A non-linear solution to the $S_8$ tension?

Weak galaxy lensing surveys have consistently reported a lower amplitude for the matter fluctuation spectrum, as measured by the $S_8$ parameter, than expected in the $\Lambda{\rm CDM}$ cosmology favoured by $Planck$. However, the expansion history follows the predictions of the $Planck$ $\Lambda{\rm CDM}$ cosmology to high accuracy, as do measurements of lensing of the cosmic microwave background anisotropies. Redshift space distortion measurements also appear to be consistent with $Planck$ $\Lambda{\rm CDM}$. In this paper, we argue that these observations can be reconciled with the $Planck$ $\Lambda{\rm CDM}$ cosmology if the matter power spectrum is suppressed more strongly on non-linear scales than assumed in analyses of weak galaxy lensing. We demonstrate this point by fitting a one-parameter model, characterising a suppression of the non-linear power spectrum, to the KiDS-1000 weak lensing measurements. Such a suppression could be attributed to new properties of the dark matter that affect non-linear scales, or to a response of the matter fluctuations to baryonic feedback processes that are stronger than expected from recent cosmological simulations. Our proposed explanation can be tested using measurements of the amplitude of the matter fluctuation spectrum on linear scales, in particular via high precision redshift space distortion measurements from forthcoming galaxy and quasar redshift surveys.