Polarization modes of gravitational waves in generalized Proca theory

In this paper, we study polarization modes of gravitational waves in generalized Proca theory in the homogeneous and isotropic Minkowski background. The results show that the polarizations of gravitational waves depend on the parameter space of this gravity theory and can be divided into quite rich cases by parameters. In some parameter space, it only allows two tensor modes, i.e., the $+$ and $\times$ modes. In some parameter space, besides tensor modes, it also allows one scalar mode, or two vector (vector-$x$ and vector-$y$) modes, or both one scalar mode and two vector modes. The scalar mode is a mixture mode of a breathing mode and a longitudinal mode, or just a pure breathing mode. Interestingly, it is found that the amplitude of the vector modes is related to the speed of the tensor modes. This allows us to give the upper bound of the amplitude of the vector modes by detecting the speed of the tensor modes. Specifically, if the speed of tensor modes is strictly equal to the speed of light, then the amplitude of vector modes is zero.