Scalar modes of polarization and speed of gravitational waves in $f(R)$ gravity

We explore the gravitational waves (GWs) within the framework of the $f(R)$ gravity model represented by $f(R)=R^{1+\delta}/R^\delta_c$ in the weak field approximation. In this scenario, gravitational waves exhibit an additional polarization mode beyond the standard transverse-traceless (TT) tensor modes. We show that the polarization characteristics of these waves are connected to the scalaron mass and the effective potential derived from the function $f(R)$. Furthermore, the study of the speed of gravitational waves ($c_g$) within the Horndeski theory, particularly using the $f(R)$ model, reveals an intriguing feature about the equality of the speed of gravitational waves to that that of electromagnetic waves. This equivalence arises due to the modification introduced in the Ricci scalar within the $f(R)$ model.