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.
