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

The GW example is of the physical spherical surface of a 'ball bearing' with all its 'tiny defects', and an a priori assumed fit by the standard round geometry of the two-sphere. As a model of the actual surface they take a polyhedral sphere $({{\mathbb{S}}}^{2},{g}_{\mathrm{pol}})$ and consider how it is approximated by the (model) metric of a standard round sphere $({{\mathbb{S}}}^{2},{g}_{\mathrm{can}})$ ('can' for 'canonical'). The analogy they wish to draw is of an approximately FLRW metric such that the difference of the real and model metrics obeys the assumptions in their section 2. We discuss these assumptions and their consequences at length in section 3.4 and in appendix B.

Green and Wald assume $({{\mathbb{S}}}^{2},{g}_{\mathrm{pol}})$ to be a convex polyhedron. However, convexity is not an intrinsic characteristic of $({{\mathbb{S}}}^{2},{g}_{\mathrm{pol}}).$ It comes about only when we consider $({{\mathbb{S}}}^{2},{g}_{\mathrm{pol}})$ embedded into Euclidean space. The assumption of convexity involves extrinsic curvature, whereas the fitting problem is concerned with intrinsic curvature (as it must be in the cosmological setting).

Green and Wald discuss intrinsic aspects, by referring to Abbott's Flatland [21]. They argue that when two-dimensional observers in 'Sphereland' 'use triangles that are large compared with the distance between vertices, they obtain results that are reasonably consistent with spherical geometry. Nevertheless, even when looking at phenomena on large scales, the observers find some disturbing anomalies when attempting to model Sphereland by a perfect sphere' and conclude that 'as a result of these observations, the observers might be tempted to conclude that the perfect sphere model provides a reasonably good description of Sphereland on large scales—although with some significant deviations—but an extremely poor description of Sphereland on small scales. However, the actual situation, of course, is that the metric, g_ab, of Sphereland is everywhere extremely close to the metric, ${g}_{{ab}}^{(0)},$ of a perfect sphere'.

The analogy is that cosmological observers may be similarly misled into thinking an FLRW model (without backreaction) was a poor description. This was made very clear in the recent note [22] responding to the first draft of the present paper.

As quoted above, Green and Wald consider the Spherelanders performing geodesy with larger and larger triangles, thus exploring the (global aspects of the) intrinsic geometry of $({{\mathbb{S}}}^{2},{g}_{\mathrm{pol}}).$ By this means Spherelanders might, as GW suggest, reach the apparently obvious conclusion that g_pol is indeed approximated by the round metric g_can, with some small-scale anomalies. However, the measurements of the geometry of triangles proposed by GW do not measure the metric, they measure curvature (integrated over the interiors of the triangles). Knowing the curvature and a finite number of its derivatives does locally specify a metric uniquely, but at the present time we do not know a way to use this information to determine a best-fit approximate metric: we discuss these points in section 5.4.

More relevantly, it is the 'significant deviations' on small scales that are the important feature. Let us consider the steel ball example in more detail. The metric of each two-cell in the polyhedron $({{\mathbb{S}}}^{2},{g}_{\mathrm{pol}})$ is not curved, but flat. Green and Wald recall that the ball's (Gaussian) curvature is represented by a Dirac measure with support at the vertices of $({{\mathbb{S}}}^{2},{g}_{\mathrm{pol}}),$ and geometrically represented by vertex deficit angles measuring excess or defect with respect to the $2\pi$ vertex angles of a flat geometry. The latter implies, however, that the geometry of $({{\mathbb{S}}}^{2},{g}_{\mathrm{pol}})$ is almost-everywhere flat with curvature defects supported at the vertices: the relevant background metric of $({{\mathbb{S}}}^{2},{g}_{\mathrm{pol}})$ is thus not a spherical metric, as intuition and figure 2 in GW's paper might suggest, but a flat metric (with conical singularities representing curvature).

To illustrate what knowing the curvature, as the Spherelanders might, means in the present setting, note that we may have positive curvatures as well as negative curvatures 'sprinkled' over the vertices of $({{\mathbb{S}}}^{2},{g}_{\mathrm{pol}}).$ The crucial point is that the 'sprinkling' cannot be random: the amount of overall positive and overall negative curvature over the vertices must comply with the Gauss–Bonnet theorem applied to $({{\mathbb{S}}}^{2},{g}_{\mathrm{pol}})$ [23–25]; the integrated Gaussian curvature must sum up to $4\pi ,$ the Euler number $\chi ({{\mathbb{S}}}^{2})\;=\;2.$ Thus, any two-surface which is topologically a sphere will have the same curvature, on average, as a round sphere. Only by measuring over the whole surface would Spherelanders find the correct total curvature, and anyway they would arrive at the same result (assuming the same topology) regardless of whether or not the coordinate components¹⁴ of the metric were close, either in pointwise value, or in some other suitable norm, to those of the round sphere.

To emphasize the point, we may easily construct a large (let it be convex) polyhedral surface $({{\mathbb{S}}}^{2},{g}_{\mathrm{pol}}),$ (using a dual metrical triangulation), where thousands of vertices are flat and just a few carry some (tiny) deficit angles. For the Sphereland inhabitants it would be very hard—perhaps impossible—to conclude that they actually live on a two-sphere. By probing rather large portions of their ambient space (analogous, say, to the observable Universe considered as part of an even larger spacetime) they would be more likely to conclude, potentially with an arbitrarily high degree of precision, that the geometry they live in is flat (there is no way they could intrinsically detect the edge-bending between the flat regions). Thus, guided by common sense and by such empirical evidence, they would probably consider as irrelevant to their cosmological modelling the local defects associated with a few deficit angles in an ocean of flatness. However, it is these local defects that determine the round sphere best-fit background, because it is the Euler number that determines the global properties of a two-dimensional surface. This is the only datum which would allow Spherelanders to draw GW's desired conclusions on the best-fitting metric in this analogy, and it is thus the curvature defects that constitute the only relevant property.

We remark that in the mathematical literature this issue has been treated in much greater depth than GW provide, giving more detailed presentations of polyhedral manifolds and of the resulting approximation techniques, as indicated in appendix A.

The steel ball analogy cannot say much about backreaction effects, especially on their evolution, since the topological constraint makes any two-dimensional example trivial. Because of the Gauss–Bonnet theorem, there is a conservation law for the integrated intrinsic curvature in two-dimensions [8, 26], just as there is such a conservation law in the standard FLRW cosmological model in 3 + 1 dimensions. As a consequence, if one imagined the two-dimensional polyhedral model as representing the spatial surfaces in a three-dimensional spacetime, the curvature evolution would be decoupled from backreaction.

Generally, backreaction effects may become strong when inhomogeneities dynamically couple to the averaged intrinsic curvature (which does not, in general, admit a conservation law in three-dimensions [27]). This averaged curvature–backreaction coupling only arises in more than two-dimensions (for details on the coupling of inhomogeneities to curvature and conservation laws, see [1]).

We have argued that the fitting problem cannot be solved by declaring, dictated by common sense, that the best-fitting metric is g_can with localized defects. Considering curvature fluctuations as being unconstrained and assuming that they average out to zero is tautological since this assumes that curvature defects globally contribute nothing and, thus, assumes that the geometry $({{\mathbb{S}}}^{2},{g}_{\mathrm{can}})$ is the best-fitting geometry without actually characterizing the explicit map between $({{\mathbb{S}}}^{2},{g}_{\mathrm{pol}})$ and $({{\mathbb{S}}}^{2},{g}_{\mathrm{can}})$ (for discussion of such a map see appendix A). Moreover, the geometry the Spherelanders experience in their surroundings (the physical geometry) is the almost-everywhere flat g_pol, not the everywhere round g_can. In particular, the measured physical curvature defects are with respect to $({{\mathbb{S}}}^{2},{g}_{\mathrm{pol}})$ and not with respect to $({{\mathbb{S}}}^{2},{g}_{\mathrm{can}}).$

From this brief analysis we see why Green and Wald obtain a highly constrained result for the possible backreaction in the steel ball model.

The term backreaction, as it is usually understood in Newtonian or relativistic cosmology, may be generally defined as deviation of spatial average properties of an inhomogeneous Universe model from the values predicted by a homogeneous-isotropic Universe model.

In Newtonian cosmology the average expansion rate is affected by the non-local term [28]

$$\begin{eqnarray}&&{{\mathcal{Q}}}_{{\mathcal{D}}}:=\displaystyle \frac{2}{3}\left({\left\langle{\theta }^{2}\right\rangle}_{{\mathcal{D}}}-{\left\langle\theta \right\rangle}_{{\mathcal{D}}}^{2}\right)-2{\left\langle{\sigma }^{2}\right\rangle}_{{\mathcal{D}}}+2{\left\langle{\omega }^{2}\right\rangle}_{{\mathcal{D}}},\end{eqnarray} \tag{ 1 }$$

where θ, σ and ω are the rate of expansion, shear and vorticity for a given velocity model, respectively, and the brackets denote volume averaging on a spatial domain ${\mathcal{D}},$ an averaging that is well-defined even for tensors in Newtonian theory. The magnitude of this term has been estimated in [29] (and follow-up papers) using well-known models for large-scale structure that are in good agreement with $N$-body simulations down to scales where the dust approximation breaks down.

Interpreted in the context of a FLRW fitting model, backreaction leads to an effective stress–energy tensor that has the form of a perfect fluid with the components (see, e.g., [30],section 3.2):

$$\begin{eqnarray}{\varrho }_{{\rm{eff}}}: & = & {\left\langle\varrho \right\rangle}_{{\mathcal{D}}}+\displaystyle \frac{1}{8\pi {{Ga}}_{{\mathcal{D}}}^{2}}{\displaystyle \int }_{{t}_{i}}^{t}{{\mathcal{Q}}}_{{\mathcal{D}}}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t^{\prime} }{a}_{{\mathcal{D}}}^{2}(t^{\prime} )\;{\rm{d}}t^{\prime} ;\quad \\ {p}_{{\rm{eff}}}: & = & -\displaystyle \frac{1}{24\pi {{Ga}}_{{\mathcal{D}}}^{2}}{\displaystyle \int }_{{t}_{i}}^{t}{{\mathcal{Q}}}_{{\mathcal{D}}}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t^{\prime} }{a}_{{\mathcal{D}}}^{2}(t^{\prime} )\;{\rm{d}}t^{\prime} -\displaystyle \frac{1}{12\pi G}{{\mathcal{Q}}}_{{\mathcal{D}}},\end{eqnarray} \tag{ 2 }$$

where ${a}_{{\mathcal{D}}}$ is the volume scale factor, defined such that the volume of the averaging domain is proportional to ${a}_{{\mathcal{D}}}^{3}$ [31]. The local stress–energy tensor satisfies the weak energy condition, since $\varrho \geqslant 0$ (implying $\varrho +p\geqslant 0$ as we only consider dust, with p = 0). However, the effective stress–energy tensor does not necessarily satisfy these conditions, and neither does its backreaction part¹⁵ . Also, the trace is not in general zero. Both of these features are in contradiction with GW's claims, specifically their theorems 1 and 2 [9, 12], considered in the Newtonian limit. We shall come back to the energy conditions in section 5.2.

Note that the magnitude of backreaction depends on the scale of averaging. In particular, already in Newtonian cosmology it is significant on intermediate scales, below the cosmological homogeneity scale¹⁶ , but well above the scales of 'black holes or neutron stars', excluded by Green and Wald¹⁷ .

Since the trace part of backreaction cannot turn on just in the Newtonian limit, backreaction cannot be trace-free in general in GR either without violation of at least one of the theory's principles. This remark makes the existence of trace parts clear, since the calculation in the Newtonian framework is unambiguous. Green and Wald emphasize in their abstract that 'Newtonian cosmologies provide excellent approximations to cosmological solutions to the Einstein equations (with dust and a cosmological constant) on all scales'. If that is the case, then—especially in view of their 'dictionary' [10]—a claim of trace-free backreaction cannot be correct.

General relativity adds the new feature that this backreaction term (represented by extrinsic curvature invariants here) is coupled to the averaged scalar curvature ${\left\langle{\mathcal{R}}\right\rangle}_{{\mathcal{D}}}$ [31]

$$\begin{eqnarray}&&\displaystyle \frac{1}{{a}_{{\mathcal{D}}}^{6}}{\left({{\mathcal{Q}}}_{{\mathcal{D}}}{a}_{{\mathcal{D}}}^{6}\right)}^{\overset{\bullet }{}}+\displaystyle \frac{1}{{a}_{{\mathcal{D}}}^{2}}{\left({\left\langle{\mathcal{R}}\right\rangle}_{{\mathcal{D}}}{a}_{{\mathcal{D}}}^{2}\right)}^{\overset{\bullet }{}}=0.\end{eqnarray} \tag{ 3 }$$

This coupling is important for backreaction in relativistic cosmology, as it can also affect the global geometrical properties. This latter is impossible in Newtonian cosmology and it is also suppressed in quasi-Newtonian relativistic perturbation theory with periodic boundary conditions due to the fact that a conservation law for the intrinsic curvature holds.

Green and Wald assume that the actual metric has the structure:

$$\begin{eqnarray}&&{g}_{{ab}}\;=\;{g}_{{ab}}(0)\;+\;{\gamma }_{{ab}},\end{eqnarray} \tag{ 4 }$$

where g(0) (denoted by ${g}^{(0)}$ in GW) is a background metric assumed to provide the averaged background modelling the Universe (the indices are spacetime indices). Green and Wald stress that in most of their computations g(0) can be any given background metric.

Carrying out computations in such a general framework (and in particular trying to interpret them geometrically) is a daunting, if not impossible, task. We are not aware of any similar analysis in Riemannian geometry (where things are definitively easier than in the Lorentzian case). Hence, Green and Wald tackle a less ambitious task by considering a perturbative formalism where the symmetric tensor γ, representing the deviations from the background, is small. However, they do not assume that the first derivatives of γ are small and even allow second derivatives of γ to be unboundedly large, to deal with very large fluctuations in curvature around the g(0) background.

We point out already here, with detailed explanations in appendix B, that the assumption in equation (4) together with the non-perturbative implications for the metric derivatives outlined above, inherits similar problems to those we explained for the steel ball analogy. Their statement that 'in summary, the geometry of Sphereland is described by a metric of the form ${q}_{{ab}}(0)+{s}_{{ab}},$ where ${q}_{{ab}}(0)$ is the metric of a perfect sphere and $| {s}_{{ab}}| \ll | {q}_{{ab}}(0)| ,$ but first derivatives of s_ab are not small, and second derivatives of s_ab may be enormous. In an exactly similar manner, the spacetime metric of our Universe takes the form ${g}_{{ab}}={g}_{{ab}}(0)+{\gamma }_{{ab}},$ where ${g}_{{ab}}(0)$ has the FLRW symmetry ...' [12] is problematic. Modelling the perturbation ${\gamma }_{{ab}}$ with the same underlying rationale as that for the perturbation s_ab of the steel ball example implies that the potentially unbounded second derivatives of the metric may give rise to curvature distributional defects in the weak-limit, see appendix B, that cannot be dismissed, since they are as crucial as the curvature defects in the steel ball example.

The scheme adopted by GW adds a stress–energy tensor to a scheme developed by Burnett [33] for vacuum spacetimes, which replaces the averaging operation in Isaacson's approach [34, 35] by a weak-limit procedure. (Burnett [33] finds a trace-free effective stress–energy tensor by applying a weak-limit scheme to gravitational waves, which are physically trace-free. He also discussed a non-vanishing stress–energy–tensor, but for the case of electromagnetic radiation, which is again physically trace-free.) For another calculation of backreaction in the weak-limit of gravitational waves emitted from a Schwarzschild black hole see [36]. We shall demonstrate that the weak-limit operation does not implement a true averaging procedure, and that the assumptions made by GW are too restrictive to allow for backreaction.

Green and Wald's weak-limit scheme is technically realized, as in the familiar perturbation formalism for GR, by assuming that there is a class of coordinate systems $\{{x}^{a}\}$ and a one-parameter family of Lorentzian metrics $(0,1]\;\ni \;\lambda \;\longmapsto \;g(\lambda ),$ such that we can locally write the metric components ${g}_{{ab}}(\lambda )$ as

$$\begin{eqnarray}&&(0,1]\;\ni \;\lambda \;\longmapsto \;{g}_{{ab}}(\lambda )\;=\;{g}_{{ab}}(0)\;+\;{\gamma }_{{ab}}(\lambda ).\end{eqnarray} \tag{ 5 }$$

These metric components are further required to satisfy the following assumptions: (see $({\rm{i}})$–$(\mathrm{iv})$ of GW):

• (i)  
  For all $\lambda \in (0,1]$ one assumes that the Einstein equations hold

  $$\begin{eqnarray}&&{G}_{{ab}}(g(\lambda ))\;+\;{\rm{\Lambda }}\;{g}_{{ab}}(\lambda )\;=\;8\pi \;{T}_{{ab}}(\lambda ),\end{eqnarray} \tag{ 6 }$$

  where ${T}_{{ab}}(\lambda )$ is a one-parameter family of stress–energy tensors that locally satisfy the weak energy condition.
• (ii)  
  There exists a smooth positive function ${C}_{1}(x)$ on $(M,g(0))$ such that we may write¹⁸

  $$\begin{eqnarray}&&\left|{\gamma }_{{ab}}(x,\lambda )\right|\;\leqslant \;\lambda \;{C}_{1}(x).\end{eqnarray} \tag{ 7 }$$

  This is the usual way of controlling the perturbing tensor γ in standard perturbation theory, implying that, as $\lambda \searrow 0,$ $g(\lambda )$ approaches g(0).
• (iii)  
  There exists a smooth positive function ${C}_{2}(x)$ on $(M,g(0))$ such that we may write

  $$\begin{eqnarray}&&\left|{{\rm{\nabla }}}_{c}{\gamma }_{{ab}}(x,\lambda )\right|\;\leqslant \;{C}_{2}(x),\end{eqnarray} \tag{ 8 }$$

  where ${\rm{\nabla }}$ denotes the covariant derivative w.r.t. $(M,g(0)).$ This requirement does not imply that the first derivatives of γ are small and they do not necessarily become small as $\lambda \searrow 0.$
• (iv)  
  Finally, they assume that there exists a smooth tensor field ${\mu }_{{abcdef}}(x)$ on $(M,g(0))$ representing the weak-limit of the product of first covariant derivatives ${\rm{\nabla }}\gamma (\lambda ){\rm{\nabla }}\gamma (\lambda )$ as $\lambda \searrow 0,$ i.e.

  $$\begin{eqnarray}&&\underset{\lambda \searrow 0}{{\rm{w}}-\text{lim}}\;\left|{{\rm{\nabla }}}_{a}{\gamma }_{{cd}}(x,\lambda ){{\rm{\nabla }}}_{b}{\gamma }_{{ef}}(x,\lambda )\right|\;=\;{\mu }_{{abcdef}}(x).\end{eqnarray} \tag{ 9 }$$

Here, the weak limit is intended in a strict weak sense, i.e., requiring that the functions ${\mu }_{{abcdef}}$ are locally summable (see GW, equations (5) and (6)). (It should be noted that this is more restrictive than the usual distributional interpretation of the weak-limit, where ${\mu }_{{abcdef}}$ could be represented by a singular distribution. We examine this issue in depth in appendix B.) This condition forces good behaviour of the potentially problematic quadratic product ${\rm{\nabla }}\gamma (\lambda ){\rm{\nabla }}\gamma (\lambda )$ as $\lambda \searrow 0.$

The condition $(\mathrm{iv})$ is not motivated by any transparent and natural physical argument: as it stands we do not consider $(\mathrm{iv})$ to be a robust hypothesis underlying backreaction analysis. Even for the high-frequency limit regime of gravitational waves there are physically transparent alternative approaches to this that do not need to assume condition $(\mathrm{iv}),$ see [37], ch III, paragraph 12 and ch XI, (this reference is also interesting for the issue of gauge dependence, discussed further below), and also [38–40] and references therein.

A delicate issue with GW's perturbation approach is the validity of assumption $(\mathrm{iv})$ together with assumption $({\rm{i}})$ that the Einstein equations are required to be satisfied along the curve of metrics, for all λ except in the limit $\lambda \searrow 0$. Thus, backreaction is supposed to turn on just in the limit (see section 3.7 for details). This situation will turn out to be problematic.

We first point out an issue that arises in taking due care of the subtleties generated by the potentially unbounded second derivatives of the metric fluctuations. There may be particular spacetimes for which their procedure can be carried out, but typically one would get distributional curvature tensors under the GW hypotheses in the limit where $\lambda \searrow 0,$ which makes their effective stress tensor ${t}_{{ab}}(0),$ see section 3.7, just a formal expression, not containing functions but distributions, which are not easily interpreted from the geometrical and physical point of view. In particular the trace-free conditions would generally be false. Even if it were possible to get rid of any non-regular distributional part, as GW implicity assume, their approach strongly relies on a very delicate non-uniform boundedness (in λ) hypothesis on $\left|{{\rm{\nabla }}}_{c}{{\rm{\nabla }}}_{d}{\gamma }_{{ab}}(x,\lambda )\right|$ (see the discussion surrounding equation (B.19)). This feature is imported from Burnett's high-frequency limit in gravitational wave theory, and it is precisely this non-uniformity that generates a backreaction only in the $\lambda \searrow 0$ limit. However, whereas in gravitational wave theory a non-uniform high-frequency limit is not a surprising feature of oscillating phenomena, in cosmology it does not correspond to the physical situation. In the real Universe, the matter distribution does not oscillate on arbitrarily small scales, but there is a hierarchy of finite averaging length scales that are physically defined by the gravitational systems we want to average. It is important to stress that if we remove this specific non-uniformity requirement from the GW formalism, for example, instead of requiring the second derivatives of γ to be uniformly bound in λ, then the effective stress–energy tensor ${t}_{{ab}}(0)$ vanishes. We give the proof in appendix B. Moreover, in a real averaging procedure, backreaction terms must already be present for non-zero λ. We shall explain this remark in the following two subsections.

We also note that known and well-controlled metric flows as investigated in the mathematical literature on cosmological backreaction are generally not required to satisfy the Einstein equations during metric deformations. Metric flows are typically (weakly) parabolic and, hence, in general they do not commute with the Einstein evolution which is (weakly) hyperbolic with elliptic constraints; 'weakly' here refers to the fact that these flows must comply with the appropriate form of equivariance under the action of the diffeomorphism group (see [41]). Geometric flows are often gradient flows (with respect to some Hilbertian positive definite inner product), whereas the Einstein evolution is Hamiltonian. In particular, in a cosmological setting, rescaling flows (such as the Ricci–Hamilton–Perelman flow) work in a 3D hypersurface, rescaling the metric in the direction of the spatial Ricci tensor (see [42]), which a priori is unrelated to the Einstein flow that deforms the initial data set along the (symplectically conjugated) linearization of the Einstein constraints in the direction of the lapse and shift. In other words, the former relate to flows in space and the latter to time evolution.

While Green and Wald aimed at formulating the weak-limit formalism in a mathematically 'completely general' manner, they noted that 'it would be extremely difficult to formulate mathematically precise criteria for the validity of applying our formalism to a given spacetime'. They argued that they simply 'resort to plausibility arguments to obtain [their] conclusions' regarding the applicability of the formalism to the real Universe [9]. They then argued that it is plausible that the weak-limit corresponds to spacetime averages over local regions, provided that the size of the regions corresponds to the homogeneity scale, which is much smaller than the curvature scale of the background Universe. However, arguments about limits that may be plausible for smooth functions do not immediately translate to singular distributions.

Formally, GW are not averaging the fluctuations represented by the tensor γ. They consider the weak-limit in a singular perturbation scheme for GR around a FLRW spacetime (or whatever geometry g(0) represents). In their words (GW): 'the notion of 'weak-limit', which appears in assumption (iv), corresponds roughly to taking a local spacetime average, and then taking the limit as $\lambda \searrow 0$'. This is not averaging in the sense used in discussing backreaction. The weak-limit of a sequence of smooth functions is typically a (singular) distribution. An example (on a compact Riemannian manifold) is provided by the heat kernel: $K(x,y;t),$ $t\gt 0.$ This is ${C}^{\infty }$ for all $t\gt 0,$ so smooth that its convolution against nasty functions tames all their defects. Nonetheless the weak-limit of $K(x,y;t),$ as $t\searrow 0,$ is a singular distribution, the Dirac measure ${\delta }_{y}$ supported at the point y. The operation might correspond to averaging if the compactly supported function that GW used to define the weak-limit had a form of translation symmetry built in (say, $\phi (x-y)$ in ${{\mathbb{R}}}^{n},$ or if we consider a large symmetry group, as in FLRW), so that one is actually regularizing (the distribution) via a convolution product, not just a generic weak-limit.

Green and Wald consider a singular perturbation theory around FLRW, not averaging or taking convolutions. This strategy is quite common in quantum field theory where one needs to tame the spectrum of fluctuations of a quantum field around a given background configuration and describe how these fluctuations may dress the bare background¹⁹ .

Dismissing the potential distributional nature of curvature fluctuations in such a setting is akin to the situation we encountered in the steel ball example. The analogy is (i) here the background metric $(M,g(0))$ plays the role of the almost-everywhere flat background metric in the steel ball model; and (ii) the distributional part plays the role of the physical curvature fluctuations described by the conical vertices in the steel ball model. As in that case, there is no reason why they should disappear when performing a real averaging procedure. (In the two-dimensional steel ball model the Gauss–Bonnet theorem implies that they cannot disappear.)

Clearly, there is nothing wrong in addressing the backreaction issue by studying a singular perturbation scheme around FLRW as GW do. It is a potentially useful idea. After all, in relativistic cosmology, unlike the averaging of scalars (which we will come to below), the averaging of tensors is not yet well-established, at least in a rigorous sense. The real point is to understand what you can get out of such an approach, how it is technically realized, and how it can be physically interpreted.

We finally remark that, even if not gauge invariant, a singular limit may be given physical sense locally, e.g., where Newtonian spacetime is obtained as a singular limit of a sequence of Einstein spacetimes (defined by appropriate initial data expressing the Newtonian situation so that it can only be defined near the limit) [44].

Even if one were to extend the GW perturbation scheme to rigorously calculate the trace from the distributional curvature using the procedure outlined in appendix B, there is still another issue: Green and Wald assume the Einstein equations to hold for all λ, except for $\lambda =0.$ This is shown in their assumption $({\rm{i}}),$ recall equation (6):

$$\begin{eqnarray*}&&{G}_{{ab}}(g(\lambda ))\;+\;{\rm{\Lambda }}\;{g}_{{ab}}(\lambda )\;=\;8\pi \;{T}_{{ab}}(\lambda );\quad \lambda \in (0,1],\end{eqnarray*}$$

and in their terminology for the limit (GW, equation (1)):

$$\begin{eqnarray*}&&{G}_{{ab}}(g(0))\;+\;{\rm{\Lambda }}\;{g}_{{ab}}(0)\;=\;8\pi \left({T}_{{ab}}(0)+{t}_{{ab}}(0)\right).\end{eqnarray*}$$

Green and Wald refer to the averaging and fitting problems as they were raised in Ellis [17] and Ellis and Stoeger [18] (GW's figure 1 is reproduced from [17]). Those references emphasize that averaging or smoothing the inhomogeneous metric and taking derivatives (and their products) are in general non-commuting operations, which implies that, in general, the Einstein equations do not hold at any smoothing level²⁰ . In other words, although the actual model obeys the Einstein equations, there need not be a one-parameter family of solutions approaching the background that also obey the Einstein equations. This issue lies at the basis of backreaction, and the first-principle argument of [17] can be realized, e.g., with the scalar averaging formalism discussed in section 5, or (directly related to a smoothing procedure) with Ricci flow techniques [46].

Putting this argument formally and adopting GW's notation, an averaging operation, however it is defined (and here specialized to a one-parameter family of metrics), leads to effective terms, symbolically put into the tensor ${\tau }_{{ab}}$ ²¹ :

$$\begin{eqnarray}&&{G}_{{ab}}(g(\lambda ))\;+\;{\rm{\Lambda }}\;{g}_{{ab}}(\lambda )\;=\;8\pi \left({T}_{{ab}}(\lambda )+{\tau }_{{ab}}(\lambda )\right);\quad \lambda \in (0,1).\end{eqnarray} \tag{ 10 }$$

Green and Wald correctly quote the starting point of Ellis' argument [17] that the Einstein equations are assumed to hold in the unaveraged case ($\lambda =1$), but they then assume that the Einstein equations also hold for $\lambda \lt 1$ down to the limit, where suddenly the backreaction term t_ab turns on²² .

In the framework of the weak-limit scheme, starting from (10), the calculation of the weak-limit gives:

$$\begin{eqnarray}&&{G}_{{ab}}(g(0))\;+\;{\rm{\Lambda }}\;g(0)\;=\;8\pi \left({T}_{{ab}}(0)+{t}_{{ab}}(0)+\underset{\lambda \searrow 0}{{\rm{w}}-\mathrm{lim}}\;[{\tau }_{{ab}}(\lambda )]\right),\end{eqnarray} \tag{ 11 }$$

where the last term is the weak-limit of the λ-dependent backreaction term, which is absent in GW's calculation. We will give an argument in section 4.2 and a proof in appendix B that the effective stress–energy tensor ${t}_{{ab}}(0),$ the trace-free 'backreaction term' of GW, is expected to vanish for a uniform convergence to the background metric.

We now illustrate the above issues in terms of examples.

It was emphasized by Geroch [48] that, if we restrict our attention to metrics parametrized by some λ only, then the limit is not uniquely defined.

Geroch considers the family of parametrized Schwarzschild metrics:

$$\begin{eqnarray}&&{\rm{d}}{s}^{2}=\left(1-\displaystyle \frac{2}{{\lambda }^{3}r}\right){\rm{d}}{t}^{2}-{\left(1-\displaystyle \frac{2}{{\lambda }^{3}r}\right)}^{-1}{\rm{d}}{r}^{2}-{r}^{2}({\rm{d}}{\theta }^{2}+{\mathrm{sin}}^{2}\theta {\rm{d}}{\phi }^{2}),\end{eqnarray} \tag{ 12 }$$

where $\lambda ={m}^{-1/3}.$ In this form there is no limit as $\lambda \searrow 0.$ Consider, however, the coordinate transformation: $\tilde{r}=\lambda r,\;\;\tilde{t}={\lambda }^{-1}t,\;\;\tilde{\rho }={\lambda }^{-1}\theta .$ Then, our family of metrics takes the form

$$\begin{eqnarray}&&{\rm{d}}{s}^{2}=\left({\lambda }^{2}-\displaystyle \frac{2}{\tilde{r}}\right){\rm{d}}{\tilde{t}}^{2}-{\left({\lambda }^{2}-\displaystyle \frac{2}{\tilde{r}}\right)}^{-1}{\rm{d}}{\tilde{r}}^{2}-{\tilde{r}}^{2}({\rm{d}}{\tilde{\rho }}^{2}+{\lambda }^{-2}{\mathrm{sin}}^{2}(\lambda \tilde{\rho }){\rm{d}}{\phi }^{2}),\end{eqnarray} \tag{ 13 }$$

and the limit exists—it is the Kasner metric with Kasner exponents $-1/3,$ $2/3,$ $2/3:$

$$\begin{eqnarray}&&{\rm{d}}{s}^{2}=-\displaystyle \frac{2}{\tilde{r}}{\rm{d}}{\tilde{t}}^{2}+\displaystyle \frac{\tilde{r}}{2}{\rm{d}}{\tilde{r}}^{2}-{\tilde{r}}^{2}({\rm{d}}{\tilde{\rho }}^{2}+{\tilde{\rho }}^{2}{\rm{d}}{\phi }^{2}).\end{eqnarray} \tag{ 14 }$$

However, applying a different coordinate transformation, $x=r+{\lambda }_{0}{\lambda }^{-4},\;\;\rho ={\lambda }_{0}{\lambda }^{-4}\theta$ (for an arbitrary constant ${\lambda }_{0}$) to our original metric (12) and taking the limit $\lambda \searrow 0$ yields the flat Minkowski metric.

Geroch's discussion illustrates that the outcome of the weak-limit procedure depends on the choice of coordinates. Moreover, it can be argued that instead of fixing the background metric, it should emerge from a coordinate-independent averaging or rescaling procedure. We finally remark that the characterization by curvature invariants can be used to clarify Geroch's example and avoid the coordinate dependence of his methods (see [49]). Such ideas could work whenever there is some identifiable way of taking a weak-limit of the set of curvature invariants.

The issue of gauge dependence has been addressed in [3]. A related discussion can be found in [50]. In this latter paper it is demonstrated that even the spacetime integrals of 4D scalars are in general gauge-dependent if the domain of integration is fixed. The domain should rather be defined intrinsically: the behaviour of the domain under gauge transformations should compensate the effects of the same gauge transformations applied to the integrated object and, thus, the domain must be coordinate- and λ-dependent (the latter requirement comes from the parametrization of the metric and its volume element). Given such an x- and λ-dependent domain, it would no longer be allowed to exchange the limits and the integration operators as is done by GW in the case of first derivatives of metric deviations. This will then imply a non-commutativity of the two operations and would give rise to backreaction terms (for the consequences of non-commutativity see [31] for the Einstein flow, [46] for three-dimensional metric flows, and [45] for a general discussion). Then, the weak-limit operator would match the authors' expectations of first performing an average and then taking a limit. This $\lambda$-dependence has also been emphasized in a recent paper by Kopeikin and Petrov [51].

We move now to another issue. As pointed out by Szybka [47], one of the key ingredients of the GW formalism is Burnett's restriction that any λ-dependent coordinate transformation must be reduced to the identity when $\lambda \searrow 0.$ This is essential and shows that the procedure cannot work for arbitrary $\lambda$-dependent coordinate transformations and is therefore not covariant. The Geroch coordinate transformations would be excluded with this condition. However, even if we take such 'allowed' coordinate transformations we have a path dependence in the weak-limit procedure as discussed in the next section.

In [11], Green and Wald look at a subclass of vacuum Gowdy metrics on a torus with metric

$$\begin{eqnarray}&&{\rm{d}}{s}^{2}={{\rm{e}}}^{(\tau -\alpha )/2}\left(-{{\rm{e}}}^{-2\tau }{\rm{d}}{\tau }^{2}+{\rm{d}}{\vartheta }^{2}\right)+{{\rm{e}}}^{-\tau }\left({{\rm{e}}}^{P}{\rm{d}}{\sigma }^{2}+{{\rm{e}}}^{-P}{\rm{d}}{\delta }^{2}\right),\end{eqnarray} \tag{ 15 }$$

which depends on the function $P(\tau ,\vartheta ).$ A one-parameter family of metrics parametrized by a discrete parameter²³ N is chosen as follows:

$$\begin{eqnarray}&&{P}_{N}=\displaystyle \frac{A}{\sqrt{N}}{J}_{0}(N{{\rm{e}}}^{-\tau })\mathrm{sin}(N\vartheta ),\end{eqnarray} \tag{ 16 }$$

so that, like $\lambda \mathrm{sin}(x/\lambda ),$ for larger and larger N (corresponding to smaller and smaller λ) it oscillates with a shorter and shorter wavelength and a smaller and smaller amplitude; the limit $N\to \infty$ corresponds to $\lambda \searrow 0.$

The weak-limit gives a non-zero result only for functions such as $f(x)=x\mathrm{sin}(1/x)$ (as in example (4.2) of [11]), which have a singular limit as they oscillate ever more finely without limit as $x\to 0.$ Any non-zero result depends essentially on taking the infinite limit of those oscillations as the homogeneous model is approached; it will be zero if they cease at any value of λ, no matter how small. This is what is called in [3] an ultra-local limit: the oscillations must not even stop at atomic scales.

But the path chosen to link ${g}_{{ab}}(x,N=1)$ to ${g}_{{ab}}(x,0)$ is arbitrary: there is no reason why the specific path of equation (16) is preferred over others. One might equally well choose

$$\begin{eqnarray}&&{P}_{N}=\displaystyle \frac{A}{N}{J}_{0}(N{{\rm{e}}}^{-\tau })\mathrm{sin}(N\vartheta )\quad {\rm{or}}\quad {P}_{N}=\displaystyle \frac{A}{{N}^{2}}{J}_{0}(N{{\rm{e}}}^{-\tau })\mathrm{sin}(N\vartheta )\end{eqnarray} \tag{ 17 }$$

for example; these are both solutions of the field equations, and these paths will give ${t}_{{ab}}(0)=0$. The physical entity is the endpoint ${g}_{{ab}}(x,N=1);$ the intervening family is arbitrary, as is the parameter choice.

The result obtained from equation (16) only has physical meaning if it gives the same answer for different such parametrizations. This procedure does not.

One could try

$$\begin{eqnarray}&&{P}_{N}=\displaystyle \frac{A}{f(N)}{J}_{0}(N{{\rm{e}}}^{-\tau })\mathrm{sin}(N\vartheta ),\end{eqnarray} \tag{ 18 }$$

and see for what $f(N)$ one gets ${t}_{{ab}}(0)\ne 0$ as well as the amplitude $A/f(N)$ going to zero. It is non-zero only for very special choices of $f(N).$

We repeat that the problem is the ultra-local requirement that perturbations in the envisaged family continue down to indefinitely small wavelengths without limit; otherwise one gets a zero answer. For very small distances in realistic cosmological modelling the solutions should behave like $(A/N)$ rather than like $(A/N)\mathrm{sin}(N\vartheta )$ (equivalent to using $f(x)=x$ for very small scales as $x\to 0$ rather than $x\mathrm{sin}(1/x)$). Then, the quantity ${t}_{{ab}}(0)$ will be zero. Thus, equations (2.7) and (2.8) of [11] are true in realistic situations only because ${t}_{{ab}}(0)=0.$

The pathological nature of the limit is emphasized by the fact that in the example given, for each $N\lt \infty ,$ the metric is a solution of the vacuum field equations, but the limiting metric (3.11) of [11] is not a solution. The backreaction term turns on in a delta-function way, in the exact limit only. Genuine backreaction effects should not have a delta function discontinuity of this sort: the backreaction term should be non-zero for every metric where $\lambda \geqslant 0$ in any realistic sequence approaching the background, as we have already pointed out. In the real Universe, inhomogeneities have a finite amplitude and length scale, which should correspond to some $\lambda ={\lambda }_{0}\gt 0.$ One could use a Taylor series to estimate the effect at $\lambda ={\lambda }_{0}$ from the data at $\lambda =0,$ but that will not work for this singular family of perturbations.

These examples demonstrate that ${t}_{{ab}}(0)$ should vanish for a realistic sequence of cosmological perturbations, which should be cut off at a finite wavelength before the amplitude goes to zero, inter alia because the Sun exists; real perturbations do not continue to zero wavelength. Indeed, as we show in appendix B, the quantity ${t}_{{ab}}(0)$ is only non-vanishing if we impose further restrictions beyond the conditions imposed by GW and, especially, if we require non-uniform convergence so that this term only arises in the singular limit (see the discussion surrounding equation (B.19)). It is worth noting that any uniformly convergent limit of a family of spacetimes will inherit some of the properties of the family. These properties are called hereditary [48]. A set of vacuum solutions with a non-vacuum limit would not respect Geroch's hereditary properties for limits, reflecting the points made above about the GW scheme being singular.

We will not list here the many investigations of models and exact solutions illustrating or quantifying backreaction effects. These works may be found in the reviews mentioned in the introduction. We will only point out one recent work by Korzyński on nonlinear effects from multi-scale structure [52] because it closely follows the arguments of GW, it precisely illustrates why their arguments do not apply, and it quantifies the GR inhomogeneity effect with the help of an exact nested structure model.

Korzyński shows that it is possible for small local deviations to produce a large global effect. The procedure is closely related to the idea of metric smoothing by removing density ripples. His result is not a counter-example to the claimed results of GW, because, as he states, his model violates GW's assumption (ii) according to which perturbations should be small not just locally, but with reference to a single global background. In Korzyński's example, the smoothing procedure picks up effects from all intermediate scales, arising as a cumulative effect, and relies on the 'depth' of structure, i.e., on the ratio between the homogeneity scale and the scale of the smallest ripples (recall from section 4.2 that no lower cut-off exists for the inhomogeneities in the GW formalism). Korzyński's example thus illustrates the non-local nature of backreaction. In line with this he also has shown in [53] that backreaction in a system of many compact sources can be large even if the metric is close to FLRW almost everywhere. The important issue there is the clustering properties of matter.

Yes, because backreaction is non-local.

Green and Wald state, and this opinion is shared by others since it seems plausible, that: '... one can give an example of this sort [13] wherein the 'backreaction' is so large that one obtains acceleration of the representative FLRW Universe, even though each of the (disjoint)²⁶ components of the actual Universe is decelerating'. (Our emphasis.)

The physical effect under question has often been explained in the literature (see, e.g., [5, 64], section 5.2): the fraction of the volume of faster expanding regions grows more rapidly, so the average expansion rate rises. Formally, we see this by looking at the Raychaudhuri equation that governs the local expansion rate of dust matter: only a positive cosmological constant and vorticity can lead to an acceleration of the local expansion rate θ (in the sense that the time-derivative of θ is positive). Putting the cosmological term and the vorticity (which is active on small scales only) to zero, at first sight it seems implausible that a collection of such decelerating fluid elements can lead to acceleration of some patch of the Universe. Let us look at the difference between the local and the volume-averaged Raychaudhuri equation for irrotational dust and vanishing cosmological constant:

$$\begin{eqnarray}&&\dot{\theta }=-4\pi G\varrho +2{\rm{II}}-{{\rm{I}}}^{2}\;\;\;;\;\;\;\;\langle \theta {\rangle }_{{\mathcal{D}}}^{{\boldsymbol{\cdot }}}=-4\pi G{\left\langle\varrho \right\rangle}_{{\mathcal{D}}}+2{\left\langle{\rm{II}}\right\rangle}_{{\mathcal{D}}}-{\left\langle{\rm{I}}\right\rangle}_{{\mathcal{D}}}^{2},\end{eqnarray} \tag{ 19 }$$

where we defined the principal scalar invariants of the expansion tensor ${{\rm{\Theta }}}_{{ij}},$ $2{\rm{II}}:= 2/3{\theta }^{2}-2{\sigma }^{2}$ and ${\rm{I}}:= \theta ;$ averaging is performed on a spatial domain ${\mathcal{D}},$ and we have used the commutation rule: $\langle \theta {\rangle }_{{\mathcal{D}}}^{{\boldsymbol{\cdot }}}-\langle \dot{\theta }{\rangle }_{{\mathcal{D}}}=\langle {\theta }^{2}{\rangle }_{{\mathcal{D}}}-\langle \theta {\rangle }_{{\mathcal{D}}}^{2}=\langle {(\theta -\langle \theta {\rangle }_{{\mathcal{D}}})}^{2}{\rangle }_{{\mathcal{D}}}.$ Clearly, by shrinking the averaging domain to a point, the two equations agree. However, evaluating the local and averaged invariants

$$\begin{eqnarray}&&2{\rm{II}}-{{\rm{I}}}^{2}=-\displaystyle \frac{1}{3}{\theta }^{2}-2{\sigma }^{2};\\ &&2{\left\langle{\rm{II}}\right\rangle}_{{\mathcal{D}}}-{\left\langle{\rm{I}}\right\rangle}_{{\mathcal{D}}}^{2}=-\displaystyle \frac{1}{3}{\left\langle\theta \right\rangle}_{{\mathcal{D}}}^{2}-2{\left\langle\sigma \right\rangle}_{{\mathcal{D}}}^{2}+\displaystyle \frac{2}{3}{\left\langle{(\theta -{\left\langle\theta \right\rangle}_{{\mathcal{D}}})}^{2}\right\rangle}_{{\mathcal{D}}}\\ &&\quad -2{\left\langle{(\sigma -{\left\langle\sigma \right\rangle}_{{\mathcal{D}}})}^{2}\right\rangle}_{{\mathcal{D}}},\end{eqnarray} \tag{ 20 }$$

gives rise to two additional, positive-definite fluctuation terms, where that for the averaged expansion variance enters with a positive sign.

Thus, the time-derivative of an averaged expansion may be positive even if the time-derivative of the expansion is negative at every point in ${\mathcal{D}}.$ This is, technically, a consequence of the non-commutativity of averaging and time-evolution and, physically, of the non-local nature of averaging that takes correlations into account. Applying this fact to a model that is the union of disjoint FLRW submanifolds with spatial boundaries implies that the difference in expansion rates of the respective sections is a positive-definite expansion variance term (see references to such models in section 5.3).

Again, the answer is yes. It is a possible consequence of what has been said above. Green and Wald state a theorem that the effective energy momentum tensor ${t}_{{ab}}(0)$ has to obey the weak energy condition (see, e.g., [11], equation (2.8) in theorem 2), where it is written in the abstract that 'the leading effect of small scale inhomogeneities on large scale dynamics is to produce a trace-less effective stress–energy tensor that itself satisfies the weak energy condition'. Green and Wald present in [11] (section 4) an example for backreaction violating the weak energy condition (where they imply that this must be a consequence of the violation of the local weak energy condition), and where 'the limiting metric has an effective stress–energy tensor which is not trace-less'. We comment on this example in appendix C.

Green and Wald 'emphasize the importance of imposing energy conditions on matter in studies of backreaction', but we emphasize that the effective stress–energy tensor may, in general, violate energy conditions: the non-local fluctuation terms discussed above may lead to violation of energy conditions upon averaging, although locally they are satisfied.

This has also been stressed in the literature with regard to the mapping of inhomogeneity effects to an effective scalar field that plays the role of a quintessence field in standard models for dark energy (see, e.g., [1, 30]). While it is known that a fundamental scalar field (e.g., a quintessence field) must violate the weak energy condition and other physical properties (e.g., [65]) to explain observational data, an effective scalar field need not. Since, as we have shown in section 3.6, the framework in which the problem is addressed by GW is local, the resulting effective stress–energy tensor would then have to obey the local energy conditions only, and only in the limit $\lambda \searrow 0$ (recall that there is no effective stress–energy tensor in the GW scheme for $\lambda \in (1,0)$), i.e., their theorem 2 does not apply to an averaged stress–energy tensor. Addressing the real averaging problem would not deliver a local expression. We remind the reader that there is currently no agreed way to average tensors in GR.

Green and Wald's theorem states that the effective stress–energy tensor emerging from a weak-limit of spacetimes obeying the Einstein equations has to obey the weak energy condition. Such a result would be fine but irrelevant for backreaction.

Green and Wald question the framework where backreaction is discussed in terms of spatial averages of scalar quantities. They assume that such a procedure assigns an averaged FLRW metric to the averaging region, 'and the main flaw with such approaches' is that the metric thus obtained may be far from the real metric, even when the latter is close to a FLRW metric, generating 'entirely spurious' backreaction terms [12].

However, this criticism is based on a fundamental misinterpretation of the scalar averaging formalism [31, 54]: it does not involve any notion of average metric (and does not refer to geodesic deviations [22]); only averaged scalars are considered.

Explicitly, Green and Wald criticise the scalar averaging approach by considering two disjoint²⁷ dust FLRW universes in different stages of expansion. In [12] they state 'the Buchert prescription would represent this [disjoint] Universe as a single FLRW Universe, which provides a bad approximation to the actual metric everywhere'. And: '... the Buchert procedure with the above choice of hypersurface instructs us to represent the Minkowski metric with an 'averaged' FLRW metric g(0) that is an extremely poor approximation to g_ab on all scales'. Again, we emphasize that in the scalar averaging framework there is no such 'instruction' on a metric, apart from the spacetime split of the four-metric (on which we comment below). As explained at the beginning of this section, the formalism defines spatial average properties among scalar variables that depend on second derivatives of the metric. The metric itself is not specified.

We note that there are indeed investigations in the literature that study so-called template metrics [57, 66–69], that are intended to be compatible with the exact average properties (although these are not the papers referred to by GW). However, even these papers do not 'represent this (disjoint) Universe as a single FLRW Universe' (the metric which the standard model exactly assumes). On the contrary, backreaction effects are often illustrated on the basis of models with two disjoint Universe sections and the main effect is identified as being the result of the differences in expansion rates (see, e.g., [68, 70–72]).

The FLRW metric itself can be viewed as a global template metric. If we find that the global average spatial curvature today is not zero, then the flat FLRW template will constitute a poor approximation of the spatial sections, and small deviations thereof will represent large deviations with respect to the physical background [73]. This is exactly the problem that a template metric, even a single global template, is supposed to correct for. Moreover, as we argued in the context of the steel ball model analogy and the weak-limit framework: curvature inhomogeneities are not required to average out on an assumed background, as GW a priori impose to be true.

Green and Wald also argue that the average expansion rate ${\left\langle\theta \right\rangle}_{{\mathcal{D}}}$ is not physically meaningful, and that deviations of ${\left\langle\theta \right\rangle}_{{\mathcal{D}}}$ from the FLRW value due to large ${{\mathcal{Q}}}_{{\mathcal{D}}}$ are 'entirely spurious', and that 'in realistic cosmological situations [...] the comoving synchronous hypersurfaces of the Buchert construction will provide a poor choice for approximating the hypersurfaces with nearly FLRW symmetry'. (Compare here our discussion of the backreaction term ${{\mathcal{Q}}}_{{\mathcal{D}}}$ as it is evaluated in Newtonian cosmology in section 3.1.)

It is well-known that the average expansion rate depends on the choice of hypersurface, and the issue has been discussed at length in the literature, where it has been argued that the physically relevant averaging hypersurface is the one of statistical homogeneity and isotropy, see, e.g., [5, 7, 56, 64, 74–80]. It is irrelevant that there are hypersurfaces that are not physically interesting, it only matters that averages on some hypersurfaces give physically meaningful results and can be formulated in a covariant way. (For discussion of covariance and gauge-invariance of scalar averaging, see [50, 81, 82].) The average expansion rate evaluated on some hypersurface is a useful quantity so far as it gives an approximate description of what is observed, which is indeed the case for the hypersurface of statistical homogeneity and isotropy. However, for realistic situations, the difference of averages taken on hypersurfaces of statistical homogeneity and isotropy from averages taken on the hypersurface generated by the fluid's rest frame and measured by observers comoving with the matter component of the Universe, well modelled by dust, is expected to be negligible [79], but this should still be demonstrated in more detail. A nontrivial example is provided by the Swiss Cheese model of [83], where the average expansion rate describes the redshift (and to a lesser extent the luminosity distance) well, even in a solution that is far from the FLRW case. We emphasize that, in practice, observations of quantities such as the expansion rate and density always involve spatial averages (or averages over null geodesics, which can be related to spatial averages [79, 80, 83, 84]). Averaging is not a mathematical artifact, it is a feature of real observations that has to be properly modelled.

The averaged metric may be a poor approximation to the g(0) of the GW formalism (recall equation (4)). Even though the averaging formalism does not refer to an explicit metric, Green and Wald argue that this 'poor approximation' is an issue. But it is misleading to base an argument on the fact that the metric coefficient functions are 'far from' the values of a background metric. It may be demonstrated that such an argument is coordinate dependent. Consider as an example a flat metric and write it in different coordinates by introducing a diffeomorphism ${x}^{a}={f}^{a}({X}^{b})$ such that the metric coefficients can be transformed to the Minkowski coefficients: ${g}_{{ab}}{\rm{d}}{X}^{a}\otimes {\rm{d}}{X}^{b}={\delta }_{{cd}}^{M}{f}_{\;| a}^{c}{f}_{\;| b}^{d}{\rm{d}}{X}^{a}\otimes {\rm{d}}{X}^{b}={\delta }_{{cd}}^{M}{\rm{d}}{x}^{c}\otimes {\rm{d}}{x}^{d}$ (the last equality uses the inverse of the transformation f^a, and a vertical slash denoting a partial derivative with respect to the local coordinates X^a). The values of the coefficients ${g}_{{ab}}={\delta }_{{cd}}^{M}{f}_{\;| a}^{c}{f}_{\;| b}^{d}$ may be 'far from' the values of the Minkowski metric ${\delta }_{{ab}}^{M},$ but in fact they are the same metric arising from reparametrizing a flat space.

We conclude that any comparison of metrics by just using metric coefficients is unphysical, since it depends on the coordinates used. The question of 'how far' one metric is from another is a geometrical question that might perhaps be addressed quantitatively through curvature invariants. Comparing two metrics in GR is quite an involved task: we can employ Cartan scalars to test for isometry of metrics, see, e.g., [85, 86], but, as Cartan showed, this may require up $n(n+1)/2$ covariant differentiations of the Riemann tensor in n dimensions. (In four-dimensional spacetimes, one only needs at most seven derivatives [87].)

Such a test is justified by theorem 9.1 of [85] which tells us that the geometry of a sufficiently smooth manifold is locally uniquely determined by the curvature and its derivatives to some finite order. But (a) the arguments by GW do not include comparing derivatives and (b) we are not aware of any 'almost' version of this theorem, i.e. a statement that, if in some region the curvature of the metric g and its derivatives are close, in a suitable topological space, e.g., some Sobolev space (see appendix B), to those of some metric g(0), then g is close to g(0) (again, in some suitable topological space). Note in this context (compare sections 4.1 and 4.2) that GW's (and Burnett's) assumption (ii), see section 3.4, is in some coordinate components, so to be useful here the missing theorem would have to include something about how the coordinates are chosen to get a limit as GW want and the GW prescription would have to say more about the derivatives of the curvature.

Another way to deal with the issue of closeness of metrics for spatial averaging might be an appropriate generalization of the methods used in [88] which allow identification of the Schwarzschild metric given the data on any suitably smooth hypersurface.

Finally, we turn to the issue of whether there is any reason to think that the standard cosmological model, which is based on work nearly a century old [89, 90] (and unlike the standard particle physics model is the simplest conceivable cosmological solution), might not be the best description of our Universe.

It is important to take into account large metric derivatives and corresponding changes of observational data interpretation. Green and Wald discuss this, underpinned by the claim that Newtonian notions fully capture these changes with the help of their dictionary, which is restricted to an assumed near-FLRW situation. We agree with their statement 'moreover, the time evolution of the shear and convergence of a bundle of geodesics depends on the Riemann curvature (i.e., second derivatives of the metric), which can be very large', but it is not clear that a 'Newtonianly perturbed FLRW model' can fully account for this. This should be regarded as an open question until backed by concrete general calculations. The issue is whether the actual GR solution or aspects of it can be constructed via a 'dictionary' from a Newtonian solution (not shown), not whether a quasi-Newtonian GR solution can be constructed via some rules from a Newtonian solution.

Green and Wald argue that a mapping of solutions of Newtonian cosmological equations with periodic boundary conditions and certain properties to GR solutions given in [10] 'should yield an extremely accurate general relativistic description of our Universe, and, in particular, it provides the leading order backreaction effects at large scales produced by small scale inhomogeneities'. However, there are two different issues. One is whether certain Newtonian solutions are the limit of some GR solutions; the other is whether the GR solution that describes the real Universe is at all times close to a corresponding Newtonian solution²⁸ (see also [105]).

Reference [10] addresses the former question, but it is the latter issue that is relevant for providing an accurate GR description of our Universe and evaluating backreaction. Even if a particular GR solution starts from initial conditions close to a Newtonian solution, this does not imply that the GR solution would remain close to the Newtonian solution. As a simple example, in Newtonian gravity an isolated two-body system with an elliptic orbit is a stable configuration, whereas in GR the orbit will decay, and the system will be driven far from the Newtonian solution²⁹ . The difference in the evolution of the orbit is related to the difference between the Weyl tensor and the corresponding quantity in Newtonian theory, the Newtonian tidal tensor. A general GR solution (even if the matter is dust) does not correspond to any Newtonian solution. This is related to the fact that in Newtonian theory the tidal tensor, corresponding to the electric part of the Weyl tensor, does not have an evolution equation, and the tensor corresponding to the magnetic part of the Weyl tensor is zero. In GR, both are in general non-zero and have evolution equations, see [78, 108–119] for discussion. In [10] the arbitrariness of the tidal tensor is fixed by the assumption of periodic boundary conditions. As the Newtonian equations are elliptic and do not have a well-defined initial value problem, the boundary conditions are essential [28]. This is quite different from GR, where there is a well-defined initial value problem. Changing the evolution of the boundary at distances much larger than the GR particle horizon would not impact the GR solutions, but can completely change the Newtonian solutions. A lack of backreaction in GR cannot be established by starting from the assumption that the Universe is well-described by Newtonian theory.

A common argument against backreaction, sometimes ascribed to [120] and repeated by GW is that 'by flux conservation, the average of the apparent luminosity of sources (including multiple images) must match the FLRW value to a high degree of accuracy'. However, if the area element is different from FLRW, then the luminosity distance will also be different. Green and Wald assert that it 'is a simple fact' [22] that if two metrics are perturbatively close, then the average apparent luminosity of sources is a close match. Actually, this is incorrect. A counterexample is provided by Enqvist et al [121] in which, while perturbations (and their first derivatives) around an Einstein–de Sitter background remain small, the luminosity distance can even match that of a ΛCDM model with ${{\rm{\Omega }}}_{{\rm{\Lambda }}}=0.7.$ It is also straightforward to construct exact spherically symmetric counterexamples to the flux claim, such as the one in [122] (see also [123, 124]). This issue has been discussed in section 2.1.2 of [71] and section 4 of [83]. Even on large scales statistical isotropy and homogeneity is not enough to reduce the luminosity distance to its FLRW value, as shown in [83]. Indeed, the violation of the FLRW relationship between the expansion rate and the luminosity distance can be used as a test of the importance of inhomogeneities [79, 80, 83, 125–127].

As Green and Wald stress, second derivatives of the metric (i.e., spacetime curvature) have to be large in realistic models of the Universe. Averaging or smoothing these derivatives leads to an effective stress–energy tensor that is in general not trace-free, as we have demonstrated. Hence, this furnishes an argument for the potential importance of backreaction.

The Einstein equations dictate inhomogeneous curvature for inhomogeneous sources. A physical cosmology has to capture both inhomogeneities in the sources and inhomogeneities in the geometry. Idealizing the latter by assuming a homogeneous FLRW metric globally, leads to a missing geometrical piece on the left-hand side of the Einstein equations that shows up as missing sources on the right-hand side of the Einstein equations in the standard model. 'Backreaction' takes the inhomogeneities in geometry into account and so provides a more realistic average description of the left-hand side of the Einstein equations.

From first principles, any realistic cosmological model emerging from a 3 + 1 spacetime split should evolve spatial curvature for evolving sources, even if the spatial curvature is set to zero at some initial time. The oversimplified standard model keeps curvature at this non-generic value, while backreaction models evolve curvature, even if it is set to zero at some initial time [31].

We have to distinguish average properties and fluctuations. Fluctuations may remain small on large scales, but that is not the issue: the issue is the background about which these fluctuations are small, and this is what is addressed by backreaction models. Metrical deviations from an (unknown) average metric may be small, but it is an assumption that this average metric is dynamically equivalent to a homogeneous (and flat) FLRW solution. In other words, small metric perturbations do not imply small distortions of a flat geometry. In 1916, Einstein already clarified this and explained that the perturbations could equally well be small deviations from a large-scale curved space. (Einstein, however, reached this conclusion with calculations on which we comment, below his quote, from a modern perspective.) This can be considered a variant of the steel ball model of GW. In Einstein's words [128]:

But it is conceivable that our Universe differs only slightly from a Euclidean one, and this notion seems all the more probable, since calculations show that the metrics of surrounding space is influenced only to an exceedingly small extent by masses even of the magnitude of our Sun. We might imagine that, as regards geometry, our Universe behaves analogously to a surface which is irregularly curved in its individual parts, but which nowhere departs appreciably from a plane: something like the rippled surface of a lake. Such a Universe might fittingly be called a quasi-Euclidean Universe. As regards its space it would be infinite. But calculation shows that in a quasi-Euclidean Universe the average density of matter would necessarily be nil. Thus such a Universe could not be inhabited by matter everywhere; it would present to us that unsatisfactory picture ... If we are to have in the Universe an average density of matter which differs from zero, however small may be that difference, then the Universe cannot be quasi-Euclidean. On the contrary, the results of calculation indicate that, if matter be distributed uniformly, the Universe would necessarily be spherical (or elliptical). Since in reality the detailed distribution of matter is not uniform, the real Universe will deviate in individual parts from the spherical, i.e. the Universe will be quasi-spherical. But it will be necessarily finite.

Einstein emphasized that the small deviations require a more careful description of the global spatial curvature. (Einstein mentions a positively curved background, but it could as well be negatively curved.) We argue that the deviations should be studied with respect to a background that is defined by the actual average distribution (the physical background), as it is done in other physical disciplines that investigate fluctuation theories, e.g., in solid state physics, not by a priori assumption.

In a modern perspective, but only in Newtonian cosmology, we can represent the Universe through a finite flat three-torus model with Newtonian inhomogeneities on a factored out relativistic FLRW model, see [28]. In Newtonian cosmology the a priori assumption of a background that is equivalent with the average distribution can be realized: the spatially averaged model can be assumed to be an FLRW model on the imposed scale of the three-torus due to the globally vanishing backreaction, which assures that the inhomogeneities average out to zero, see [28]; in general relativity such a construction is not possible, and the average model in general does not comply with an FLRW model [27]: the physical background will not coincide with an assumed fixed background, since it is determined through the dynamics of inhomogeneities.

Discussions restricting the regime of applicability to regions 'far from black holes or neutron stars' overlook the fact that on cosmological scales the curvature can be tiny but still dominant over the source contributions. Even in homogeneous cosmology, the Einstein cosmos [89] provides an example of a curvature-dominated model, its curvature radius being smaller than the Schwarzschild radius if the former is greater than about 3.3 Gpc (assuming a mean density of the present estimate of the total baryonic plus dark matter densities and the observed value of the Hubble constant). In other words, this example shows that if the curvature is cosmologically significant small metric deviations from a flat metric would run into contradictions.

More generally, even if fields are weak then an important consideration is the calibration of the asymptotic rulers and clocks of the close to spatially flat metric around bound structures relative to a generally non-FLRW cosmological average. Since no systems are truly isolated, this may generally require a notion of 'finite infinity' [17, 131, 129, 130] to replace the exactly asymptotically flat geometry to which many definitions of gravitational mass are tied. The problems of averaging and coarse-graining may therefore be intimately related to fundamental unsolved problems concerning gravitational mass–energy [6].

Finally, in this paper we focussed on backreaction effects in general relativity, as this is also the framework addressed by Green and Wald in their papers. However, there could be more degeneracies relating observations with theoretical predictions: we did not touch upon issues related to different theoretical frameworks such as the possibility of a non-symmetric connection, the role of torsion, non-metrical theories or, generally, modified gravitational theories that contain general relativity as a low-energy limit. A general assessment of backreaction effects would also incorporate studies of possible deviations from classical general relativity.

Let us consider Szybka et al's [47] example in relation to GW's claim that backreaction is trace-free. The example is based on the Wainwright–Marshman metric:

$$\begin{eqnarray}&&{\rm{d}}{s}^{2}={t}^{2m}{{\rm{e}}}^{n}\left(-{\rm{d}}{t}^{2}+{\rm{d}}{z}^{2}\right)+{t}^{1/2}\left[{\rm{d}}{x}^{2}+\left(t+{w}^{2}\right){\rm{d}}{y}^{2}+2w\;{\rm{d}}x{\rm{d}}y\right],\end{eqnarray} \tag{ C.1 }$$

where $t\gt 0,-\infty \lt x,y,z,\lt +\infty ,$ m is a free parameter, n and w are functions of a single variable $u:=t-z,$ and the field equation

$$\begin{eqnarray}&&{\rm{d}}n/{\rm{d}}u={({\rm{d}}w/{\rm{d}}u)}^{2}\end{eqnarray} \tag{ C.2 }$$

helps to simplify the Einstein tensor. The stress–energy tensor is that of a perfect fluid with equation of state

$$\begin{eqnarray}&&\varrho =p=\displaystyle \frac{1}{8\pi }(m+3/16){t}^{-2(m+1)}{{\rm{e}}}^{-n},\end{eqnarray} \tag{ C.3 }$$

so the weak energy condition holds for $m\geqslant -3/16.$ Szybka et al [47] set $w:= \lambda \mathrm{sin}(u/\lambda )$ which gives $n=\frac{1}{2}\left(u+\frac{1}{2}\lambda \mathrm{sin}(2u/\lambda )\right).$ In the limit: ${\mathrm{lim}}_{\lambda \searrow 0}{\rm{w}}=0$ and ${\mathrm{lim}}_{\lambda \searrow 0}n=\frac{1}{2}u,$ hence (C.2) no longer holds for the background.

The Ricci scalar of the metric ${g}_{{ab}}(\lambda )$ is $R(\lambda )=-\frac{1}{8}\left(16m+3\right){t}^{-2(m+1)}{{\rm{e}}}^{-n(\lambda )},$ where for the background metric ${g}_{{ab}}(0),$ n should be substituted by ${\mathrm{lim}}_{\lambda \searrow 0}n.$ This can be used as follows to show that, although the derivatives of metric deviations and quadratic products thereof in this example are singular in the limit as λ goes to zero, the Ricci scalar and the stress–energy tensor are not and thus cannot produce backreaction from inhomogeneities. First, let us subtract the Einstein equations for $\lambda \gt 0,$

$$\begin{eqnarray}&&{R}_{{ab}}(g(\lambda ))-\displaystyle \frac{1}{2}{g}_{{ab}}(\lambda )R(g(\lambda ))=8\pi {T}_{{ab}}(\lambda ),\end{eqnarray} \tag{ C.4 }$$

from the background dynamical equation; for $\lambda =0$ (using the notation ${t}_{{ab}}(0),$ see section 3.7), we have:

$$\begin{eqnarray}&&{R}_{{ab}}(g(0))-\displaystyle \frac{1}{2}{g}_{{ab}}(0)R(g(0))=8\pi \left({T}_{{ab}}(0)+{t}_{{ab}}(0)\right).\end{eqnarray} \tag{ C.5 }$$

We take the ${\rm{w}}-\mathrm{lim}$ of the difference (C.5)−(C.4) (recall that ${t}_{{ab}}(0)$ remains unaffected by the weak-limit operator):

$$\begin{eqnarray}&&{t}_{{ab}}(0)=\underset{\lambda \searrow 0}{{\rm{w}}-\mathrm{lim}}\left({R}_{{ab}}(g(0))-{R}_{{ab}}(g(\lambda ))\right),\end{eqnarray} \tag{ C.6 }$$

because ${\rm{w}}-\mathrm{lim}\left(R(g(0))-R(g(\lambda ))\right)=0,$ ${\rm{w}}-\mathrm{lim}\left({g}_{{ab}}(0)R(g(0))-{g}_{{ab}}(\lambda )R(g(\lambda ))\right)=0,$ and ${\rm{w}}-\mathrm{lim}\left({T}_{{ab}}(0)-{T}_{{ab}}(\lambda )\right)=0.$

Introducing the trace-free Ricci tensor ${S}_{{ab}}={R}_{{ab}}-\frac{1}{4}{g}_{{ab}}R,$ we get:

$$\begin{eqnarray}&&{t}_{{ab}}(0)=\underset{\lambda \searrow 0}{{\rm{w}}-\mathrm{lim}}\left({S}_{{ab}}(g(0))-{S}_{{ab}}(g(\lambda ))\right),\end{eqnarray} \tag{ C.7 }$$

since again ${{\rm{w}}-\mathrm{lim}}_{\lambda \searrow 0}\left({g}_{{ab}}(0)R(g(0))-{g}_{{ab}}(\lambda )R(g(\lambda ))\right)=0.$ From this we see that the backreaction term t_ab entirely emerges from the trace-less λ-dependent curvature (${S}_{{ab}}(g(\lambda ))$). The Ricci scalar and the stress–energy tensor, where the density inhomogeneities are encoded, cancel out. Since the Wainwright–Marshman spacetimes are interpreted as cosmological models with gravitational waves, we can attribute all the backreaction present in this example to gravitational radiation, not to density inhomogeneities.

Now we consider the example provided by GW in [11] (section 4). We recall GW's assumption (i) [9], i.e., that there exists a family of metrics ${g}_{{ab}}(\lambda )$ and a smooth function C₁ such that:

• ${g}_{{ab}}(1)$ represents the real Universe, while ${g}_{{ab}}(0)$ is the background, averaged metric;
• ${G}_{{ab}}(g(\lambda ))=8\pi {T}_{{ab}}(\lambda )$ for all $\lambda \gt 0$ and ${T}_{{ab}}(\lambda )$ obeys the weak energy condition; and
• ${h}_{{ab}}(\lambda )=| {g}_{{ab}}(0)-{g}_{{ab}}(\lambda )| \lt \lambda {C}_{1}.$

What does it imply for ${g}_{{ab}}(\lambda )$ to be a solution of the Einstein equations? For $\lambda =1$ this is true by definition, since we assume that the real Universe is well-described by general relativity. However, while we have a straightforward prescription of how to construct ${G}_{{ab}}(g(\lambda ))$ for $1\gt \lambda \gt 0,$ the situation with ${T}_{{ab}}(\lambda )$ is not so clear.

We can distinguish two reasonable approaches:

• (i)  
  we calculate ${G}_{{ab}}(g(\lambda ))$ and define ${T}_{{ab}}(\lambda ):= \frac{1}{8\pi }{G}_{{ab}}(\lambda )$ (this is the option chosen by Green and Wald); or
• (ii)  
  we calculate T_ab for a given λ from its definition in terms of ${{\mathcal{L}}}_{{\rm{m}}},$ the non-gravitational part of the Lagrangian density of the Einstein–Hilbert action for matter, and its functional derivative, i.e.

  $$\begin{eqnarray}&&{T}_{{ab}}:= 2\;\displaystyle \frac{\delta {{\mathcal{L}}}_{{\rm{m}}}}{\delta {g}_{{ab}}}+{g}^{{ab}}\;{{\mathcal{L}}}_{{\rm{m}}}.\end{eqnarray} \tag{ C.8 }$$

To see that these two possibilities are, in general, different, let us consider Green and Wald's example for both of them. By assumption, there is an FLRW background metric ${g}_{{ab}}(0)$ and a conformally related family of metrics:

$$\begin{eqnarray}&&{g}_{{ab}}(\lambda )={{\rm{\Omega }}}^{2}(\lambda ){g}_{{ab}}(0),\quad {\rm{where}}\quad \mathrm{ln}{\rm{\Omega }}(\lambda )=\lambda A\left(\mathrm{sin}\displaystyle \frac{x}{\lambda }+\mathrm{sin}\displaystyle \frac{y}{\lambda }+\mathrm{sin}\displaystyle \frac{z}{\lambda }\right).\end{eqnarray} \tag{ C.9 }$$

The Einstein tensor built out of ${g}_{{ab}}(\lambda ),$ i.e., ${G}_{{ab}}(g(\lambda )),$ is related to the FLRW Einstein tensor ${G}_{{ab}}(g(0))$ by a purely geometrical formula:

$$\begin{eqnarray}{G}_{{ab}}(g(\lambda )) & = & {G}_{{ab}}(g(0))-\left(2{{\rm{\nabla }}}_{a}{{\rm{\nabla }}}_{b}\mathrm{ln}{\rm{\Omega }}-2{g}_{{ab}}(0){g}^{{cd}}(0){{\rm{\nabla }}}_{c}{{\rm{\nabla }}}_{d}\mathrm{ln}{\rm{\Omega }}\right.\\ & & \left.-2({{\rm{\nabla }}}_{a}\mathrm{ln}{\rm{\Omega }})({{\rm{\nabla }}}_{b}\mathrm{ln}{\rm{\Omega }})-{g}_{{ab}}(0){g}^{{cd}}(0)({{\rm{\nabla }}}_{c}\mathrm{ln}{\rm{\Omega }})({{\rm{\nabla }}}_{d}\mathrm{ln}{\rm{\Omega }})\right).\end{eqnarray} \tag{ C.10 }$$

Green and Wald show (see (4.4) in [11]) that the effective stress–energy tensor in the weak-limit reads:

$$\begin{eqnarray}&&{t}_{{ab}}(0)=\displaystyle \frac{1}{8\pi }{G}_{{ab}}(g(0))-{T}_{{ab}}(0)=\displaystyle \frac{1}{8\pi }\underset{\lambda \searrow 0}{{\rm{w}}-\mathrm{lim}}\left({G}_{{ab}}(g(0))-{G}_{{ab}}(g(\lambda ))\right),\end{eqnarray} \tag{ C.11 }$$

and that the backreaction found by the GW prescription has a trace corresponding to a $P=-\frac{5}{3}\rho$ fluid.

Now we have to choose in which way we define the stress–energy tensor for $0\lt \lambda \lt 1.$

On the one hand from (i):

$$\begin{eqnarray}8\pi {T}_{{ab}}(\lambda ) & = & {G}_{{ab}}(g(\lambda ))\end{eqnarray} \tag{ C.12 }$$

into which (C.10) has to be substituted while on the other hand from (ii):

$$\begin{eqnarray}&&{T}_{{ab}}(\lambda )={{\rm{\Omega }}}^{-2}{T}_{{ab}}(0).\end{eqnarray} \tag{ C.13 }$$

The reason for the transformation rule (C.13) is conformal invariance of the matter action:

$$\begin{eqnarray}&&S(\lambda )=\int {{\mathcal{L}}}_{{\rm{m}}}(\lambda )\sqrt{-g(\lambda )}{{\rm{d}}}^{4}x=\int {{\mathcal{L}}}_{{\rm{m}}}(0)\sqrt{-g(0)}{{\rm{d}}}^{4}x=S(0),\end{eqnarray} \tag{ C.14 }$$

where the Lagrangian densities transform according to

$$\begin{eqnarray}&&{{\mathcal{L}}}_{{\rm{m}}}(\lambda )={{\rm{\Omega }}}^{-4}{{\mathcal{L}}}_{{\rm{m}}}(0),\end{eqnarray} \tag{ C.15 }$$

so that (C.8) gives

$$\begin{eqnarray}{T}^{{ab}}(\lambda ) & = & \displaystyle \frac{2}{\sqrt{-g(\lambda )}}\;\;\displaystyle \frac{\delta }{\delta {g}_{{ab}}(\lambda )}\left(\sqrt{-g(\lambda )}{{\mathcal{L}}}_{{\rm{m}}}(\lambda )\right)\\ & = & {{\rm{\Omega }}}^{-4}\displaystyle \frac{2}{\sqrt{-g(0)}}\;\;\displaystyle \frac{\partial {g}_{{cd}}(0)}{\partial {g}_{{ab}}(\lambda )}\;\;\displaystyle \frac{\delta }{\delta {g}_{{cd}}(0)}\left(\sqrt{-g(0)}{{\mathcal{L}}}_{{\rm{m}}}(0)\right);\end{eqnarray} \tag{ C.16 }$$

lowering of indices using ${g}_{{ae}}(\lambda )$ and ${g}_{{bf}}(\lambda )$ gives (C.13) (see, e.g., [144]). In (C.14) and (C.16), a non-subscripted g indicates the determinant of the metric.

To show that case (i) is in general incompatible with case (ii) we proceed as follows. Let us assume that for $\lambda =1,$ the metric and stress–energy tensor represent the 'real' Universe or, as in the case presented, the toy Universe model that we wish to average. Define $f:= {\rm{\Omega }}(\lambda ){{\rm{\Omega }}}^{-1}(\lambda =1).$

We have for case (i):

$$\begin{eqnarray*}&&{G}_{{ab}}(g(1))=8\pi {T}_{{ab}}(1);{g}_{{ab}}(1)={{\rm{\Omega }}}^{2}(\lambda =1){g}_{{ab}}(0);{g}_{{ab}}(\lambda )={{\rm{\Omega }}}^{2}(\lambda ){g}_{{ab}}(0),\end{eqnarray*}$$

which implies that

$$\begin{eqnarray}&&{g}_{{ab}}(\lambda )={{\rm{\Omega }}}^{2}(\lambda ){{\rm{\Omega }}}^{-2}(\lambda =1){g}_{{ab}}(1).\end{eqnarray} \tag{ C.17 }$$

Thus, ${g}_{{ab}}(\lambda \lt 1)$ and ${g}_{{ab}}(1)$ are conformally related via ${g}_{{ab}}(\lambda )={f}^{2}{g}_{{ab}}(1),$ since f is a ratio of exponentials of smooth real-valued functions (see (C.9)), and thus smooth and real-valued.

For case (ii):

$$\begin{eqnarray}&&{T}_{{ab}}(\lambda )={f}^{-2}{T}_{{ab}}(1),\end{eqnarray} \tag{ C.18 }$$

$$\begin{eqnarray}{G}_{{ab}}(g(\lambda )) & = & {G}_{{ab}}(g(1))-\left(2{\tilde{{\rm{\nabla }}}}_{a}{\tilde{{\rm{\nabla }}}}_{b}\mathrm{ln}f-2{g}_{{ab}}(1){g}^{{cd}}(1){\tilde{{\rm{\nabla }}}}_{c}{\tilde{{\rm{\nabla }}}}_{d}\mathrm{ln}f\right.\\ & & -\left.2({\tilde{{\rm{\nabla }}}}_{a}\mathrm{ln}f)({\tilde{{\rm{\nabla }}}}_{b}\mathrm{ln}f)-{g}_{{ab}}(1){g}^{{cd}}(1)({\tilde{{\rm{\nabla }}}}_{c}\mathrm{ln}f)({\tilde{{\rm{\nabla }}}}_{d}\mathrm{ln}f)\right),\end{eqnarray} \tag{ C.19 }$$

where the covariant derivative $\tilde{{\rm{\nabla }}}$ is compatible with the metric ${g}_{{ab}}(1)$ and the geometrical formula (C.19) is obtained directly from (C.10), and thus is also valid in case (ii). Combining (C.18) and (C.19) and imposing the Einstein equations for $\lambda \gt 0$ we obtain:

$$\begin{eqnarray}8\pi {T}_{{ab}}(1) & = & \displaystyle \frac{1}{(1-{f}^{-2})}\left(2{\tilde{{\rm{\nabla }}}}_{a}{\tilde{{\rm{\nabla }}}}_{b}\mathrm{ln}f-2{g}_{{ab}}(1){g}^{{cd}}(1){\tilde{{\rm{\nabla }}}}_{c}{\tilde{{\rm{\nabla }}}}_{d}\mathrm{ln}f\right.\\ & & -\left.2({\tilde{{\rm{\nabla }}}}_{a}\mathrm{ln}f)({\tilde{{\rm{\nabla }}}}_{b}\mathrm{ln}f)-{g}_{{ab}}(1){g}^{{cd}}(1)({\tilde{{\rm{\nabla }}}}_{c}\mathrm{ln}f)({\tilde{{\rm{\nabla }}}}_{d}\mathrm{ln}f)\right),\end{eqnarray} \tag{ C.20 }$$

which is a contradiction since the left-hand side is fixed and the right-hand side depends on λ (lengthy but easy calculations show that the dependence on λ does not cancel).

We conclude that, within the GW formalism (as their explicit example shows), it is not in general possible to build a family of metric-dependent tensors using the method (ii) that obeys the Einstein equations. This gives us a strong indication of the nature of the ${T}_{{ab}}(\lambda )$ family obtained with method (i).

To see this explicitly let us look at GW's choice (i): using the purely geometrical relation (C.19) and the Einstein equations for $\lambda \gt 0,$ we have³⁶ :

$$\begin{eqnarray}8\pi {T}_{{ab}}(\lambda ) & = & 8\pi {T}_{{ab}}(1)-\left(2{\tilde{{\rm{\nabla }}}}_{a}{\tilde{{\rm{\nabla }}}}_{b}\mathrm{ln}f-2{g}_{{ab}}(1){g}^{{cd}}(1){\tilde{{\rm{\nabla }}}}_{c}{\tilde{{\rm{\nabla }}}}_{d}\mathrm{ln}f\right.\\ & & -\left.2({\tilde{{\rm{\nabla }}}}_{a}\mathrm{ln}f)({\tilde{{\rm{\nabla }}}}_{b}\mathrm{ln}f)-{g}_{{ab}}(1){g}^{{cd}}(1)({\tilde{{\rm{\nabla }}}}_{c}\mathrm{ln}f)({\tilde{{\rm{\nabla }}}}_{d}\mathrm{ln}f)\right).\end{eqnarray} \tag{ C.21 }$$

Now we take the weak-limit of both sides.

The weak-limit of ${T}_{{ab}}(g(1))$ is (where ${f}^{{ab}}$ is a test tensor field):

$$\begin{eqnarray}&&\underset{\lambda \searrow 0}{{\rm{w}}-\mathrm{lim}}{T}_{{ab}}(g(1))=\int {{\rm{d}}}^{4}x\sqrt{-g(0)}{T}_{{ab}}(g(1)){f}^{{ab}},\end{eqnarray} \tag{ C.22 }$$

which we can naturally associate with an averaged stress–energy tensor usually denoted $\langle {T}_{{ab}}\rangle$ and denoted ${T}_{{ab}}(0)$ by GW (according to GW: 'we may interpret ${T}_{{ab}}(0)$ as representing the matter stress–energy tensor averaged over small scale inhomogeneities', [9] p 13). Since the weak-limit of the remaining terms on the right-hand side does not disappear, we see that ${\rm{w}}-\mathrm{lim}{T}_{{ab}}(\lambda )\ne {T}_{{ab}}(0).$ As a consequence, in a more realistic case we should not expect ${T}_{{ab}}(0)$ to match the FLRW stress–energy tensor obtained by averaging the density inhomogeneities. It seems that the ${T}_{{ab}}(\lambda )$ defined via the Einstein field equations already contains a backreaction effect and thus a part of backreaction is absorbed into the definition of ${T}_{{ab}}(0).$ In other words, forcing the existence of stress–energy tensors that obey the Einstein equations makes the weak-limit of this family of tensors distinct from the averaged stress–energy tensor.

In summary, we are not aware of an example of a metric family satisfying the GW conditions that satisfactorily describes backreaction from matter inhomogeneities.