Polarized gravitational waves in the parity violating scalar-nonmetricity theory

There has been increasing interest in investigating the possible parity violating features in the gravity theory and on the cosmological scales. In this work, we consider a class of scalar-nonmetricity theory, of which the Lagrangian is polynomial built of the nonmetricity tensor and a scalar field. The nonmetricity tensor is coupled with the scalar field through its first order derivative. Besides the monomials that are quadratic order in the nonmetricity tensor, we also construct monomials that are cubic order in the nonmetricity tensor in both the parity preserving and violating cases. These monomials act as the non-canonical (i.e., non-quadratic) kinetic terms for the spacetime metric, and will change the behavior in the propagation of the gravitational waves. We find that the gravitational waves are generally polarized, which present both the amplitude and velocity birefringence features due to the parity violation of the theory. Due to the term proportional to $1/k$ in the phase velocities, one of the two polarization modes suffers from the gradient instability on large scales.