Analyzing clustering of astrophysical gravitational-wave sources: Luminosity-distance space distortions

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.