Modified Gravity Theories Offer New Insights into Dark Energy and Wormholes.

Modified theories of General Relativity represent a significant area of contemporary cosmological research, driven by the enduring mysteries of dark energy, dark matter, and the theory’s inherent limitations. Kavya N. S. and Venkatesha, both from Kuvempu University, systematically investigate several such modifications utilising the mathematical language of differential geometry – a branch of mathematics concerned with the properties of curves and surfaces – to explore alternative gravitational frameworks. Their work centres on extending the Einstein-Hilbert action, the foundational equation of General Relativity, by incorporating higher-order terms or modifying the underlying geometric structure through concepts like torsion and non-Levi-Civita connections. This thesis presents a detailed analysis of various modified models, ranging from parameterisations of the deceleration parameter within f(Q) gravity, where Q represents a geometrical quantity related to non-metricity, to investigations of wormhole solutions and the implications for Big Bang Nucleosynthesis within f(T) gravity, where T denotes torsion. The research employs observational data and analytical techniques to constrain these models and assess their viability against cosmological observations.

This thesis investigates modifications to General Relativity, addressing limitations in explaining phenomena such as dark energy and dark matter. The research employs differential geometry as a foundational framework, exploring models that alter the gravitational Lagrangian through the introduction of torsion or non-Levi-Civita connections. Specifically, the work examines several modified theories, including f(Q), f(Q, L_m), f(R, L_m), and f(T) gravity, where Q represents non-metricity, R the Ricci scalar, L_m the matter Lagrangian, and T the torsion scalar.

The study develops a novel parametrisation of the deceleration parameter, constrained using Markov Chain Monte Carlo methods and observational data, and applies this to f(Q) gravity. Furthermore, the LambdaCDM model is embedded within f(Q, L_m) gravity, yielding analytical solutions and observational validation through cosmographic analysis. Anisotropic behaviour is investigated using Bianchi-I spacetime within f(R, L_m) gravity, again with observational constraints. The research extends to the exploration of wormhole solutions in both f(Q, T) and f(R, L_m) gravity, incorporating conformal symmetries and non-commutative geometry to analyse shape functions, energy conditions, and stability. Finally, the implications of f(T) gravity for Big Bang Nucleosynthesis are assessed, with hybrid models constrained using early- and late-time data and validated through cosmography.

These investigations demonstrate the potential of modified gravity theories to address cosmological challenges and provide alternative explanations for observed phenomena. Future work could focus on refining these models with more precise observational data, exploring the connections between different modified gravity theories, and investigating the implications for gravitational wave astronomy and black hole physics. The continued development of these theoretical frameworks, coupled with ongoing observational efforts, promises to deepen our understanding of gravity and the universe.