Cosmology intertwined III: fσ8 and S8

The standard Cold Dark Matter cosmological model provides a wonderful fit to current cosmological data, but a few statistically significant tensions and anomalies were found in the latest data analyses. While these anomalies could be due to the presence of systematic errors in the experiments, they could also indicate the need for new physics beyond the standard model. In this Letter of Interest we focus on the tension between Planck data and weak lensing measurements and redshift surveys, in the value of the matter energy density and the amplitude (or the growth rate ) of cosmic structure. We list a few promising models for solving this tension, and discuss the importance of trying to fit multiple cosmological datasets with complete physical models, rather than fitting individual datasets with a few handpicked theoretical parameters.