I've changed the page to make it less confusing to the casual reader. The critical density in general relativity is the 3/8 factor you mention, which creates the unconfirmed "dark matter" prediction for density.
Notice I have a list of predictions near the top of the page, which includes the fact that the critical density (based on the assumption that there is no mechanism of gravity) is out by 0.5e^3 which is a factor of 10.
What general relativity does is to omit a mechanism. This assumes that the furthest galaxies, receding at light speed, are being pushed back towards us (slowed down by gravity).
In 1996 via Electronics World I pointed out that the furthest galaxies are not going to be slowed down, because the push of gravity requires that you are surrounded by a large amount of receding matter. The universe is like a 10^55 megatons space burst nuclear explosion, so far as gravity is concerned.
The 'warped space' comes from the radial contraction of masses due to this space pressure, not from mathematical equations, which are just modelling what occurs. Popular accounts of general relativity tend to use non-causal interpretations. I have pointed out that space fabric does not expand. What happens to it in the big bang? It moves inwards, pushing things together, gravity.
The furthest receding matter or radiation is not slowed down by gravity because gravity is not a pulling effect. It is purely pushing, and that requires that you have an expanding universe around you, which doesn't apply to the outer most edge of an space burst explosion.
Thus, the most distant supernovae continue unimpeded. This was discovered experimentally by Dr Perlmutter in 1998, about a year after the prediction. However, the scientific journals still wanted to ignore the mechanism.
What has happened is that instead of admitting that gravity doesn't slow down the most distant expansion, they have instead invented a new force that offsets the supposed gravity pull back, allowing the supernovae to continue as if there were no gravity.
In the same way, when the Earth centred universe failed to account for the motion of planets, epicycles were invented to "fix" the problem.