Does the lumpiness of the universe have a large impact on the metric?

UPDATE July 30, 2019: Green and Wald are wrong. See our latest post.

---

https://arxiv.org/abs/1505.07800

Is there proof that backreaction of inhomogeneities is irrelevant in cosmology?

T. Buchert, M. Carfora, G.F.R. Ellis, E.W. Kolb, M.A.H. MacCallum, J.J. Ostrowski, S. Räsänen, B.F. Roukema, L. Andersson, A.A. Coley, D.L. Wiltshire

(Submitted on 28 May 2015 (v1), last revised 15 Oct 2015 (this version, v2))

"No. In a number of papers Green and Wald argue that the standard FLRW model approximates our Universe extremely well on all scales, except close to strong field astrophysical objects. In particular, they argue that the effect of inhomogeneities on average properties of the Universe (backreaction) is irrelevant. We show that this latter claim is not valid. ..."


There seems to be an ongoing debate of the relevance of galaxy clusters on the large scale (> 100 megaparsecs) structure and development of the universe.

Syksy Räsänen et al. argue, that the impact on the metric might be relevant and even be the sole explanation for the accelerated expansion. No dark energy would be needed.

https://arxiv.org/abs/1506.06452

Stephen R. Green and Robert M. Wald have introduced a model which, they claim, shows that the effect of lumpiness is negligible.

One would think that by now there would be numerical simulations which decide the impact that lumpiness has on the universe.

This is an interesting dilemma. Which of the camps is right?

Does there exist a perturbed solution of an FLRW universe at all?

We know that the standard FLRW universe is an exact solution of the Einstein equations. It has a perfectly uniform mass-energy distribution.

If we start from the age where the cosmic microwave background was born, then the differences in mass-energy density were of the order 1 / 100,000.

Does there exist a perturbed FLRW solution which would have the characteristics of the early universe?

Since the Einstein equations are very strict, even a small perturbation might make the FLRW solution to diverge, so that there is no solution at all.

The question is not just what magnitude corrections does the lumpiness cause in the standard FLRW model - the question is if a solution exists at all.

The camp of Syksy Räsänen et al. has observed the fact that the corrections might blow up.

In this blog we have suspected that the Einstein equations are too strict, so that no solution exists at all for realistic mass distributions. A symptom of that would be that corrections to a known symmetric solution would blow up when we try to perturb it moderately.

Stephen R. Green and Robert M. Wald, if they are right, have to present a mathematical proof that the 1 / 100,000 perturbation does not make the corrections blow up - that is equivalent to proving that a solution of the Einstein equations exist. Mathematically, this is a very hard task. Has anyone made progress on this? Could the technique of Christodoulou and Klainerman work in an FLRW universe, too?

The stability of QED versus the stability of general relativity

https://www.physicsforums.com/insights/struggles-continuum-part-6/

Freeman Dyson observed in 1952 that the perturbation series of quantum electrodynamics probably diverges, even though the partial sums of the first terms give very accurate predictions for natural phenomena.

In general relativity, the so-called post-newtonian approximation gives accurate results for binary pulsars. We do not know if LIGO uses a similar technique.

If a numerical approximation series converges, then the limit might be an exact solution of the Einstein equations.

By studying numerical approximation algorithms we may get heuristic information about the existence of a solution for the Einstein equations. Divergence of an approximation series may be a symptom that no solution exists. However, QED shows that the series may appear to converge even though it is divergent.

We need to check what approximation methods Räsänen, Wald, etc. use and do the methods appear to converge.

It is possible that the FLRW model combined with some approximation method produces accurate results. Then we would have a practical model. To show that the model really is a result of general relativity, we need to show that general relativity has a solution and that the model approximates that solution.

We have suggested in this blog that general relativity should be replaced with a more flexible rubber sheet model of gravity. Then the existence of solutions might be very easy to prove. Furthermore, the rubber model might show its validity by predicting the properties of neutron stars better than general relativity.