Metric-Affine Gravity and Cosmology/Aspects of Torsion and non-Metricity in Gravity Theories

TL;DR

This thesis explores Metric-Affine Gravity theories, focusing on torsion and non-metricity, providing new methods to solve for affine connections and applying these to cosmological models with detailed equations.

Contribution

It introduces a novel step-by-step method to solve for affine connections in non-Riemannian geometries and applies this to cosmology, including a new general form of the Raychaudhuri equation.

Findings

Derived the most general form of the Raychaudhuri equation with torsion and non-metricity.

Developed a systematic approach to solve for affine connections in non-Riemannian geometries.

Analyzed a conformally invariant f(R) gravity model with an undetermined scalar degree of freedom.

Abstract

This Thesis is devoted to the study of Metric-Affine Theories of Gravity and Applications to Cosmology. The thesis is organized as follows. In the first Chapter we define the various geometrical quantities that characterize a non-Riemannian geometry. In the second Chapter we explore the MAG model building. In Chapter 3 we use a well known procedure to excite torsional degrees of freedom by coupling surface terms to scalars. Then, in Chapter 4 which seems to be the most important Chapter of the thesis, at least with regards to its use in applications, we present a step by step way to solve for the affine connection in non-Riemannian geometries, for the first time in the literature. A peculiar f(R) case is studied in Chapter 5. This is the conformally (as well as projective invariant) invariant theory f(R)=a R^{2} which contains an undetermined scalar degree of freedom. We then turn our attention to Cosmology with torsion and non-metricity (Chapter 6). In Chapter 7, we formulate the necessary setup for the $1+3$ splitting of the generalized spacetime. Having clarified the subtle points (that generally stem from non-metricity) in the aforementioned formulation we carefully derive the generalized Raychaudhuri equation in the presence of both torsion and non-metricity (along with curvature). This, as it stands, is the most general form of the Raychaudhuri equation that exists in the literature. We close this Thesis by considering three possible scale transformations that one can consider in Metric-Affine Geometry.