Gravitational-wave polarizations in generic linear massive gravity and generic higher-curvature gravity

We study the polarizations of gravitational waves (GWs) in two classes of extended gravity theories. First, we formulate the polarizations in linear massive gravity (MG) with generic mass terms of non-Fierz-Pauli type by identifying all the independent variables that obey Klein-Gordon-type equations. The dynamical degrees of freedom (dofs) in the generic MG consist of spin-2 and spin-0 modes, the former breaking down into two tensor (helicity-2), two vector (helicity-1) and one scalar (helicity-0) components, while the latter just corresponding to a scalar. We find convenient ways of decomposing the two scalar modes of each spin into distinct linear combinations of the transverse and longitudinal polarizations with coefficients directly expressed by the mass parameters, thereby serving as a useful tool in measuring the masses of GWs. Then we analyze the linear perturbations of generic higher-curvature gravity (HCG) whose Lagrangian is an arbitrary polynomial of the Riemann tensor. On a flat background, the linear dynamical dofs in this theory are identified as massless spin-2, massive spin-2, and massive spin-0 modes. As its massive part encompasses the identical structure to the generic MG, GWs in the generic HCG provide six massive polarizations on top of the ordinary two massless modes. In parallel to MG, we find convenient representations for the scalar-polarization modes directly connected to the parameters of HCG. In this analysis, we employ two distinct methods; One takes full advantage of the partial equivalence between the generic HCG and MG at the linear level, whereas the other relies upon a gauge-invariant formalism. We confirm that the two results agree. We also discuss methods to determine the theory parameters by GW-polarization measurements. Our method does not require measuring the propagation speeds or the details of the waveforms of the GWs. [Abridged]