Probing the speed of gravitational waves beyond general relativity from CMB observations

In modified gravity theories, gravitational waves can propagate differently from general relativity and their propagating speed can be either constant or acquire a time dependence. We consider the constant models first and update the constraints on cosmological parameters from the combinations of $\mathrm{\text{Planck}}18+\mathrm{\text{BK}}18$ datasets. In this case, excluding superluminal propagation, we obtain the lower limit on the speed $c_g>0.22$ at 95% C.L. A non-trivial propagating speed impacts the amplitude of tensor spectrum by adding a factor $c_g^{n_t-1}$, where $n_t$ is the tensor tilt. We find that the value of $c_g$ has positive correlation with the tensor-to-scalar ratio and anti-correlation with the factor $c_g^{n_t-1}$. Then we explore a time-dependent speed which contains the resonance of the stochastic gravitational wave background. If the speed of gravitational waves oscillates at primordial era, resonance continuously enhances stochastic gravitational wave background which produces observable effects on tensor power spectra. We derive the constraints on the amplitude of oscillatory speed and tensor parameters from the combinations of $\mathrm{\text{Planck18}}+\mathrm{\text{BK18}}$ datasets. The numerical results show that the speed resonance of the stochastic gravitational wave background is sensitive to CMB observations.