Modified Gravity with Torsion

We are at a point in time when alternative gravitational theories are beginning to be constrained by high precision cosmological and astrophysical data. The work in this thesis focuses on applications of a particular modified gravity theory, the extended Weyl Gauge Theory (eWGT), that was recently developed by Prof. Lasenby and Prof. Hobson. The applications corresponds to subsets of the full theory that are applicable to different cosmological and astrophysical sectors. We start by investigating a simplified scenario that simulates a famous alternative theory of gravity, Weyl$^2$ Gravity. Recently a couple of issues have been raised regarding the validity of the theory. Starting from a gauge theory perspective we bring a fresh contribution to the debate. We argue against the classical formulation by showing that the theory cannot support astrophysical matter (introduced by a perfect fluid). Furthermore we extend the theory and show that even if we allow torsion to be present we cannot reach a physical setup. In this process we have discovered interesting properties of the torsion field that could play an important role in generalised cosmological setups. In the next application we consider a cosmology dictated by a Riemann$^2$ Lagrangian that can accommodate only radiation. We find new physical behaviour in the perturbed regime that discretises the power spectrum. We prove that the setup admits gravitational waves. Finally, we construct a new Lagrangian theory for spinning fluids. We show that it is compatible with current literature for a flat space time. Considering its extensibility we believe that it can be widely used in future research