Polarization modes of gravitational waves in general modified gravity: General metric theory and general scalar-tensor theory

In this paper, we establish a unified parametrized framework for analyzing the polarization modes of gravitational waves in the general metric theory (where gravity is only described by the metric) and the general scalar-tensor theory (where gravity is described by the metric and an additional scalar field). Specifically, we study the polarization modes of gravitational waves in the most general metric theory and general scalar-tensor theory that satisfy the following conditions: (1) Spacetime is four-dimensional; (2) The theory satisfies the principle of least action; (3) The theory is generally covariant; (4) The action describing a free particle is $\int ds$. We find that the polarization modes of gravitational waves in the theory satisfying the above conditions depend on the selection of parameters in the framework, and the theory allows for up to all six polarization modes. Once we have established our framework, the analysis of the polarizations of gravitational waves in specific theories will depend on determining the corresponding parameters within our framework. We also find that the polarization modes of gravitational waves in the general metric theory and the general scalar-tensor theory that satisfy the conditions also have some interesting universal properties.