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 H₀ and ₈ problems

In this paper we explore the existing tensions in the local cosmological expansion rate, H-0, and amplitude of the clustering of the large-scale structure at 8h(-1) Mpc, sigma(8), as well as models that claim to alleviate these tensions. We consider seven models: evolving dark energy (wCDM), extra radiation (N-eff), massive neutrinos, curvature, primordial magnetic fields (PMF), self-interacting neutrino models, and early dark energy (EDE). We test these models against three datasets 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) root Omega(m)/0.3). We find that four models are above the "strong" threshold in Bayesian model selection, wCDM, N-eff, PMF, and EDE. However, only EDE also relieves the H-0 tension in the full datasets to below 2 sigma. We discuss how the S-8/sigma(8) tension is reduced in recent observations. However, even when adopting a strong tension dataset, no model alleviates the S-8/sigma(8) tension, nor does better than Lambda CDM in the combined case of both H-0 and S-8/sigma(8) tensions.