On the degeneracies between baryons, massive neutrinos, and $f(R)$ gravity in Stage IV cosmic shear analyses

Modelling nonlinear structure formation is essential for current and forthcoming cosmic shear experiments. We combine the halo model reaction formalism, implemented in the REACT code, with the COSMOPOWER machine learning emulation platform, to develop and publicly release REACTEMU-FR (https://github.com/cosmopower-organization/reactemu-fr), a fast and accurate nonlinear matter power spectrum emulator for $f(R)$ gravity with massive neutrinos. Coupled with the state-of-the-art baryon feedback emulator BCEMU, we use REACTEMU-FR to produce Markov Chain Monte Carlo forecasts for a cosmic shear experiment with typical Stage IV specifications. We find that the inclusion of highly nonlinear scales (multipoles between $1500\leq \ell \leq 5000$) only mildly improves constraints on most standard cosmological parameters (less than a factor of 2). In particular, the necessary modelling of baryonic physics effectively damps most constraining power on the sum of the neutrino masses and modified gravity at $\ell \gtrsim 1500$. Using an approximate baryonic physics model produces mildly improved constraints on cosmological parameters which remain unbiased at the $1\sigma$-level, but significantly biases constraints on baryonic parameters at the $> 2\sigma$-level.