Gravitational wave source clustering in the luminosity distance space with the presence of peculiar velocity and lensing errors

GW number count can be used as a novel tracer of the large scale structure (LSS) in the luminosity distance space (LDS), just like galaxies in the redshift space. It is possible to obtain the DL−DA duality relation with clustering effect. However, the peculiar velocity dispersion error of the host galaxy and the foreground lensing magnification will contaminate the GW luminosity distance measurement, and will degrade the GW clustering from a spectroscopic-like data down to a photometric-like data. In this paper, we investigate how these LSS induced distance errors modify our cosmological parameter precision inferred from the LDS clustering for the Big Bang Observatory (BBO) and the Einstein Telescope (ET). We forecast the parameter estimation errors on the angular diameter distance DA, luminosity distance space Hubble parameter HL and structure growth rate fLσ8 with a Fisher matrix method. We find that for BBO, it is possible to constrain the cosmological parameters with a relative error of 10−3 to 10−2 below DL<5 Gpc. The velocity dispersion error is dominant in the low luminosity distance range, while the lensing magnification error is the bottleneck in the large luminosity distance range. To reduce the lensing error, we assumed two different delensing efficiencies, namely 50% (optimistic) and 10% (conservative). Even with the optimistic assumption, the fractional error increased to O(1) at luminosity distance DL=25 Gpc. The results for ET are similar as those from BBO. Due to the GW source number in ET is less than that from BBO, the corresponding results also get a bit worse.