  The recent Advanced LIGO and Advanced Virgo joint observing runs have not
claimed a stochastic gravitational-wave background detection, but one expects
this to change as the sensitivity of the detectors improves. The challenge of
claiming a true detection will be immediately succeeded by the difficulty of
relating the signal to the sources that contribute to it. In this paper, we
consider backgrounds that comprise compact binary coalescences and additional
cosmological sources, and we set simultaneous upper limits on these
backgrounds. We find that the Advanced LIGO, Advanced Virgo network, operating
at design sensitivity, will not allow for separation of the sources we
consider. Third generation detectors, sensitive to most individual compact
binary mergers, can reduce the astrophysical signal via subtraction of
individual sources, and potentially reveal a cosmological background. Our
Bayesian analysis shows that, assuming a detector network containing Cosmic
Explorer and Einstein Telescope and reasonable levels of individual source
subtraction, we can detect cosmological signals $\Omega_{\rm{CS}}
(25\,\rm{Hz})=4.5 \times 10^{-13}$ for cosmic strings, and $\Omega_{\rm
BPL}(25\,\rm{Hz})= 2.2 \times 10^{-13}$ for a broken power law model of an
early universe phase transition.

General Relativity and Quantum Cosmology