Cosmological gravitational-wave backgrounds are an exciting science target for next-generation ground-based detectors, as they encode invaluable information about the primordial Universe. However, any such background is expected to be obscured by the astrophysical foreground from compact-binary coalescences. We propose a novel framework to detect a cosmological gravitational-wave background in the presence of binary black holes and binary neutron star signals with next-generation ground-based detectors, including Cosmic Explorer and the Einstein Telescope. Our procedure involves first removing all the individually resolved binary black hole signals by notching them out in the time-frequency domain. Then, we perform joint Bayesian inference on the individually resolved binary neutron star signals, the unresolved binary neutron star foreground, and the cosmological background. For a flat cosmological background, we find that we can claim detection at $5\sigma$ level when ${\mathrm{\Omega }}_{\mathrm{ref}}\ge 2.7\times {10}^{-12}/\sqrt{T_{\mathrm{obs}}/\mathrm{yr}}$, where $T_{\mathrm{obs}}$ is the observation time (in years), which is within a factor of $\lesssim 2$ from the sensitivity reached in the absence of these astrophysical foregrounds.

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• Cosmology
• General relativity
• Gravitational waves
• Astronomical black holes
• Neutron stars & pulsars
• Gravitational wave detectors
• Numerical simulations in gravitation & astrophysics