  With about a hundred binary black hole (BBH) mergers detected by
LIGO-Virgo-KAGRA, and with several hundreds expected in the current O4 run, GWs
are revolutionizing our understanding of the universe. Some BBH sources are too
faint to be individually detected, but collectively they may give rise to a
stochastic GW background (SGWB). In this paper, we calculate the SGWB
associated with BBH mergers dynamically assembled in dense star clusters, using
state-of-the-art numerical models. We discuss the role of modeling the
evolution of the mass distribution of BBH mergers, which has significant
implications for model selection and parameter estimation, and could be used to
distinguish between different channels of BBH formation. We demonstrate how the
birth properties of star clusters affect the amplitude and frequency spectrum
of the SGWB, and show that upcoming observation runs of ground-based GW
detectors may be sensitive enough to detect it. Even in the case of a
non-detection, we find that GW data can be used to constrain the highly
uncertain cluster birth properties, which can complement direct observations of
young massive clusters and proto-star clusters in the early universe by JWST.
