• bayesian hierarchical inference
• cosmology
• Einstein Telescope
• gravitational waves
• gravitational-wave background (GWB)
• Hubble constant
• independent catalog
• joint analysis
• O5 LVK
• resolved sources

Since their first detection in 2015, gravitational waves (GWs) have become a reliable and innovative probe of our Universe. One of the fields that benefits most from GW detection is cosmology. To date, measurements of the present-day expansion rate of the Universe, H0, have mainly relied on two classes of methods: late-universe and early-universe probes. These yield results that differ at the 5σ level, a discrepancy known as the Hubble tension, whose solution is one of the main goals of modern cosmology. GWs provide a new way to address the Hubble tension, offering an independent and complementary estimate of H0 through distance and redshift measurements that do not require calibration. GW-based inference has usually been performed using resolved events, such as binary black hole (BBH) mergers, either in association with an electromagnetic counterpart, through correlation with galaxy catalogs, or by exploiting peculiar features in the source mass distribution. In this work, we propose to infer H0 (along with other astrophysical parameters) from the joint analysis of resolved GW events and the unresolved gravitational-wave background (GWB) expected during the fifth observing run (O5) of the LIGOVirgo–KAGRA collaboration. We develop an independent pipeline to simulate a catalog of BBHs and extract both resolved and unresolved sources. From the unresolved population, we estimate the BBH contribution to the GWB energy density, confirming the expected power-law behavior with a peak at 100 Hz reaching an amplitude of the order of ten to the minus nine. We treat the two classes of sources as independent, leading to a simplified hierarchical likelihood expressed as the product of the individual likelihoods for the resolved events and the GWB. We perform the inference using the ICAROGW pipeline. We find that including the GWB leads to a marginal improvement in the inference of H0, reducing its uncertainty and slightly shifting the inferred value toward higher values, consistently with recent results in the literature. Moreover, the GWB allows us to place meaningful constraints on certain merger rate parameters that would remain poorly determined, if only resolved sources were considered. Finally, we investigate which approximations underlying the Bayesian approach used in this work break down in the context of third-generation detectors, such as Einstein Telescope. We then propose strategies to account for the lack of full source independence in the inference. Overall, this work shows the feasibility and current limitations of joint GW cosmology and highlights the challenges and opportunities offered by both future runs of current detectors and next-generation observatories.