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.
