Theoretical and Phenomenological Aspects of Projective-Invariant Metric-Affine Theories of Gravity

Three models of parity violating modified gravity are considered in the metric-affine approach, their formal properties are discussed and new solutions are derived, together with predictions in cosmology, black holes and gravitational waves settings. An extension to generic metric-affine spacetimes of the Holst, Nieh-Yan and Chern-Simons topological terms is provided, accounting for the presence of nonmetricity and recovering the invariance under projective transformations acting on the affine connection. While the Holst term is projective invariant and vanishing on-half shell, even in presence of nonmetricity, the latter is responsible for the breaking of projective symmetry and the spoiling of the topological character both in the Nieh-Yan and Chern-Simons case. A generalization of these two topological invariants is provided, allowing to recover both properties independently. Specific models featuring the generalized terms are considered and the role of projective symmetry in relation to the absence of dynamical instabilities is thoroughly discussed. The absence of ghost degrees of freedom is proved for all three models. Ostrogradski instabilities in the Nieh-Yan model are ruled out by its dynamical equivalence with DHOST theories while new methods to avoid higher-order derivatives introduced by the Chern-Simons modification are devised. Then, several solutions are derived. New analytical hairy black hole solutions are presented for the Holst model and their thermodynamics is analysed, highlighting modifications to the black hole entropy and the role of the solution within a system of conjectures recently proposed in literature in the extended phase space approach. In models featuring the generalized Nieh-Yan term, semi-analytic, anisotropic cosmological solutions are obtained, exhibiting a big-bounce regularization of the initial singularity. The presence of future finite-time singularities is pointed out and their physical viability in terms of geodesic completeness and well-behaved scalar perturbations is proved. In the generalized Chern-Simons theory modifications to black hole perturbations and gravitational waves are derived. The quasinormal modes of Schwarzschild black holes are computed, showing their deviations from both General Relativity and the metric version of Chern-Simons gravity. New exact cosmological solutions with non-trivial affine structure are used as background configurations to study the propagation of gravitational waves, establishing the phenomenon of gravitational birefringence in metric-affine Chern-Simons gravity. The propagation in matter is also investigated proving the existence of gravitational Landau damping. Although observable in principle, the magnitude of the effects is too low for them to be tested given the sensitivity of present day experiments.