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.

中文翻译：

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对引力波定相的修正

引力波经过很长的宇宙距离到达干涉仪，与（均匀和各向同性的）宇宙背景相互作用，所以一般来说，它的振幅和相位以某种非平凡的方式被修改。随着干涉仪灵敏度的提高，人们可以检测到对短波长近似的修正，这自然包括宇宙演化的信息。在这项工作中，计算了 Newman-Penrose 变量 $\Psi_4$ 以表明对短波长近似值有两个新的修正。一个形式出现在第一个后牛顿阶次，但被哈勃参数高度抑制；另一个发生在第五后牛顿阶，这是由于振幅的变化。第一次修正包含了宇宙的演化，但它可能不容易被发现。第二个表示当计算具有更高阶后牛顿项的更准确波形时，必须校正短波长近似值。