If we try to apply general relativity to the description of the Universe on large scales we encounter the following problem: the Einstein field equations are written in terms of local Ricci curvature that varies considerably from point to point. If we have a matter distribution that is close to being homogeneous and isotropic on large scales, but is inhomogeneous on smaller scales, it is therefore not obvious that the large-scale evolution of the cosmological average is given by applying EFE to FLRW cosmologies with simple sources. The natural thing to do is to introduce an idealised, coarse-grained (or averaged) metric tensor $g^{(0)}$, that represents the large-scale geometry of the Universe. We can then argue that this metric satisfies the effective EFE:
$$G_{ij}[g^{(0)}]=\kappa T^{(0)}_{ij},$$where $T^{(0)}_{ij}$ is defined as the effective stress–energy tensor. But this stress–energy tensor in general will be different than the simple average of the local one, $T^{(0)}_{ij}\neq \langle T_{ij}\rangle $. The difference between the two is referred in the literature as the backreaction of inhomogeneities on the large-scale cosmological evolution.
This problem of cosmological backreaction is all the more glaring because observation suggest that the largest part of $T^{0}_{ij}$ is the dark energy for which no universally agreed upon theory exists. It is thus tempting to ask is it possible to explain the dark energy (or at least some of it contribution in $T^{(0)}$) as an effect of smaller-scale inhomogeneities. A good introduction to this programme is the following review:
Abstract
The effective evolution of an inhomogeneous universe model in any theory of gravitation may be described in terms of spatially averaged variables. In Einstein’s theory, restricting attention to scalar variables, this evolution can be modeled by solutions of a set of Friedmann equations for an effective volume scale factor, with matter and backreaction source terms. The latter can be represented by an effective scalar field (“morphon field”) modeling Dark Energy. The present work provides an overview over the Dark Energy debate in connection with the impact of inhomogeneities, and formulates strategies for a comprehensive quantitative evaluation of backreaction effects both in theoretical and observational cosmology. We recall the basic steps of a description of backreaction effects in relativistic cosmology that lead to refurnishing the standard cosmological equations, but also lay down a number of challenges and unresolved issues in connection with their observational interpretation. The present status of this subject is intermediate: we have a good qualitative understanding of backreaction effects pointing to a global instability of the standard model of cosmology; exact solutions and perturbative results modeling this instability lie in the right sector to explain Dark Energy from inhomogeneities. It is fair to say that, even if backreaction effects turn out to be less important than anticipated by some researchers, the concordance high-precision cosmology, the architecture of current N-body simulations, as well as standard perturbative approaches may all fall short in correctly describing the Late Universe.
The other viewpoint on the cosmological backreaction problem is that the effects of inhomogeneities are negligible on the large-scale evolution of the Universe and the FLRW cosmologies (with cosmological constant and dark matter) explain observations extremely well. This approach is developed in a series of influential papers by Green & Wald the latest of which is:
Abstract:
Extremely well! In the $\Lambda$CDM model, the spacetime metric, $g_{ab}$, of our universe is approximated by an FLRW metric, $g_{ab}^{(0)}$, to about $1$ part in $10^4$ or better on both large and small scales, except in the immediate vicinity of very strong field objects, such as black holes. However, derivatives of $g_{ab}$ are not close to derivatives of $g_{ab}^{(0)}$, so there can be significant differences in the behavior of geodesics and huge differences in curvature. Consequently, observable quantities in the actual universe may differ significantly from the corresponding observables in the FLRW model. Nevertheless, as we shall review here, we have proven general results showing that — within the framework of our approach to treating backreaction — the large matter inhomogeneities that occur on small scales cannot produce significant effects on large scales, so $g_{ab}^{(0)}$ satisfies Einstein's equation with the averaged stress-energy tensor of matter as its source. We discuss the flaws in some other approaches that have suggested that large backreaction effects may occur. As we also will review here, with a suitable “dictionary”, Newtonian cosmologies provide excellent approximations to cosmological solutions to Einstein's equation (with dust and a cosmological constant) on all scales. Our results thereby provide strong justification for the mathematical consistency and validity of the $\Lambda$CDM model within the context of general relativistic cosmology.
The other side of this debate argues that some of the underlying assumptions made by Green & Wald are unfounded and their criticism on backreaction frameworks are already addressed in newer publications. This is summarized in a paper by a number of influential relativists:
- Buchert, T., Carfora, M., Ellis, G. F., Kolb, E. W., MacCallum, M. A., Ostrowski, J. J., ... & Wiltshire, D. L. (2015). Is there proof that backreaction of inhomogeneities is irrelevant in cosmology? Classical and quantum gravity, 32(21), 215021, doi:10.1088/0264-9381/32/21/215021, arXiv:1505.07800.
Abstract:
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. Specifically, we demonstrate, referring to their recent review paper, that (i) their two-dimensional example used to illustrate the fitting problem differs from the actual problem in important respects, and it assumes what is to be proven; (ii) the proof of the trace-free property of backreaction is unphysical and the theorem about it fails to be a mathematically general statement; (iii) the scheme that underlies the trace-free theorem does not involve averaging and therefore does not capture crucial non-local effects; (iv) their arguments are to a large extent coordinate-dependent, and (v) many of their criticisms of backreaction frameworks do not apply to the published definitions of these frameworks. It is therefore incorrect to infer that Green and Wald have proven a general result that addresses the essential physical questions of backreaction in cosmology.
A follow-up by Green & Wald (preprint only, already addressed in the published version of the Buchert et al. paper):
So the debate continues …
As an example of more recent entry in this debate here is a paper from a few days ago (thanks to SE user Kevin Kostlan for bringing this to attention)):
Croker, K. S., & Weiner, J. L. (2019). Implications of Symmetry and Pressure in Friedmann Cosmology. I. Formalism. The Astrophysical Journal, 882(1), 19.
doi:10.3847/1538-4357/ab32da
