On the role of shear in cosmological averaging

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) similar or equal to 0 and expansion rate H(0)t(0) similar or equal to 1, while the walls are regions with matter density Omega(0) similar or equal to 1 and expansion rate H(0)t(0) similar or equal to 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.