Gravitational-wave polarization modes and stability analysis in Weyl geometry gravity

We investigate the polarization modes and stability properties of gravitational waves in Weyl geometry gravity on a Minkowski background endowed with a nonvanishing Weyl gauge field. Starting from the Weyl-invariant gravitational action, we perform a linear perturbation analysis and construct gauge-invariant variables through scalar–vector–tensor decomposition. This framework enables a systematic classification of the propagating degrees of freedom and their associated gravitational-wave polarizations. Our analysis shows that the tensor sector contains the two standard plus- and cross-polarization modes, both propagating at the speed of light, as in general relativity. Although the vector sector possesses a propagating dynamical degree of freedom, it does not generate observable vector polarization modes. The scalar sector, in contrast, features a mixture of breathing and longitudinal modes associated with a single scalar degree of freedom. This scalar mode exhibits superluminal propagation and an intrinsic damping of amplitude during propagation, both originating from the background Weyl gauge field. We further estimate the observational prospects of detecting this scalar signal and discuss the conditions under which it may be distinguishable in future gravitational-wave observations. To assess the theoretical consistency of the theory, we derive the quadratic actions for tensor, vector, and scalar perturbations and analyze their stability properties. While the tensor and vector sectors are free from ghost and Laplacian instabilities, a Hamiltonian analysis reveals the presence of an Ostrogradsky ghost in the scalar sector. These results clarify the distinctive gravitational-wave signatures of Weyl geometry gravity and provide a theoretical basis for testing this framework with future multi-messenger and gravitational-wave experiments.