The Averaging Problem in Cosmology and Macroscopic Gravity

The dynamics of the universe are traditionally modelled by employing cosmological so- lutions to the Einstein field equations. In these solutions, the matter distribution is taken to be averaged over cosmological scales, and hence, the Einstein tensor needs to be av- eraged as well. To construct such an averaged theory of gravity, one needs a covariant averaging procedure for tensor fields. Macroscopic gravity (MG) is one such theory. It gives the macroscopic Einstein field equations (mEFEs) where the effects due to averag- ing are encapsulated in a correction term, the so-called back-reaction. This additional term accounts for the non-commutativity between the averaging operation and the calculation of Einstein tensor. In this dissertation, we analyse how to deal with inhomogeneities within macroscopic gravity. First, we model the inhomogeneities as linear perturbations around the spa- tially homogeneous Friedmann-Robertson-Lemaître-Walker (FLRW) geometry. Then, we analyse exact inhomogeneous models with plane and spherically symmetric geometries. We calculate the back-reaction in these models and analyse how it modifies observations done within them. First, we explore the application of the MG formalism to an almost-FLRW model. Namely, we find solutions to the field equations of MG taking the averaged universe to be almost- FLRW modelled using a linearly perturbed FLRW metric. We study several solutions with different functional forms of the metric perturbations including plane waves ansatzes. We find that back-reaction terms are present not only at the background level but also at perturbed level, reflecting the non-linear nature of the averaging process. To analyse how observations get modified by the back-reaction, we derive the expres- sions for distance measures in MG. We analyse two cases. In the first one, the back- reaction modifies distances only through the expansion history. In the second one, the back-reaction density parameter enters the distance formulae in such a way that, phe- nomenologically, it is degenerate with a spatial curvature. Turning to the perturbations, we derive an equation for growth of structure and analyse how back-reaction modifies the linear growth rate. Thus, the averaging effect can extend to both the expansion and the growth of structure in the universe. Then, we turn our attention to inhomogeneous models with plane and spherically sym- metric geometries. We calculate the MG correction term for such models and find that it takes the form of an anisotropic fluid with a qualitative behaviour of an effective cur- vature in the field equations. We categorise the solutions according to the source for the space-time – vacuum, dust and perfect fluid. Within these three categories, we treat, in detail, the cases of the static spherically symmetric vacuum solution (Schwarzschild exte- rior), the static spherically symmetric perfect fluid solutions (Schwarzschild interior and Tolman VII) and the non-static spherically symmetric dust solution (Lemaître-Tolman- Bondi (LTB)). This is a first step towards analysing back-reaction in inhomogeneous cos- mology with MG.