we analyze their respective contributions. Among space-based detectors, Taiji provides the most stringent constraints compared with LISA and TianQin. Regarding ground-based detectors, LIGO excels in vector modes while ET offers comprehensive constraints across all polarization modes. In network scenarios, $\mathrm{LISA}+\mathrm{TJm}$ performs best, and ET surpasses second-generation detector combinations. Furthermore, multiband observations effectively mitigate scalar mode degeneracies thereby significantly enhancing the performance of ground-based detectors. Ultimately, combined space- and ground-based observations provide robust constraints on GW polarizations that advance tests for deviations from GR. Our findings underscore the potential of future GW missions in refining our understanding of gravitational physics through precise measurements of polarizations.