Damping of gravitational waves in f(R) gravity

We investigate the damping of gravitational waves (GW) in $f(R)$ gravity by matter. By applying the kinetic theory, we examine the first-order approximation of the relativistic Boltzmann equation. In the flat spacetime, we derive the evolution equations for waves in $f(R)$ gravity and demonstrate that Landau damping is absent while collision damping is present. In the Friedmann-Robertson-Walker (FRW) cosmology, we also examine the dynamical equations for the two modes. Furthermore, in the model $f(R) = R + \alpha R^2$, we investigate the effect of the mass term on wave amplitude decay within the neutrino system. We observe that the tensor mode with $m = 1 \, \text{eV}$ exhibits faster decay compared to other cases, while the scalar mode with $m = 1 \, \text{eV}$ appears to suppress decay.