Measuring the modified gravitational waves propagation beyond general
  relativity from observations
  In modified gravity theories, the gravitational waves propagation are
presented in nonstandard ways. We propose a modification to the damping rate
different from GR and measure the modified gravitational waves propagation from
observations. The modified tensor perturbations generate anisotropies and
polarization which show corresponding signatures in the tensor power spectra.
We explore how modified gravitational waves affect CMB power spectra and
explore the constraints on the cosmological parameters with modified damping
rate from Planck+BK15+BAO datasets. In the $\Lambda$CDM+$r$ model, the impacts
on the tensor-to-scalar ratio are slightly. We also measure the tilt of tensor
power spectra from gravitational waves observations within LIGO, LISA, IPTA,
FAST and SKA projects. We find that gravitational waves observations can
improve the constraints on the tensor tilt significantly if the amplitude of
the tensor power spectrum is within the detectable range. In particular,
compared with LIGO and LISA, SKA may provide a much better constraint on the
positive part of the tensor tilt.
