We present and test a framework that models the three-dimensional distribution of mass in the Universe as a function of cosmological and astrophysical parameters. Our approach combines two different techniques: a rescaling algorithm that modifies the cosmology of gravity-only N-body simulations, and a baryonification algorithm which mimics the effects of astrophysical processes induced by baryons, such as star formation and AGN feedback. We show how this approach can accurately reproduce the effects of baryons on the matter power spectrum of various state-of-the-art hydro-dynamical simulations (EAGLE, Illustris, Illustris-TNG, Horizon-AGN, and OWLS,Cosmo-OWLS and BAHAMAS), to percent level from very large down to small, highly nonlinear scales, k= 5 h/Mpc, and from z=0 up to z=2. We highlight that, thanks to the heavy optimisation of the algorithms, we can obtain these predictions for arbitrary baryonic models and cosmology (including massive neutrinos and dynamical dark energy models) with an almost negligible CPU cost. Therefore, this approach is efficient enough for cosmological data analyses. With these tools in hand we explore the degeneracies between cosmological and astrophysical parameters in the nonlinear mass power spectrum. Our findings suggest that after marginalising over baryonic physics, cosmological constraints inferred from weak gravitational lensing should be moderately degraded.
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