Investigating the Hubble Constant Tension -- Two Numbers in the Standard Cosmological Model

Abstract

The current Hubble constant tension is usually presented by comparing constraints on $H_0$ only. However, the post-recombination background cosmic evolution is determined by two parameters in the standard $Λ$CDM model, the Hubble constant ($H_0$) and today's matter energy fraction ($Ω_{\rm{m}}$). If we therefore compare all constraints individually in the $H_0$-$Ω_{\rm{m}}$ plane, (1) various constraints can be treated as independently as possible, (2) single-sided constraints are easier to consider, (3) compatibility among different constraints can be viewed in a more robust way, (4) the model dependence of each constraint is clear, and (5) whether or not a nonstandard model is able to reconcile all constraints in tension can be seen more effectively. We perform a systematic comparison of different constraints in the $H_0$-$Ω_{\rm{m}}$ space based on a flat $Λ$CDM model, treating them as separately as possible. Constraints along different degeneracy directions consistently overlap in one region of the space, with the local measurement from Cepheid variable-calibrated supernovae being the most outlying, followed by the time-delay strong-lensing result. Considering the possibility that some nonstandard physics may reconcile the constraints, we provide a general discussion on nonstandard models with modifications at high, mid, or low redshifts, and the effect of local environmental factors. Due to the different responses of individual constraints to a modified model, it is not easy for nonstandard models to reconcile all constraints if none of them have unaccounted-for systematic effects.