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 σ8 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 Mpc, σ 8 , as well as models that claim to alleviate these tensions. We consider seven models: evolving dark energy ( w CDM), 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 σ 8 (and its related parameter S 8 = σ 8 (cid:112) Ω m / 0 . 3). We find that four models are above the “strong” threshold in Bayesian model selection, w CDM, N eff , PMF, and EDE. However, only EDE also relieves the H 0 tension in the full datasets to below 2 σ . We discuss how the S 8 /σ 8 tension is reduced in recent observations. However, even when adopting a strong tension dataset, no model alleviates the S 8 /σ 8 tension, nor does better than ΛCDM in the combined case of both H 0 and S 8 /σ 8 tensions.