We present a formulation of observed number density fluctuations of gravitational-wave (GW) sources in a three dimensional space. In GW observations, redshift identification for each GW source is a challenging issue, in particular, for high redshift sources. The use of observed luminosity distance as a distance indicator will be a simple yet optimal way for measuring the clustering signal. We derive the density fluctuations of GW sources estimated from observed luminosity distance and sky position of each source. The density fluctuations are distorted as similar to the so-called redshift space distortions in galaxy surveys but with several differences. We then show the two-point correlation function and multipole power spectrum in the presence of the distortion effect. We find that the line-of-sight derivative of the lensing convergence, which does not appear in the redshift-space distortions, leads to significant distortions in the observed correlation function. In addition, the lensing effect affects higher-order multipole power spectra and its signal-to-noise at high redshifts.

我们提出了在三维空间中观察到的重力波（GW）源数密度波动的公式。在GW观测中，每个GW源的红移识别是一个具有挑战性的问题，特别是对于高红移源。使用观测到的光度距离作为距离指示器将是一种简单但最优的测量聚类信号的方法。我们从观测到的每个光源的光度距离和天空位置推导出了GW光源的密度波动。密度波动像银河系测量中的所谓的红移空间畸变一样失真，但有一些差异。然后，我们显示了存在失真效应时的两点相关函数和多极功率谱。我们发现透镜收敛的视线导数，不会出现在红移空间畸变中的信号会导致观察到的相关函数发生明显的畸变。此外，透镜效应会影响高阶多极功率谱及其在高红移下的信噪比。