Polarization modes of gravitational waves in general Einstein-vector theory

We study the polarization modes of gravitational waves in general Einstein-vector theory with an arbitrary constant background vector field under a Minkowski background. We compare these polarization modes with those of other vector-tensor theories and constrain the parameter spaces based on the gravitational-wave event GW170817 with its electromagnetic counterpart GRB170817A and observations of pulsar timing arrays. The presence of the background vector field leads to the anisotropy of space and a rich variety of gravitational wave contents. Our results reveal that the polarization modes of gravitational waves depend on the parameter spaces. There are at least two and at most five independent polarization modes in one parameter space. In different parameter spaces, some mixture modes are allowed, including tensor-vector, tensor-scalar, tensor-vector-scalar, vector-scalar, and scalar-scalar mixture modes, as well as five independent modes (excluding $P_l$): $P_+$, $P_{\times}$, $P_x$, $P_y$, and $P_b$. In all regions of the parameter spaces, there are always two tensor modes, which can be either independent of or mixed with other modes. The independent $P_b$ mode consistently exhibits the same speed as light. If the speed of tensorial gravitational waves strictly equals that of light, only the $P_+$, $P_{\times}$, and $P_b$ modes are permitted. Furthermore, through comparisons of some vector-tensor theories and based on the observations of gravitational waves, the $P_b$ mode is expected to be allowed.