Averaging, Backreaction, and Model Dependence

“Backreaction” can name several inequivalent operations: averaging scalar evolution equations on spatial slices, taking a short-wavelength limit of the metric, propagating photons through an inhomogeneous geometry, or fitting a homogeneous model to those photons. The operations need not commute and their outputs are not the same observable. This page develops two rigorous analytic controls—Buchert’s scalar averaging and the Green–Wald weak-limit framework—then states the extra calculation required before either can support a claim about observed cosmic acceleration.

Required background. Geodesics and Jacobi Fields supplies the null propagation and observer measurements needed to turn a spacetime model into light-cone data.

Helpful background. Einstein Equations supplies the local constraints and evolution equations; Relativistic Matter and Compact Objects supplies matter-model limits; Relativistic Cosmology supplies the homogeneous comparison model; and Probability and Statistical Inference supplies the fitted-model and likelihood distinction.

Four questions hidden inside one word

It helps to separate four maps.

1. Spatial averaging: choose a foliation and a domain $\mathcal D$ on one slice, then average scalar fields over its proper volume.
2. Dynamical backreaction: ask how fluctuation terms enter equations for the domain volume and averaged curvature.
3. Light-cone averaging: average quantities sampled by null rays over directions, sources, or redshift bins.
4. Fitting: infer parameters of a restricted model, often FLRW, from a likelihood for observed fluxes, redshifts, angles, and counts.

Only the first two are directly described by the Buchert equations below. The Green–Wald construction instead controls a family of exact spacetimes in a small-amplitude, short-scale limit. Neither a domain scale factor $a_{\mathcal D}(t)$ nor a weak-limit background metric is automatically the FLRW model that best fits a supernova Hubble diagram. In particular, a spatial average does not directly equal a past-light-cone observable; a coincidence can be established only in a stated approximation after ray propagation and selection have been included.

Buchert’s scalar equations

Take irrotational pressureless dust with geodesic four-velocity orthogonal to synchronous hypersurfaces, while the dust flow remains smooth and has not developed shell crossing,

$$ds^2=-dt^2+g_{ij}(t,\boldsymbol X)\,dX^i dX^j.$$

Let $\mathcal D$ be a compact comoving portion of a slice. For a scalar $S$, define

$$\langle S\rangle_{\mathcal D} =\frac{1}{V_{\mathcal D}}\int_{\mathcal D}S\,dV, \qquad a_{\mathcal D} =\left(\frac{V_{\mathcal D}}{V_{\mathcal D0}}\right)^{1/3}.$$

The local volume element obeys $\dot{dV}=\theta\,dV$. Integrating over the comoving domain therefore gives $\dot V_{\mathcal D}/V_{\mathcal D}=\langle\theta\rangle_{\mathcal D}=3H_{\mathcal D}$, where $H_{\mathcal D}=\dot a_{\mathcal D}/a_{\mathcal D}$. Differentiating both the numerator and denominator of the average shows why averaging and evolution fail to commute:

$$\frac{d}{dt}\langle S\rangle_{\mathcal D} -\langle\dot S\rangle_{\mathcal D} =\langle S\theta\rangle_{\mathcal D} -\langle S\rangle_{\mathcal D}\langle\theta\rangle_{\mathcal D}.$$

This covariance is the origin of fluctuation terms, not a failure of local Einstein dynamics Buchert 2000, §2.2, Eqs. (9a)–(9f).

With $\sigma^2\equiv\tfrac12\sigma_{ij}\sigma^{ij}$, define the kinematical backreaction

$$\mathcal Q_{\mathcal D} =\frac{2}{3}\left( \langle\theta^2\rangle_{\mathcal D} -\langle\theta\rangle_{\mathcal D}^2 \right)-2\langle\sigma^2\rangle_{\mathcal D}.$$

Spatially averaging Raychaudhuri’s equation, the Hamiltonian constraint, and rest-mass conservation gives

$$\begin{aligned} 3\frac{\ddot a_{\mathcal D}}{a_{\mathcal D}} &=-4\pi G\langle\rho\rangle_{\mathcal D}+\Lambda+\mathcal Q_{\mathcal D},\\ 3H_{\mathcal D}^2 &=8\pi G\langle\rho\rangle_{\mathcal D}+\Lambda -\frac12\left(\langle{}^{(3)}R\rangle_{\mathcal D}+\mathcal Q_{\mathcal D}\right),\\ \dot{\langle\rho\rangle}_{\mathcal D} &+3H_{\mathcal D}\langle\rho\rangle_{\mathcal D}=0. \end{aligned}$$

These are exact scalar identities under the stated dust, foliation, and domain assumptions Buchert 2000, §2.2, Eqs. (10a)–(12e). Expansion variance contributes positively to $\mathcal Q_{\mathcal D}$, shear negatively. Thus “inhomogeneous” does not by itself fix even the sign.

Integrability is not closure

The averaged Hamiltonian and acceleration equations must remain compatible. Their integrability condition is

$$\frac{d}{dt}\!\left(a_{\mathcal D}^{6}\mathcal Q_{\mathcal D}\right) +a_{\mathcal D}^{4}\frac{d}{dt}\!\left( a_{\mathcal D}^{2}\langle{}^{(3)}R\rangle_{\mathcal D} \right)=0.$$

It couples averaged curvature to backreaction Buchert 2000, §2.2, Eqs. (13a)–(13b). Treating $\langle{}^{(3)}R\rangle_{\mathcal D}$ as an independently conserved FLRW curvature term while also allowing arbitrary $\mathcal Q_{\mathcal D}$ generally violates this equation.

But integrability supplies one relation for two new histories; it does not close the averaged system. Closure requires an exact spacetime, a controlled approximation, or an additional relation whose domain and foliation dependence can be tested. Even if $\ddot a_{\mathcal D}\gt0$, this means acceleration of a chosen proper volume. It does not yet predict $D_L(z)$, shear, number counts, or the parameters recovered by an FLRW likelihood.

The Green–Wald short-scale limit

Green and Wald ask a different question. Consider exact solutions $g_{ab}(\lambda)=g^{(0)}_{ab}+h_{ab}(\lambda)$ for $\lambda\gt0$, with matter satisfying the weak energy condition. Their hypotheses require $h_{ab}=O(\lambda)$ uniformly on compact sets, bounded first derivatives, and an existing weak limit of quadratic derivative products,

$$\underset{\lambda\to0}{\operatorname{w-lim}} \left(\nabla_a h_{cd}(\lambda)\nabla_b h_{ef}(\lambda)\right) =\mu_{abcdef}.$$

The limiting metric obeys

$$G_{ab}[g^{(0)}]+\Lambda g^{(0)}_{ab} =8\pi G\left(T^{(0)}_{ab}+t^{(0)}_{ab}\right),$$

where $t^{(0)}_{ab}$ is an effective contribution from nonlinear metric derivatives Green and Wald 2011, §II, Eqs. (5)–(16). Under those hypotheses it is traceless and itself satisfies the weak energy condition. If both the limiting metric and averaged matter have FLRW symmetry, the effective term has $p=\rho/3$ and cannot imitate a cosmological constant Green and Wald 2011, §II, Theorems 1–2 and Eqs. (30), (67).

This is a strong negative result inside its model class. It is not a theorem that every possible averaging, long-wavelength void configuration, finite-domain fit, or light-cone effect is negligible. Conversely, the existence of $\mathcal Q_{\mathcal D}$ does not evade the theorem: the two frameworks use different limiting families, averages, variables, matter assumptions, and questions. Calling them contradictory would require first constructing the same one-parameter spacetimes and showing that both prescriptions target the same quantity.

Results and limitations through 2026-08-19

Whether relativistic inhomogeneity produces an order-unity change in a fitted background expansion remains unresolved in general. The assigned sources establish exact analytic controls, not a universal observational verdict.

Evidence or result	Assumptions and regime	Scope or limitation
Buchert 2000 exact averaged scalar equations	Irrotational geodesic dust; flow-orthogonal slicing; compact comoving domain	Establishes noncommutation, $\mathcal Q_{\mathcal D}$, curvature coupling, and nonclosure; does not calculate a light-cone likelihood
Green–Wald 2011 Theorems 1 and 2	Small-amplitude one-parameter family, bounded derivatives, quadratic weak limit, matter weak energy condition	Effective term is traceless and positive-energy; rules out a cosmological-constant-like short-scale term only within these hypotheses
Relativistic-simulation bridge	Evolve one resolved spacetime, then compute domain averages and null observables from the same metric	Neither assigned framework paper supplies this matched numerical experiment; no simulation result is reproduced here
Observational or null comparison	Generate fluxes, redshifts, directions, selection, and covariance; fit the same model used on survey data	Neither analytic theorem is an official data release or empirical null test; observed acceleration cannot be read directly from $a_{\mathcal D}$ or $t^{(0)}_{ab}$

Competing positions therefore include Buchert scalar averaging, explicit light-cone or observational averaging, fully relativistic simulations, weak-field effective descriptions, timescape or void phenomenology, and results finding negligible backreaction in a specified regime. The conclusion depends on the matter model, gauge choice or foliation, domain, scale range, closure, and fitted observables. An “order unity” conclusion cannot be transferred from one formulation to another without a common spacetime and common inference target.

What could change the conclusion. Reassess the status when a controlled relativistic simulation or theorem ties gauge-invariant light-cone observables to an order-unity change in fitted background evolution while checking foliation, domain, conservation, resolution, and selection dependence.

A comparison that could decide something

Choose a family of constraint-satisfying inhomogeneous initial data and evolve it at increasing resolution. On each saved spacetime, compute $a_{\mathcal D}$, $\mathcal Q_{\mathcal D}$, and $\langle{}^{(3)}R\rangle_{\mathcal D}$ for several comoving domains in the dust-orthogonal slicing; verify the Hamiltonian equation and integrability residual. Comparing other slicings requires the corresponding lapse, tilt, and matter terms rather than reusing the dust equations unchanged. From the same observer worldlines, ray-trace a catalog with redshift, flux, shear, angular position, and a documented source selection. Fit the synthetic catalog with the same FLRW likelihood used on real data, then compare fitted parameters with the spatial averages.

This design separates discretization error from physical dependence and reveals whether an apparent acceleration is a volume effect, a photon-propagation effect, a selection effect, or a fitting effect. Neither such data nor an executable simulation accompanies this page, so no numerical magnitude is claimed.

Diagnostic: variance is not an observation

As an algebraic illustration at one instant, assign equal volumes and zero shear to two regions with expansion scalars $\theta_A=3H(1+\epsilon)$ and $\theta_B=3H(1-\epsilon)$. Compute $\mathcal Q_{\mathcal D}$. If it is positive, does that establish observed acceleration?

Solution

The domain mean is $\langle\theta\rangle_{\mathcal D}=3H$, while

$$\langle\theta^2\rangle_{\mathcal D} -\langle\theta\rangle_{\mathcal D}^2 =9H^2\epsilon^2.$$

Because the assigned shear vanishes, $\mathcal Q_{\mathcal D}=6H^2\epsilon^2\gt0$ for $\epsilon\ne0$. This arithmetic does not construct an exact dust spacetime: the constraints and the matching between regions must also be satisfied, and a transition region can contribute shear. In an admissible spacetime, the averaged domain accelerates only if $\mathcal Q_{\mathcal D}\gt4\pi G\langle\rho\rangle_{\mathcal D}-\Lambda$; positivity alone is insufficient. Even satisfying that inequality would establish only $\ddot a_{\mathcal D}\gt0$ in the chosen foliation and domain. An observational claim still requires null propagation, source selection, and a fit to synthetic $D_L(z)$ or equivalent light-cone data.

References

• Buchert, Thomas. 2000. “On Average Properties of Inhomogeneous Fluids in General Relativity. I: Dust Cosmologies.” General Relativity and Gravitation 32: 105–125. DOI. Open PDF.
• Green, Stephen R., and Robert M. Wald. 2011. “A New Framework for Analyzing the Effects of Small Scale Inhomogeneities in Cosmology.” Physical Review D 83: 084020. DOI. Open PDF.

Dated Research assessment. For open questions and recent results, follow the link below. Each assessment states the date through which it considers results and the limits of its conclusions.

• Cosmological Backreaction and Apparent Acceleration