Inferring astrophysics and cosmology with individual compact binary coalescences and their gravitational-wave stochastic background

Gravitational waves (GWs) from compact binary coalescences (CBCs) provide a new avenue to probe the cosmic expansion, in particular the Hubble constant $H_0$. The spectral sirens method is one of the most used techniques for GW cosmology. It consists of obtaining cosmological information from the GW luminosity distance, directly inferred from data, and the redshift that can be implicitly obtained from the source frame mass distribution of the CBC population. With GW detectors, populations of CBCs can be either observed as resolved individual sources or implicitly as a stochastic gravitational-wave background (SGWB) from the unresolved ones. In this manuscript, we study how resolved and unresolved sources of CBCs can be employed in the spectral siren framework to constrain cosmic expansion. The idea stems from the fact that the SGWB can constrain additional population properties of the CBCs thus potentially improving the measurement precision of the cosmic expansion parameters.We show that with a 5-detector network at O5-designed sensitivity, the inclusion of the SGWB will improve our ability to exclude low values of $H_0$ and the dark matter energy fraction $\Omega_m$, while also improving the determination of a possible CBC peak in redshift. Although low values of $H_0$ and $\Omega_m$ will be better constrained, we obtain that most of the precision on $H_0$ will be provided by resolved spectral sirens. Instead, for the 4-detector network, the population posterior is entirely dominated by resolved sources, and the inclusion of the SGWB does not lead to any noticeable improvement in the precision of $H_0$ across its range.We also performed a spectral siren analysis for 59 resolved binary black hole sources detected during the third observing run with an inverse false alarm rate higher than 1 per year jointly with the SGWB. We obtain that with current sensitivities, the cosmological and population results are not impacted by the inclusion of the SGWB.