Corrections to the gravitational wave phasing

The gravitational wave, traveling a long cosmological distance to reach interferometers, interacts with the (homogeneous and isotropic) cosmological background, so generally speaking, its amplitude and phase are modified in some nontrivial way. As the sensitivity of interferometers is improved, one may detect corrections to the short-wavelength approximation, which naturally includes the information of cosmological evolution. In this work, the Newman-Penrose variable $\Psi_4$ has been calculated to show that there are two new corrections to the short-wavelength approximation. One formally occurs at the first post-Newtonian order but is highly suppressed by the Hubble parameters; the other occurs at the fifth post-Newtonian order, which is due to the variation of the amplitude. The first correction contains the evolution of the Universe, but it may not be easily detected. The second one indicates that the short-wavelength approximation has to be corrected, when the more accurate waveforms with the higher-order post-Newtonian terms are calculated.