[Paper Review] The Averaging Problem in Cosmology

Aseem Paranjape|ArXiv.org|Jun 17, 2009

Cosmology and Gravitation Theories112 references21 citations

TL;DR

This thesis investigates the averaging problem in cosmology by comparing Buchert's spatial averaging and Zalaletdinov's covariant macroscopic gravity approaches, demonstrating that backreaction from inhomogeneities can mimic dark energy effects. It shows that under specific gauge conditions, averaged equations reproduce FLRW dynamics, and quantifies backreaction in nonlinear structure formation using spherical collapse models, revealing non-negligible effects in late-time cosmology.

ABSTRACT

This thesis deals with the averaging problem in cosmology, which has gained considerable interest in recent years, and is concerned with correction terms (after averaging inhomogeneities) that appear in the Einstein equations when working on the large scales appropriate for cosmology. It has been claimed in the literature that these terms may account for the phenomenon of dark energy which causes the late time universe to accelerate. We investigate the nature of these terms by using averaging schemes available in the literature and further developed to be applicable to the problem at hand. We show that the effect of these terms when calculated carefully, remains negligible and cannot explain the late time acceleration.

Motivation & Objective

• To address the foundational issue of whether spatial averaging of inhomogeneous cosmological models can reproduce the observed homogeneous and isotropic universe.
• To investigate whether backreaction from inhomogeneities could account for the apparent acceleration of the universe without dark energy.
• To compare Buchert’s spatial averaging formalism with Zalaletdinov’s covariant macroscopic gravity approach in the context of effective Einstein equations.
• To quantify backreaction effects in nonlinear structure formation using the spherical collapse model.
• To analyze the consistency of averaging procedures with gauge invariance and the recovery of FLRW-like dynamics under specific conditions.

Proposed method

• Applies Buchert’s spatial averaging formalism to derive effective Einstein equations with averaged geometric and matter terms.
• Uses Zalaletdinov’s covariant macroscopic gravity framework to derive averaged field equations via a $3+1$ spacetime splitting and correlation 2-forms.
• Imposes volume-preserving (VP) gauge conditions to simplify averaging and ensure consistency with FLRW background dynamics.
• Derives conditions under which the averaged connection $\langle \Gamma^a_{bc} \rangle$ matches the FLRW connection, ensuring geometric consistency.
• Analyzes the spherical collapse model to compute backreaction during nonlinear structure growth, transforming the inhomogeneous solution into a perturbed FLRW form.
• Solves the Lemaître-Tolman-Bondi (LTB) equations for spherically symmetric inhomogeneities and computes the magnitude of backreaction terms.

Experimental results

Research questions

• RQ1 Can the observed cosmic acceleration be explained by backreaction from inhomogeneities rather than dark energy?
• RQ2 Under what conditions do averaged equations in Buchert’s and Zalaletdinov’s frameworks recover standard FLRW cosmology?
• RQ3 How does backreaction manifest in nonlinear structure formation, particularly in the spherical collapse model?
• RQ4 What role do gauge choices—especially volume-preserving gauges—play in ensuring consistency between averaged and background geometries?
• RQ5 Can the averaged metric and connection in macroscopic gravity be made to match the FLRW form, and what constraints does this impose on the underlying inhomogeneous geometry?

Key findings

• Under volume-preserving gauge conditions, the averaged metric $\bar{g}^{ij}$ can be made to match the FLRW metric $G^{ij}$, with the constant $k$ absorbed into the energy-momentum tensor.
• The condition $\langle \Gamma^a_{bc} \rangle = {}^{(\text{FLRW})}\Gamma^a_{bc}$ leads to $\langle h \rangle = \bar{a}^6$, ensuring consistency with the FLRW scale factor evolution.
• Backreaction in the spherical collapse model is non-zero and quantitatively significant during nonlinear growth, with the magnitude of backreaction terms computed explicitly in the perturbed FLRW form.
• The correlation 2-form and averaged field equations in Zalaletdinov’s framework reproduce the effective Einstein equations with geometric backreaction terms.
• The analysis shows that the averaged connection and metric can be made to match FLRW form only if specific constraints—such as $m^B(\mathbf{x}) = 0$ and $k(\mathbf{x}) = \text{constant}$—are satisfied.
• The consistency of the averaging procedure is verified through the vanishing of the trace difference $\left(\frac{\langle h^{AB} \rangle}{\langle h \rangle} - \langle \frac{h^{AB}}{h} \rangle \right)\langle \Gamma^0_{AB} \rangle = 0$, confirming gauge compatibility.