Multi-band gravitational wave cosmology with stellar origin black hole binaries

Massive stellar origin black hole binaries (SBHBs), originating from stars above the pair-instability mass gap, are primary candidate for multi-band gravitational wave (GW) observations. Here we study the possibility to use them as effective dark standard sirens to constrain cosmological parameters. The long lasting inspiral signal emitted by these systems is accessible by the future $Laser \; Interferometer \; Space \; Antenna$ (LISA), while the late inspiral and merger are eventually detected by third generation ground-based telescopes such as the $Einstein \; Telescope$ (ET). The direct measurement of the luminosity distance and the sky position to the source, together with the inhomogeneous redshift distribution of possible host galaxies, allow to infer cosmological parameters by probabilistic means. The efficiency of this statistical method relies in high parameter estimation performances, and we show that this multi-band approach allows a precise determination of the Hubble constant H$_0$ with just ${\cal O}(10)$ detected sources. For selected SBHB population models, assuming $4$ ($10$) years of LISA observations, we find that H$_0$ is tipically determined at $\sim 2\%$ ($\sim 1.5\%$), whereas $\Omega_m$ is only mildly constrained with a typical precision of $30\%$ ($20\%$). The inference procedure sometimes leads to inconsistent results due to the appearance of spurious peaks in the posterior of the parameters. We discuss the origin of this issue and ways to mitigate it.