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

TL;DR

This paper investigates how large-scale structure induced errors, such as peculiar velocity and lensing, affect gravitational wave source clustering measurements in luminosity distance space for future observatories like BBO and ET, impacting cosmological parameter estimation.

Contribution

It provides a detailed forecast of the impact of velocity and lensing errors on GW clustering-based cosmology using Fisher matrix analysis for BBO and ET.

Findings

GW clustering can constrain cosmology below 5 Gpc.

Velocity errors dominate at low distances.

Lensing errors limit accuracy at large distances.

Abstract

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 $D_L-D_A$ duality relation with clustering effect. However, several LSS induced errors will contaminate the GW luminosity distance measurement, such as the peculiar velocity dispersion error of the host galaxy as well as the foreground lensing magnification. The distance uncertainties induced from these effects 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. We consider two of the next generation GW observatories, namely the Big Bang Observatory (BBO) and the Einstein Telescope (ET). We forecast the parameter estimation errors on the angular diameter distance $D_A$, luminosity distance space Hubble parameter $H_L$ and structure growth rate $f_L\sigma_8$ with a Fisher matrix method. Generally speaking, the GW source clustering data can be used for cosmological studies below $D_L<5$ Gpc, while above this scale the lensing errors will increase significantly. We find that for BBO, it is possible to constrain the cosmological parameters with a relative error of $10^{-3}$ to $10^{-2}$ below $D_L<5$ 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 a $50\%$ delensing efficiency. Even with this optimal assumption, the fractional error increased to $O(1)$ at luminosity distance $D_L=25$ 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.