Exact Solutions of Einstein Field Equations in Cosmology

Abstract

The present paper gives a detailed mathematical discussion of the exact solutions to the Einstein Field Equations (EFE) in the context of the contemporary theoretical cosmology. The extremely non-linear, coupled system of the EFE has known that it is impossible to obtain an exact analytical solution without first imposing extreme geometric symmetries on the spacetime manifold. The paper will delve into these mathematical symmetries in a systematic fashion starting with the derivation of the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric assuming the Cosmological Principle of space homogeneity and isotropy. To derive the standard Friedmann equations, we will strictly build up the corresponding metric tensors, Christoffel symbols and Ricci curvature tensors. Moreover, the article explores cosmologically constant solutions of vacuums and the exponential growth of de Sitter spacetimes and the mechanism of Anti-de Sitter theories. In order to solve the complicated dynamical problems in the early universe and the local structural anomalies, our assumptions about the Cosmological Principle are loosened to mathematically model anisotropic expansion with the Kasner metric and radial inhomogeneity through the Lemaitre-Tolman-Bondi (LTB) model. Lastly, the paper generalizes these classical solutions to Extended Theories of Gravity, i.e. the modified field equations in f(R) gravity models of late-time acceleration of the Universe. Finally, this study shows the importance of these very mathematical models as the indispensable tool to the study of the dynamical evolution and the geometry of the universe.