Yu-Qi Dong, Xiao-Bin Lai, Yu-Zhi Fan, Yu-Xiao Liu

This paper investigates the polarization modes of gravitational waves within the most general symmetric teleparallel gravity theory that allows for second-order field equations We consider both scenarios where test particles either carry or do not carry a hypermomentum charge. Our findings reveal the existence of tensor, vector, and scalar modes of gravitational waves. Firstly, the theory supports the + and $\times$ tensor modes propagating at the speed of light. Secondly, in the case where particles do not carry hypermomentum, vector modes propagating at the speed of light exist only within a very specific parameter space. However, when particles do carry hypermomentum, there are two shear modes that propagate at the speed of light, while the vector-$x$ and vector-$y$ modes emerge only under very specific conditions. Thirdly, in the presence of hypermomentum, there is always a longitudinal mode propagating at the speed of light. The universal existence of the shear modes and the longitudinal mode in the presence of hypermomentum is a key feature of symmetric teleparallel gravity, distinguishing it from the Riemannian framework through gravitational wave polarization detection. We also analyze the polarization modes in two widely studied special theories: $f(Q)$ theory and quadratic non-metricity theory. Our study reveals that, within the $f(Q)$ gravity framework, it is crucial to assume that matter fields are independent of the connection, as any dependence would lead to unphysical results.