Clustering effects on the Dark Siren determination of $H_0$: A simulation study

Abstract: Gravitational waves (GWs) offer an alternative way to measure the Hubble parameter. The optimal technique, the ``bright siren'' approach, requires the identification of an electromagnetic counterpart. However, a significant fraction of gravitational waves signals will not have counterparts. Such events can still constrain the Hubble parameter $H_0$ via statistical methods, exploiting galaxy information from the GWs sky localisation volume. In this work, we investigate the power of this method using high-resolution, cosmological simulations that include realistic clustering. We find that clustering leads to increased convergence of the $H_0$ posteriors, with clear recovery of the input value as early as $N_{\rm gw}=40$ events, compared to uniform catalogues, where the posterior remains largely unconstrained, even with $N_{\rm gw}=100$ events. In addition, we quantify the role of catalogue incompleteness. We show that catalogues with completeness levels as low as $f=25\%$ can be competitive with fully complete catalogues, confirming the impact of clustering. Completeness levels of $f=50\%$ perform statistically similar to complete catalogues with as few as $N_{\rm gw}=40$ events. This indicates the need to focus on improving gravitational waves detection capabilities, rather than obtaining more complete galaxy catalogues. Finally, we investigate additional properties of the method by taking into consideration physical weights, different observational errors, potential biases from the $H_0$ priors, a variety of detectors' horizon distances, and different methods of catalogue completion and statistical analysis.