On the role of shear in cosmological averaging

Abstract

Using the spherically symmetric inhomogeneousLemaȋtre-Tolman-Bondi dust solution, we study how the shear andthe backreaction depend on the sharpness of the spatial transitionbetween voids and walls and on the size of the voids. The voidsconsidered here are regions with matter density Ω0 ≃ 0 and expansion rate H0t0 ≃ 1, while the walls areregions with matter density Ω0 ≃ 1 and expansion rateH0t0 ≃ 2/3. The results indicate that both thevolume-average shear and the variance of the expansion rate growproportional to the sharpness of the transition and diverge in thelimit of a step function, but, for realistic-sized voids, arevirtually independent of the size of the void. However, thebackreaction, given by the difference of the variance and theshear, has a finite value in the step-function limit. By comparingthe exact result for the backreaction to the case where the shearis neglected by treating the voids and walls as separateFriedmann-Robertson-Walker models, we find that the shearsuppresses the backreaction by a factor of (r0/t0)2, thesquared ratio of the void size to the horizon size. Thisexemplifies the importance of using the exact solution for theinterface between the regions of different expansion rates anddensities. The suppression is justified to hold also for a networkof compensated voids, but may not hold if the universe isdominated by uncompensated voids.