Dark Matter Cosmology with Varying Viscosity: a Possible Resolution to the $S_8$ Tension

We study varying forms of viscous dark matter and try to address the intriguing tensions of the standard model of cosmology with recent cosmological data, including the Hubble and $S_8$ tensions. We note that by assuming the dark matter viscosity depends on the Hubble parameter, dark matter density, or both, one can improve the statistics. Although the models tend to aggravate the Hubble tension a bit, they tend to reduce the $S_8$ tension, even in comparison with the constant viscosity case. Since similar to viscosity massive neutrinos suppress the power spectrum of matter on small length scales, considering them along with the viscous dark matter, we find that the neutrino mass range is tightened.