  The formation of merging binary black holes can occur through multiple
astrophysical channels such as, e.g., isolated binary evolution and dynamical
formation or, alternatively, have a primordial origin. Increasingly large
gravitational-wave catalogs of binary black-hole mergers have allowed for the
first model selection studies between different theoretical predictions to
constrain some of their model uncertainties and branching ratios. In this work,
we show how one could add an additional and independent constraint to model
selection by using the stochastic gravitational-wave background. In contrast to
model selection analyses that have discriminating power only up to the
gravitational-wave detector horizons (currently at redshifts $z\lesssim 1$ for
LIGO-Virgo), the stochastic gravitational-wave background accounts for the
redshift integration of all gravitational-wave signals in the Universe. As a
working example, we consider the branching ratio results from a model selection
study that includes potential contribution from astrophysical and primordial
channels. We renormalize the relative contribution of each channel to the
detected event rate to compute the total stochastic gravitational-wave
background energy density. The predicted amplitude lies below the current
observational upper limits of GWTC-2 by LIGO-Virgo, indicating that the results
of the model selection analysis are not ruled out by current background limits.
Furthermore, given the set of population models and inferred branching ratios,
we find that, even though the predicted background will not be detectable by
current generation gravitational-wave detectors, it will be accessible by
third-generation detectors such as the Einstein Telescope and space-based
detectors such as LISA.
