Modeling Uncertainties in Modified Gravity Predictions for the Stochastic Gravitational-Wave Background

We investigate the impact of modified gravity on the stochastic gravitational-wave background (SGWB) generated by a cosmological population of unresolved binary black hole mergers. We consider two complementary classes of beyond-General Relativity (GR) effects: waveform-generation modifications described within the parametrized post-Einsteinian (ppE) framework and cosmological propagation effects associated with a modified gravitational-wave luminosity distance. Astrophysical uncertainties in the binary black hole population are consistently incorporated using a Power-Law plus Peak mass model combined with a Madau--Dickinson merger-rate evolution. Using SGWB forecasts for Advanced LIGO, the Einstein Telescope (ET), and Cosmic Explorer (CE), we perform injection-recovery analyses jointly varying modified-gravity and astrophysical population parameters. We show that frequency-dependent ppE corrections produce characteristic distortions in the SGWB spectral shape and can be meaningfully constrained by third-generation detectors, particularly CE. In contrast, modified propagation effects mainly induce smooth amplitude rescalings and exhibit stronger degeneracies with astrophysical uncertainties. Our results demonstrate that future SGWB observations will provide a complementary probe of gravitational physics across cosmic history and may open new avenues for testing deviations from GR beyond individually resolved compact-binary events.