Baryonic feedback effects lead to a suppression of the weak lensing angular power spectrum on small scales. The poorly constrained shape and amplitude of this suppression is an important source of uncertainties for upcoming cosmological weak-lensing surveys such as Euclid or LSST. In this first paper in a series of two, we use simulations to build a Euclidlike tomographic mock data-set for the cosmic shear power spectrum and the corresponding covariance matrix, which are both corrected for baryons following the baryonification method of Schneider et al. [1]. In addition, we develop an emulator to obtain fast predictions of the baryonic suppression effects, allowing us to perform a likelihood inference analysis for a standard ΛCDM cosmology with both cosmological and astrophysical parameters. Our main findings are the following: (i) ignoring baryonic effects leads to a greater than 5σ bias on the cosmological parameters Ω m and σ 8 ; (ii) restricting the analysis to the largest scales, that are mostly unaffected by baryons, makes the bias disappear, but results in a blow-up of the Ω m -σ 8 contour area by more than a factor of 10; (iii) ignoring baryonic effects on

An accurate modelling of baryonic feedback effects is required to exploit the full potential of future weak-lensing surveys such as Euclid or LSST. In this second paper in a series of two, we combine Euclid-like mock data of the cosmic shear power spectrum with an eROSITA X-ray mock of the cluster gas fraction to run a combined likelihood analysis including both cosmological and baryonic parameters. Following the first paper of this series, the baryonic effects (based on the baryonic correction model of Ref.[1]) are included in both the tomographic power spectrum and the covariance matrix. However, this time we assume the more realistic case of a ΛCDM cosmology with massive neutrinos and we consider several extensions of the currently favoured cosmological model. For the standard ΛCDM case, we show that including X-ray data reduces the uncertainties on the sum of the neutrino mass by ∼ 30 percent, while there is only a mild improvement on other parameters such as Ω m and σ 8 . As extensions of ΛCDM, we consider the cases of a dynamical dark energy model (wCDM), a f (R) gravity model (fRCDM), and a mixed dark matter model (ΛMDM) with both a cold and a warm/hot dark matter component. We find that combining weak-lensing with X-ray data only leads to a mild improvement of the constraints on the additional parameters of wCDM, while the improvement is more substantial for both fRCDM and ΛMDM. Ignoring baryonic effects in the analysis pipeline leads significant false-detections of either phantom dark energy or a light subdominant dark matter component. Overall we conclude that for all cosmologies considered, a general parametrisation of baryonic effects is both necessary and sufficient to obtain tight constraints on cosmological parameters.