Compact objects in modified gravity: junction conditions and other viability criteria

This thesis aims at improving our understanding of the strong-field regime of gravity, where deviations from General Relativity (GR) are expected to be found on theoretical grounds. In particular, we have been concerned with the formulation and application of junction conditions (which govern stellar collapse, black-hole formation and the dynamics of more exotic compact objects, such as thin-shell wormholes), as well as with other viability criteria necessary for modified gravity theories and their solutions to be physically admissible (for instance, compatibility with gravitational-wave observations, stability or the avoidance singularities). First, we have studied stellar collapse in \(f(R)\) gravity. By means of a systematic treatment of the relevant junction conditions, we have proven a series of stringent no-go results on the exterior space-time, which entail that some paradigmatic classes of vacuum metrics cannot represent space-time outside a collapsing dust star in metric \(f(R)\) gravity. This could lead to potentially observable deviations with respect to GR. We have also provided an exhaustive derivation of the junction conditions in bi-scalar Poincaré gauge gravity, finding that the matching interface is allowed to host surface axial spin monopoles, as well as matter thin shells and double layers. This singular structure is richer than its counterparts in GR or \(f(R)\) gravity, leading to possible interesting applications. In addition, we have had to resolve some procedural and mathematical subtleties in the junction-condition formalism. Finally, we have examined a wide class of metric \(f(R)\) models, the so-called '\(R_0\)-degenerate models,' solved by all metrics with constant scalar curvature \(R=R_0\). We have shown that they feature all sorts of shortcomings (such as previously-unforeseen strong-coupling instabilities) rendering their viability extremely limited.