Baryonic effects for weak lensing. Part II. Combination with X-ray data
  and extended cosmologies
  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 Schneider et al. 2019) are included
in both the tomographic power spectrum and the covariance matrix. However, this
time we assume the more realistic case of a $\Lambda$CDM cosmology with massive
neutrinos, and we consider several extensions of the currently favoured
cosmological model. For the standard $\Lambda$CDM case, we show that including
X-ray data reduces the uncertainties on the sum of the neutrino mass by
$\sim30$ percent, while there is only a mild improvement on other parameters
such as $\Omega_m$ and $\sigma_8$. As extensions of $\Lambda$CDM, we consider
the 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 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 $\Lambda$MDM. Ignoring baryonic effects in the analysis pipeline leads to
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.
