Implications of feedback solutions to the $S_8$ tension for the baryon fractions of galaxy groups and clusters

Recent large-scale structure (LSS) surveys have revealed a persistent tension in the value of $S_8$ compared to predictions from the standard cosmological model. This tension may suggest the need for new physics beyond the standard model, but an accurate characterisation of baryonic effects is essential to avoid biases. Although some studies indicate that baryonic effects are too small to resolve this tension, others propose that more aggressive feedback mechanisms could reconcile differences between cosmic microwave background (CMB) measurements and low-redshift LSS observations. In this paper, we investigate the role of baryonic effects in alleviating the $S_8$ tension. We extend the SP(k) model (Salcido et al. 2023), which was trained on hundreds of cosmological hydrodynamical simulations to map the suppression of the matter power spectrum to the baryon fraction in groups and clusters, to predict the required baryon fraction for a given $P(k)$ suppression. We then compare predictions from recent cosmic shear (weak lensing) analyses with the latest baryon budget measurements from X-ray and weak gravitational lensing studies. Our findings show that studies marginalising over baryonic effects while fixing cosmological parameters to a Planck-like cosmology predict strong $P(k)$ suppression and baryon fractions that are much lower than existing low-redshift baryon budget estimates of galaxy groups and clusters. Conversely, most studies that marginalise over both cosmological parameters and baryonic effects imply baryon fractions that are consistent with observations but lower values of $S_8$ than inferred from the CMB. Unless the observed baryon fractions are biased high by a factor of several, these results suggest that a mechanism beyond baryonic physics alone is required to modify or slow down the growth of structure in the universe in order to resolve the $S_8$ tension.