Connection Formulations of Cosmology and Unimodular Gravity

In this thesis we examine how chiral connection formulations of gravity can be applied in the fields of quantum cosmology and unimodular gravity.
Chiral connection formulations are reformulations of general relativity (GR) in which the central dynamical field is a gauge connection a la Yang--Mills, as opposed to a metric tensor or a tetrad.
Here, the fields are assumed to be complex, and we require further reality conditions to get solutions for Lorentzian GR.
These formulations are derived from the (chiral) Plebanski formulation by integrating out variables.
Unimodular gravity refers to formulations in which the cosmological constant arises as an integration constant, which can be achieved by fixing the value of the metric determinant with a dynamical constraint.
We construct actions for chiral connection formulations of unimodular gravity.
We then derive canonical formulations for these actions, yielding constrained Hamiltonian systems whose constraint algebras resemble somewhat modified versions of the constraint algebra from GR.
Following this, we examine the classical dynamics of the spatially homogeneous Bianchi I and IX models within a certain chiral connection formulation.
We focus on approaches to implementing reality conditions, including an approach where they are treated as second class constraints in Dirac's formalism.
We also see how one can derive Lorentzian solutions from Euclidean signature solutions through a kind of Wick rotation.
Finally, we examine the quantum cosmology of a homogeneous and isotropic FLRW type spacetime from the perspective of Krasnov's pure connection formulation of GR.
We derive an established result for a two-point function from a novel perspective.