[Paper Review] Comments on Backreaction and Cosmic Acceleration

E. W. Kolb, S. Matarrese+1 more|arXiv (Cornell University)|Nov 2, 2005

Cosmology and Gravitation Theories4 citations

TL;DR

This paper argues that cosmic acceleration may arise from kinematical backreaction in inhomogeneous cosmologies, not dark energy. By deriving effective Friedmann equations after spatial averaging, it shows that even if individual regions decelerate, the mean scale factor can accelerate due to nonlinearities in the Hubble rate, with the backreaction term $ Q_{\cal D} $ potentially exceeding the average energy density, enabling acceleration in a relativistic, inhomogeneous framework.

ABSTRACT

In this brief WEB note we comment on recent papers related to our paper "On Acceleration Without Dark Energy".

Motivation & Objective

• To investigate whether cosmic acceleration can emerge from backreaction in inhomogeneous spacetimes, rather than from dark energy or modified gravity.
• To clarify the role of spatial averaging and domain choice in defining observable average expansion dynamics.
• To challenge the claim that Newtonian approximations rule out significant backreaction effects.
• To demonstrate that acceleration of the averaged scale factor $ a_{\cal D} $ can occur even when individual fluid elements decelerate.
• To resolve apparent contradictions in the literature regarding the physical relevance of backreaction-driven acceleration.

Proposed method

• Derives effective Friedmann equations for an inhomogeneous universe after spatial averaging over a domain $ \mathcal{D} $, using Buchert's formalism.
• Defines the average scale factor as $ a_{\cal D} = (V_{\cal D})^{1/3} $, and analyzes the time evolution of its second derivative to assess acceleration.
• Expresses the average acceleration as a sum of local accelerations and a variance-like term involving $ \left\langle (\dot{a}/a)^2 \right\rangle - \left( \left\langle \dot{a}/a \right\rangle \right)^2 $, which captures the backreaction effect.
• Identifies the kinematical backreaction term $ Q_{\cal D} $ as the key driver of acceleration when $ Q_{\cal D} > 4\pi G \langle \rho \rangle_{\cal D} $.
• Uses the synchronous and comoving gauge to compute backreaction terms, showing they contain non-Newtonian, post-Newtonian contributions even in weak-field limits.
• Applies the Ergodic Theorem to justify replacing spatial averages with ensemble averages when the domain $ \mathcal{D} $ is sufficiently large.

Experimental results

Research questions

• RQ1 Can cosmic acceleration be explained by backreaction in an inhomogeneous universe without invoking dark energy?
• RQ2 Is the acceleration of the averaged scale factor $ a_{\cal D} $ physically meaningful, even if individual regions decelerate?
• RQ3 Why do Newtonian approximations fail to capture significant backreaction effects, despite their accuracy on large scales?
• RQ4 How does the choice of time slicing and domain $ \mathcal{D} $ affect the averaging procedure and the resulting dynamics?
• RQ5 What is the role of non-perturbative, relativistic effects in generating a non-zero backreaction term $ Q_{\cal D} $?

Key findings

• Acceleration of the averaged scale factor $ a_{\cal D} $ can occur even when all individual fluid elements decelerate, due to the nonlinear variance term in the effective dynamics.
• The condition $ Q_{\cal D} > 4\pi G \langle \rho \rangle_{\cal D} $ is necessary for average acceleration, and this term can be large due to inhomogeneities in Hubble rates across regions.
• Newtonian approximations fail to capture the backreaction effect because $ Q_{\cal D} $ reduces to a boundary term via Gauss's theorem, suppressing any net effect.
• In the relativistic, synchronous and comoving gauge, $ Q_{\cal D} $ contains non-perturbative, post-Newtonian terms like $ \sim H^2 \langle \delta^2 (v/c)^2 \rangle $, which are significant and non-zero.
• Shell-crossing singularities do not invalidate the backreaction approach, as they affect only a small mass fraction and can be smoothed without altering the mean expansion rate.
• The dependence on domain $ \mathcal{D} $ is unavoidable but physically meaningful, and the Ergodic Theorem allows replacement of spatial averages with ensemble averages for large enough domains.