The metallicity dependence and evolutionary times of merging binary black holes: Combined constraints from individual gravitational-wave detections and the stochastic background

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.