The redistribution of baryonic matter in massive halos through processes like
active galactic nuclei feedback and star formation leads to a suppression of
the matter power spectrum on small scales. This redistribution can be measured
empirically via the gas and stellar mass fractions in galaxy clusters, and
leaves imprints on their electron density profiles. We constrain two
semi-analytical baryon correction models with a compilation of recent Bayesian
population studies of galaxy groups and clusters sampling a mass range above
βΌ3Γ1013Mββ, and with cluster gas density profiles
derived from deep, high-resolution X-ray observations. We are able to fit all
the considered observational data, but highlight some anomalies in the
observations. The constraints allow us to place precise, physically informed
priors on the matter power spectrum suppression. At a scale of k=1h
Mpcβ1 we find a suppression of 0.042β0.014+0.012β
(0.049β0.012+0.016β), while at k=3h Mpcβ1 we find
0.184β0.031+0.026β (0.179β0.020+0.018β), depending on the model
used. We also predict at 97.5 percent credibility, that at scales k<0.37h
Mpcβ1 baryon feedback impacts the matter power less than 1%. This puts
into question if baryon feedback is the driving factor for the discrepancy
between cosmic shear and primary CMB results. We independently confirm results
on this suppression from small-scale cosmic shear studies, while we exclude
some hydro-dynamical simulations with too strong and too weak baryonic
feedback. Our empirical prediction of the power spectrum suppression shows that
studies of galaxy groups and clusters will be instrumental in unlocking the
cosmological constraining power of future cosmic shear experiments like
\textit{Euclid} and Rubin-LSST.Comment: 14 pages, 7 figures, submitted to MNRA