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An accurate modelling of baryonic feedback effects is required to exploit thefull potential of future weak-lensing surveys such as Euclid or LSST. In thissecond paper in a series of two, we combine Euclid-like mock data of the cosmicshear power spectrum with an eROSITA X-ray mock of the cluster gas fraction torun a combined likelihood analysis including both cosmological and baryonicparameters. Following the first paper of this series, the baryonic effects(based on the baryonic correction model of Schneider et al. 2019) are includedin both the tomographic power spectrum and the covariance matrix. However, thistime we assume the more realistic case of a $\Lambda$CDM cosmology with massiveneutrinos, and we consider several extensions of the currently favouredcosmological model. For the standard $\Lambda$CDM case, we show that includingX-ray data reduces the uncertainties on the sum of the neutrino mass by$\sim30$ percent, while there is only a mild improvement on other parameterssuch as $\Omega_m$ and $\sigma_8$. As extensions of $\Lambda$CDM, we considerthe cases of a dynamical dark energy model (wCDM), a $f(R)$ gravity model(fRCDM), and a mixed dark matter model ($\Lambda$MDM) with both a cold and awarm/hot dark matter component. We find that combining weak lensing with X-raydata only leads to a mild improvement of the constraints on the additionalparameters of wCDM, while the improvement is more substantial for both fRCDMand $\Lambda$MDM. Ignoring baryonic effects in the analysis pipeline leads tosignificant false-detections of either phantom dark energy or a lightsubdominant dark matter component. Overall we conclude that for all cosmologiesconsidered, a general parametrisation of baryonic effects is both necessary andsufficient to obtain tight constraints on cosmological parameters.

Baryonic effects for weak lensing. Part II. Combination with X-ray data and extended cosmologies