It is worth to stress that the problem is within measurements, on the one hand resulting from an astrophysical phenomenon (SN1A supernovae whose absolute luminosty is supposed to known – standard candles) at usualy z< 1 using the luminosity distance and redshift and on the other hand of a cosmolgical phenomenon the CMB by using the angular distance of the universe whose physical size at the time of recombination (z = 1089) is supposed to be known (in fact it is computed under cosmological hypothesis and laws of propagation of acoustic waves in a plasma) and redshift;
Notwithstanding with the detailed mathematical development, we point out a structural difference. Let us keep this in mind in our search for a solution for the discrepency between the results.
This would, likely, not question general relativity, which stipulates that the universe is totally determined by the physical nature and distribution of its elements, but that this is only a global property and that locally (at the cosmological scale, z<1 ) this would not be necessary verified (the laws may slightly be different)!
Are astrophysical and cosmological phenomena fully consistent ?
General relativity states that the universe is totally determined by the physical nature and distribution of its « astrophysical » elements, by virtue of Mach’s principle, a principle that inspired Einstein in the construction of his theory. In particular he justified the nature of inertia by the gravitational interaction between all the masses of the universe.
This implicitly invokes gravitation geometrically modeled by a manifold in general relativity. Note that the geometric model of an object is not the object itself, that is physical in nature (they are not of the same nature), but is a convenient way for describing its phenomenology.
In particular the geometric model of general relativity describes, in a very elegant (non-linear) way, the dynamics (what interests the physicist) of the objects in the system defined by the global geometry.
Does this imply that astrophysical phenomena are consistent (lead to the same results) with cosmological phenomena.
By construction, the answer of relativity is positive.
But can we verify this experimentally? It is not so simple, because the models deduced from theory and experience (the theory does not predict the astrophysical constitution of the universe, it is an experimental data, it defines the system (here the universe ) corresponding to these data.
It turns out that in the universes resulting from the application of this method, one is led to suppose dark matter and dark energy (the whole up to 95%), of unknown nature to propose solutions of general relativity satisfying the observations.
This can be a way for explaining the incompatibilities between astrophysical and cosmological measurements.
Can inhomogeneity be invoked to explain incompatibility?
Independently of the arguments developed in the previous paragraph, it should be remembered that the supposed homogeneity and isotropy of the universe, (By the Copernican principle, but also as simplifying hypothesis for getting an analytical solution) is valid only at the scale of the universe) but that locally (on the cosmological scale z< 1), this should not be fully satisfied (small disparities could exist)!
Observations and simulations show, that in fact, even at huge scales the matter is concentrated on the filaments and is far from being perfectly homogeneous and isotropic (« sponge » structure).
For example we could be in a local bubble (size z < 1) of over-density or under-density, which according to the equations would imply a difference in the value of the Hubble constant with that calculated on a much larger scale.
On the one hand, a measurement of the Hubble constant in the bubble (z < 1) where we s would be, which is effective for us and on the other hand, a measurement of the Hubble constant, for the homogeneous and isotropic universe (z>>1).
In this hypothesis, general relativity would not be in question, it would be an unadequate approximation of the parameters that would have to be invoked.
Thus the two measurements would be « correct » but would relate to two different phenomena.
In attached pdf document (to be downloaded), we recall the standard approach.