  The advent of gravitational-wave astronomy is now allowing for the study of
compact binary merger demographics throughout the Universe. This information
can be leveraged as tools for understanding massive stars, their environments,
and their evolution. One active question is the nature of compact binary
formation: the environmental and chemical conditions required for black hole
birth and the time delays experienced by binaries before they merge.
Gravitational-wave events detected today, however, primarily occur at low or
moderate redshifts due to current interferometer sensitivity, therefore
limiting our ability to probe the high redshift behavior of these quantities.
In this work, we circumvent this limitation by using an additional source of
information: observational limits on the gravitational-wave background from
unresolved binaries in the distant Universe. Using current gravitational-wave
data from the first three observing runs of LIGO-Virgo-KAGRA, we combine
catalogs of directly detected binaries and limits on the stochastic background
to constrain the time-delay distribution and metallicity dependence of binary
black hole evolution. Looking to the future, we also explore how these
constraints will be improved at the Advanced LIGO A+ sensitivity. We conclude
that, although binary black hole formation cannot be strongly constrained with
today's data, the future detection (or a non-detection) of the
gravitational-wave background with Advanced LIGO A+ will carry strong
implications for the evolution of binary black holes.
