Early or phantom dark energy, self-interacting, extra, or massive neutrinos, primordial magnetic fields, or a curved universe: An exploration of possible solutions to the $H_0$ and $\sigma_8$ problems

In this paper we explore the existing tensions in the local cosmological expansion rate, $H_0$, and amplitude of the clustering of large-scale structure at $8\, h^{-1}\mathrm{Mpc}$, $\sigma_8$, as well as models that claim to alleviate these tensions. We consider seven models: evolving dark energy ($w$CDM), extra radiation ($N_\mathrm{eff}$), massive neutrinos, curvature, primordial magnetic fields (PMF), self-interacting neutrino models, and early dark energy (EDE). We test these models against three data sets that span the full range of measurable cosmological epochs, have significant precision, and are well-tested against systematic effects: the Planck 2018 cosmic microwave background data, the Sloan Digital Sky Survey baryon acoustic oscillation scale measurements, and the Pantheon catalog of Type Ia supernovae. We use the recent SH0ES $H_0$ measurement and several measures of $\sigma_8$ (and its related parameter $S_8=\sigma_8\sqrt{\Omega_\mathrm{m}/0.3}$). We find that four models are above the "strong" threshold in Bayesian model selection, $w$CDM, $N_\mathrm{eff}$, PMF, and EDE. However, only EDE also relieves the $H_0$ tension in the full data sets to below 2$\sigma$. Contrarily, no model alleviates the $S_8/\sigma_8$ tension in the full data set, nor does better than $\Lambda$CDM in the combined case of both $H_0$ and $S_8/\sigma_8$ tensions.