[Paper Review] Accelerating Universe via Spatial Averaging

Yasusada Nambu, Masayuki Tanimoto|ArXiv.org|Jul 13, 2005

Cosmology and Gravitation Theories19 citations

TL;DR

This paper proposes that spatial averaging of an inhomogeneous Tolman-Bondi universe—featuring coexisting regions of positive and negative spatial curvature—can produce effective dark energy-like acceleration without exotic matter. When the positively curved region collapses, the spatially averaged expansion transitions to acceleration due to non-perturbative backreaction effects, with the averaged deceleration parameter turning negative under specific geometric conditions.

ABSTRACT

We present a model of an inhomogeneous universe that leads to accelerated expansion after taking spatial averaging. The model universe is the Tolman-Bondi solution of the Einstein equation and contains both a region with positive spatial curvature and a region with negative spatial curvature. We find that after the region with positive spatial curvature begins to re-collapse, the deceleration parameter of the spatially averaged universe becomes negative and the averaged universe starts accelerated expansion. We also discuss the generality of the condition for accelerated expansion of the spatially averaged universe.

Motivation & Objective

• To investigate whether spatial averaging of an inhomogeneous universe can produce effective accelerated expansion without introducing dark energy.
• To explore the role of non-perturbative inhomogeneities, particularly in the collapse phase of a closed region, in generating backreaction effects.
• To determine the geometric and dynamical conditions under which the spatially averaged universe exhibits accelerated expansion.
• To examine the generality of the backreaction mechanism in inhomogeneous cosmologies beyond perturbative treatments.

Proposed method

• Uses the exact Tolman-Bondi solution of the Einstein equations with dust, allowing for regions of both positive and negative spatial curvature.
• Defines spatial averaging via the physical volume of a comoving region D, leading to an averaged scale factor $ a_D $ and deceleration parameter $ q_D $.
• Derives the effective Friedmann equations for the averaged universe by computing the spatial average of curvature and density terms.
• Models spatial curvature as $ k(r) = \frac{1}{L^2}[2\theta(r-r_0)-1] $, creating a spatially open region ($ r < r_0 $) and a spatially closed region ($ r > r_0 $).
• Evaluates the effective energy density $ \rho_{\text{eff}} $ and pressure $ p_{\text{eff}} $ from the averaged dynamics, showing $ w_{\text{eff}} \approx -1/3 $ asymptotically.
• Analyzes the conditions for acceleration by requiring $ \rho_{\text{eff}} > 0 $ and $ \rho_{\text{eff}} + 3p_{\text{eff}} < 0 $, leading to constraints on $ c_1 $, $ c_2 $, and $ \rho_0 L^2 $.

Experimental results

Research questions

• RQ1 Can spatial averaging of an inhomogeneous universe with non-linear inhomogeneities produce effective accelerated expansion?
• RQ2 What specific geometric and dynamical conditions are required for the averaged universe to enter an accelerating phase?
• RQ3 How does the backreaction from a collapsing, positively curved region influence the large-scale dynamics of the spatially averaged universe?
• RQ4 Is the acceleration effect robust beyond perturbative treatments, and what role does non-perturbative curvature averaging play?
• RQ5 What constraints on the size of the open and closed regions are necessary for the averaged universe to accelerate before re-collapse?

Key findings

• The spatially averaged universe exhibits accelerated expansion when the region with positive spatial curvature enters the re-collapse phase.
• The deceleration parameter $ q_D $ becomes negative after the closed region begins to contract, signaling effective acceleration.
• The effective equation of state parameter asymptotically approaches $ w_{\text{eff}} \approx -1/3 $, indicating a dark energy-like behavior.
• Acceleration occurs only if the spatially open region is neither too large nor too small, with $ 0.4 < r_0/L < 0.9 $ for $ \rho_0 L^2 = 1 $, $ V_* = L^3 $.
• The necessary conditions are $ K < 0 $ (negative averaged spatial curvature) and $ 0 < 4C - \rho_* < 3(-K) $, which translate to $ c_1 < c_2/3 $ and a geometric constraint on $ r_0/L $.
• The acceleration is a non-perturbative effect arising from the coexistence of expanding and collapsing regions, unattainable via standard cosmological perturbation theory.