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On the role of shear in cosmological averaging II: large voids, non-empty voids and a network of different voids

We study the effect of shear on the cosmological backreaction in the context of matching voids and walls together using the exact inhomogeneous Lemaȋtre-Tolman-Bondi solution. Generalizing JCAP 1010 (2010) 021, we allow the size of the voids to be arbitrary and the densities of the voids and walls to vary in the range 0 ⩽ Ωv ⩽ Ωw ⩽ 1. We derive the exact analytic result for the backreaction and consider its series expansion in powers of the ratio of the void size to the horizon size, r0/t0. In addition, we deduce a very simple fitting formula for the backreaction with error less than 1% for voids up to sizes r0≳t0. We also construct an exact solution for a network of voids with different sizes and densities, leading to a non-zero relative variance of the expansion rate between the voids. While the leading order term of the backreaction for a single void-wall pair is of order (r0/t0)2, the relative variance between the different voids in the network is found to be of order (r0/t0)4 and thus very small for voids of the observed size. Furthermore, we show that even for very large voids, the backreaction is suppressed by an order of magnitude relative to the estimate obtained by treating the walls and voids as disjoint Friedmann solutions. Whether the suppression of the backreaction due to the shear is just a consequence of the restrictions of the used exact models, or a generic feature, has to be addressed with more sophisticated solutions.