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Gravitational waves (GWs) from compact binary coalescences (CBCs) canconstrain the cosmic expansion of the universe. In the absence of an associatedelectromagnetic counterpart, the spectral sirens method exploits the relationbetween the detector frame and the source frame masses to jointly infer theparameters of the mass distribution of black holes (BH) and the cosmicexpansion parameter $H_0$. This technique relies on the choice of theparametrization for the source mass population of BHs observed in binary blackholes merger (BBHs). Using astrophysically motivated BBH populations, we studythe possible systematic effects affecting the inferred value for $H_0$ whenusing heuristic mass models like a broken power law, a power law plus peak anda multi-peak distributions. We find that with 2000 detected GW mergers, theresulting $H_0$ obtained with a spectral sirens analysis can be biased up to$3\sigma$. The main sources of this bias come from the failure of the heuristicmass models used so far to account for a possible redshift evolution of themass distribution and from their inability to model unexpected mass features.We conclude that future dark siren GW cosmology analyses should make use ofsource mass models able to account for redshift evolution and capable to adjustto unforeseen mass features.

A Study of Systematics on the Cosmological Inference of the Hubble Constant from Gravitational Wave Standard Sirens