Dark energy a furphy, says new paper

[Wallace]

Thanks for the explanation Wallace.

My sense is that the Newtonian approximation deals with energy gradients arising from mass differentials (which are too small to generate significant deviations), but it does not deal with energy gradients arising from spatial curvature. I believe that the latter is a subject which must be calculated entirely by means of GR.

Spatial curvature is caused by uneven mass density in GR, you can't separate them unless you want to invent a new theory of gravity. The weak field metric is GR, so you can't say that it is not applicable and we must be GR 'entirely'. This metric has been shown to be an accurate solution of the 'entire' Einstein Equation, including any curvature gradients.

You might benefit from taking a coherent course in GR in order to learn it from the basics up. This will help guide your 'sense' much better. It's hard to learn such a theory from pop sci explanations on forums or even from published journal articles, since they assume the basics are understood. The worst trap one can fall into is having confidence in an erroneous understanding on the basics. I thoroughly recommend you work your way through a book such as Hartle's 'Gravity' (although many other books are good as well).

Since no suitable exact solution to the Einstein Field Equations is a available to us (and may not exist), in my opinion it comes down to trying to understand the Buchert averaging equations and assessing whether or not they are sound and Wiltshire is applying them properly.

It must be established whether the Buchert equations themselves are of any use which has not yet been shown. You can't apply something 'properly' if it in itself is flawed.