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Third-generation (3G) gravitational wave detectors, in particular EinsteinTelescope (ET) and Cosmic Explorer (CE), will explore unprecedented cosmicvolumes in search for compact binary mergers, providing us with tens ofthousands of detections per year. In this study, we simulate and employ binaryblack holes detected by 3G interferometers as dark sirens, to extract and infercosmological parameters by cross-matching gravitational wave data withelectromagnetic information retrieved from a simulated galaxy catalog.Considering a standard $\Lambda$CDM model, we apply a suitable Bayesianframework to obtain joint posterior distributions for the Hubble constant $H_0$and the matter energy density parameter $\Omega_m$ in different scenarios.Assuming a galaxy catalog complete up to $z=1$ and dark sirens detected with anetwork signal-to-noise ratio greater than 300, we show that a network made ofET and two CEs can constrain $H_0$ ($\Omega_m$) to a promising $0.8\%$($10.0\%$) at $90\%$ confidence interval within one year of continuousobservations. Additionally, we find that most of the information on $H_0$ iscontained in local, single-host dark sirens, and that dark sirens at $z>1$ donot substantially improve these estimates. Our results imply that a sub-percentmeasure of $H_0$ can confidently be attained by a network of 3G detectors,highlighting the need for characterising all systematic effects to a higheraccuracy.

Dark siren cosmology with binary black holes in the era of third-generation gravitational wave detectors