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On the role of shear in cosmological averaging
Cosmology and Nongalactic Astrophysics · Submitted 17 Jul 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim By comparing the exact result for the backreaction to the case where the shear is neglected by treating the voids and walls as separate Friedmann-Robertson-Walker models, we find that the shear suppresses the backreaction by a factor of (r_0/t_0)^2, the squared ratio of the void size to the horizon size.
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The abstract, sentence by sentence
Using the spherically symmetric inhomogeneous Lemaitre-Tolman-Bondi dust solution, we study how the shear and the backreaction depend on the sharpness of the spatial transition between voids and walls and on the size of the voids. The voids considered here are regions with matter density Omega ~ 0 and expansion rate Ht ~ 1, while the walls are regions with matter density Omega ~ 1 and expansion rate Ht ~ 2/3. The results indicate that both the volume-average shear and the variance of the expansion rate grow proportional to the sharpness of the transition and diverge in the limit of a step function, but, for realistic-sized voids, are virtually independent of the size of the void. However, the backreaction, given by the difference of the variance and the shear, has a finite value in the step-function limit. By comparing the exact result for the backreaction to the case where the shear is neglected by treating the voids and walls as separate Friedmann-Robertson-Walker models, we find that the shear suppresses the backreaction by a factor of (r_0/t_0)^2, the squared ratio of the void size to the horizon size. This exemplifies the importance of using the exact solution for the interface between the regions of different expansion rates and densities. The suppression is justified to hold also for a network of compensated voids, but may not hold if the universe is dominated by uncompensated voids.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Using the spherically symmetric inhomogeneous Lemaitre-Tolman-Bondi dust solution, we study how the shear and the backreaction depend on the sharpness of the spatial transition between voids and walls and on the size of the voids. | contribution 0.90 | 0.95 | 0.06 | asserted 0.79 |
| s1 | The voids considered here are regions with matter density Omega ~ 0 and expansion rate Ht ~ 1, while the walls are regions with matter density Omega ~ 1 and expansion rate Ht ~ 2/3. | method 0.92 | 0.37 | 0.02 | asserted 0.99 |
| s2 | The results indicate that both the volume-average shear and the variance of the expansion rate grow proportional to the sharpness of the transition and diverge in the limit of a step function, but, for realistic-sized voids, are virtually independent of the size of the void. | result 1.00 | 0.94 | 0.32 | hedged possibility 0.88 |
| s3 | However, the backreaction, given by the difference of the variance and the shear, has a finite value in the step-function limit. | result 1.00 | 0.92 | 0.18 | asserted 1.00 |
| s4 | By comparing the exact result for the backreaction to the case where the shear is neglected by treating the voids and walls as separate Friedmann-Robertson-Walker models, we find that the shear suppresses the backreaction by a factor of (r_0/t_0)^2, the squared ratio of the void size to the horizon size. | result 1.00 | 0.96 | 0.50 | asserted 1.00 |
| s5 | This exemplifies the importance of using the exact solution for the interface between the regions of different expansion rates and densities. | interpretation 1.00 | 0.70 | 0.10 | asserted 0.98 |
| s6 | The suppression is justified to hold also for a network of compensated voids, but may not hold if the universe is dominated by uncompensated voids. | limitation 0.60 | 0.72 | 0.18 | hedged possibility 0.93 |
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Original abstract
Using the spherically symmetric inhomogeneous Lemaitre-Tolman-Bondi dust solution, we study how the shear and the backreaction depend on the sharpness of the spatial transition between voids and walls and on the size of the voids. The voids considered here are regions with matter density Omega ~ 0 and expansion rate Ht ~ 1, while the walls are regions with matter density Omega ~ 1 and expansion rate Ht ~ 2/3. The results indicate that both the volume-average shear and the variance of the expansion rate grow proportional to the sharpness of the transition and diverge in the limit of a step function, but, for realistic-sized voids, are virtually independent of the size of the void. However, the backreaction, given by the difference of the variance and the shear, has a finite value in the step-function limit. By comparing the exact result for the backreaction to the case where the shear is neglected by treating the voids and walls as separate Friedmann-Robertson-Walker models, we find that the shear suppresses the backreaction by a factor of (r_0/t_0)^2, the squared ratio of the void size to the horizon size. This exemplifies the importance of using the exact solution for the interface between the regions of different expansion rates and densities. The suppression is justified to hold also for a network of compensated voids, but may not hold if the universe is dominated by uncompensated voids.
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References
How the literature uses it
6 arXiv papers cite this in 14 sentences
Papers citing it · 8 sentences without a confident label
Contests it 1 paper
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We have also shown (sections 11 and 12) that the existence of effective acceleration is very restrictive for domains in collapsing configurations, which disproves an earlier claim by Räsänen in [ref] that gravitational collapse favors this acceleration (this was also proven to be mistaken in [ref] ).
Back-reaction and effective acceleration in generic LTB dust models · Final discussion. · contest 0.94
Builds on it 1 paper
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We have generalized the analytic approach of Ref. [ref] or paper I in studying the role of shear in the cosmological backreaction problem.
On the role of shear in cosmological averaging II: large voids, non-empty voids and a network of different voids · Summary of the results · 0.94
Cites it as background 1 paper
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More recently, the role of shear in estimating the magnitude of back–reaction was examined by Mattsson and Mattsson for central voids in hyperbolic models in [ref] , while conditions on the sign of the back–reaction were given by Sussman for generic LTB models in [ref] and on the effective acceleration in restricted situations in [ref] .
Back-reaction and effective acceleration in generic LTB dust models · Introduction. · 0.93
What it cites · 37 sentences without a confident label
Uses the method or tool of 2 papers
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By applying the averaging ([ref]) to the scalar parts of the Einstein equation in an irrotational dust universe, we obtain the Buchert equations [ref] : [equation] where the regional scale factor is defined as [equation] and the backreaction term [equation] quantifies the difference of the time evolution of the averages relative to the homogeneous quantities in an FRW dust universe.
On average properties of inhomogeneous fluids in general relativity I: dust cosmologies · Scalar averaging · 0.96
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Let us compare the exact solution of the void-wall configuration to a model where the walls and voids are treated as separate FRW regions, following the idea outlined in Ref. [ref] .
Accelerated expansion from structure formation · Comparison with a model that consists of two disjoint FRW solutions · 0.83
Cites as background 2 papers
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The exact spherically symmetric dust solution of general relativity was discovered by Lemaître in 1933 [ref] and is now commonly known as the LTB metric: [equation] where $A'(r,t) ≡∂_r A(r,t)$, $k(r)$ is related to the curvature of the spatial sections and $A(r,t)$ is determined by the Friedmann-like evolution equation [ref] [equation] where $H(r,t) ≡∂_t A(r,t)/A(r,t) ≡Ȧ(r,t)/A(r,t)$, $H_0(r) ≡H(r,t_0)$ and $Ω_0(r)$ are boundary condition functions specified on a spatial hypersurface $t=t_0$ that determine the radial inhomogeneity profile, while the freedom to choose the function $A_0(r) ≡A(r,t_0)$ corresponds to the scaling of the $r$-coordinate, used in this work to set [equation] The curvature function $k(r)$ in the metric ([ref]) is related to these by [equation] and the boundary condition function $Ω_0(r)$ is related to the physical matter density $ρ(r,t)$ on the $t=t_0$ hypersurfac
The effect of inhomogeneous expansion on the supernova observations · LTB solution · 0.91
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An unresolved issue in cosmology is the effect of the structure formation on the cosmological observations beyond perturbative analysis of the Friedmann-Robertson-Walker or FRW models [ref] .
Applicability of the linearly perturbed FRW metric and Newtonian cosmology · Introduction · 0.99
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