Investigating the Hubble Constant Tension -- Two Numbers in the Standard
  Cosmological Model
  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 $\Lambda$CDM model,
the Hubble constant ($H_0$) and today's matter energy fraction
($\Omega_{\rm{m}}$). If we therefore compare all constraints individually in
the $H_0$-$\Omega_{\rm{m}}$ 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 $H_0$-$\Omega_{\rm{m}}$ space based on a flat $\Lambda$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.
