Systematic bias in dark siren statistical methods and its impact on Hubble constant measurement

The advent of multimessenger cosmology, marked by the detection of GW170817, demonstrated that standard sirens are a valuable cosmological probe. In the absence of an electromagnetic counterpart identification, gravitational waves carry valuable information through the dark siren approach, where the source redshift is estimated using galaxy catalogs of potential hosts within the localisation volume. However, the DS analysis can be affected by galaxy catalog incompleteness at the limits of gravitational-wave detectability, potentially introducing biases in the constraints on cosmological parameters. Focusing on GWs from binary black holes detected by the LVK collaboration, we explore the possible systematic biases in the measurement of the $H_0$. These biases may arise from (1) the incompleteness of catalogs due to the apparent magnitude thresholds of optical telescope sensitivity, and (2) the use of incorrect weighting schemes for each potential host. We found that an unbiased estimate of $H_0$ can be obtained when the corrected weighting scheme is applied to a complete or volume-limited catalog. We use a complete galaxy catalog covering 90% of the localisation probability for each GW detection, employing stellar mass as a tracer. Our results show that a sample of 100 binary black hole events with $A_{90\%}$<10 deg$^2$ and measured luminosity distances below 1600(2500) Mpc, detected by the LVK at O4(O5) sensitivity, can provide a percent-level measurement of $H_0$, with a precision of 3%(1%). This number of detections is expected to be accumulated after approximately 8 and 3 years of observations with the LVK at O4 and O5 sensitivity, respectively. The O5 run provides a reduction in the $H_0$ uncertainty by 1.34 km/s/Mpc compared to the O4-like configuration. The $H_0$ precision increases to approximately 6% when it is assumed that every galaxy has an equal probability of being the host.