The Study on Modified Theories of General Relativity: A Differential Geometric Approach (2507.04031v1)

Abstract: The introduction of General Relativity (GR) in 1915 revolutionized our understanding of gravity, but over time, its limitations in explaining phenomena like dark energy, dark matter, and quantum gravity have motivated alternative theories. Early modifications, such as Weyl's 1919 proposal, focused on adding higher-order terms to the Einstein-Hilbert action. GR's non-renormalizability further strengthened the case for extending it. A central theme of modern gravity research is modifying the geometric structure, often by changing the gravitational Lagrangian. This leads to theories such as teleparallel and symmetric teleparallel gravity, utilizing torsion or non-Levi-Civita connections, with differential geometry providing the essential framework. This thesis explores several modified gravity models. Chapter 1 introduces necessary mathematical tools. Chapter 2 develops a novel parametrization of the deceleration parameter, constrained using MCMC and observational data, and applies it to f(Q) gravity. Chapter 3 embeds the LambdaCDM model into f(Q, L_m) gravity with non-minimal coupling, producing analytic solutions and matching observations through cosmographic analysis. Chapter 4 considers Bianchi-I spacetime in f(R, L_m) gravity with observational constraints to measure anisotropy. Chapter 5 presents wormhole solutions in f(Q, T) gravity with conformal symmetries. Chapter 6 explores wormholes in f(R, L_m) with non-commutative geometry, analyzing shape functions, energy conditions, and stability. Chapter 7 studies Big Bang Nucleosynthesis in f(T) gravity, constraining hybrid models using early- and late-time data, and validating intermediate epochs via cosmography.