The strong gravitational field regime of compact objects beyond General Relativity

The maturity of experimental probes of the strong field regime of gravity, both in terms of sensitivity and number of observations, offers hope in discriminating General Relativity (GR) from alternative theories of gravity in the near future. At present, such probes include gravitational-wave observations with ground-based interferometers, 1.3 mm electromagnetic observations with Very-Large-Baseline Interferometry (VLBI) and binary pulsar observations. For these, a prominent role is played by compact objects, which source strong (and in some cases highly dynamical) gravitational fields, and therefore make up the main target of such observations. Most efforts so far have focused on computing strong field predictions for compact objects in GR. However, if one aims to fairly discriminate among them, the same predictions need to be obtained for theories beyond GR. The purpose of this Thesis is to explore various aspects of compact objects in theories beyond GR where strong gravitational fields are relevant. We begin by studying k-essence, a scalar-tensor theory motivated as a viable dynamical explanation for Dark Energy. Derivative self-interactions provide (through a kinetic screening mechanism) the suppression of the extra scalar force needed to satisfy local gravitational constraints. We first explore ways to ensure that the theory admits a well-posed initial-value problem, a mathematical property that is essential for obtaining predictions in the strong field with numerical relativity. We then show how some of the lessons learned for k-essence can also be applied to self-interacting massive vector theories. In addition, we explore the resilience of kinetic screening with different possibilities in which the scalar can couple to the matter sector. We then turn our attention to the question of whether gravity can be constrained with black hole images from VLBI, and whether this can be done in spite of uncertainties in the astrophysical modelling of the system. We present a proof-of-principle demonstration of a theory-agnostic framework to reconstruct simultaneously both the underlying geometry and accretion behind these images. Our framework makes use of a general parametrization for these properties and of a Principal Component Analysis to mitigate the degeneracies linked to the presence of a large number of parameters. Finally, we consider the question of whether quantum gravity can provide a resolution to the issue of singularities inside black holes. We do so in the context of (2+1) projectable Hořava gravity, a Lorentz-violating quantum gravity candidate that has been shown to be renormalizable (beyond power counting) and ultraviolet complete. We obtain all circularly-symmetric stationary solutions and show that, in spite of naive expectations, solutions that reduce to BHs at low energies remain singular in the interior.