Testing the nature of black holes with gravitational waves

What is the nature of the dark, massive ultracompact objects that populate our Universe? This question is the guiding line for the work developed in this thesis. Black holes are an extremely successful model to describe these compact objects and are supported by a solid theoretical foundation and consistent with all astrophysical observations. However, black holes have several theoretical problems that are not fully understood and could potentially be an issue for the self-consistency of General Relativity. One way to evade their outstanding issues is to consider that some new physics prevents the full collapse to a black hole and an exotic compact object is formed. Quantifying the evidence for horizons is, therefore, an essential and pressing task in future gravitational-wave observations. In this thesis, we investigate a wide variety of phenomenological effects of exotic compact objects and put them to test against the behavior of black holes. Some particularly relevant quantities that we study here are the multipole moments of the object, which may differ significantly from those of black holes and provide a good way to test the nature of compact objects in a binary. We study in particular the multipole moments of (i) fuzzballs - a multicenter microstate geometry arising from string theory - finding that the multipolar structure is much richer than Kerr and can be used to put constraints on these models with future gravitational wave observations; (ii) soft" exotic compact objects - for which the curvature at the surface is comparable with the corresponding curvature at the horizon, and finding that the multipole moments that are not spin-induced are strongly suppressed in the black-hole limit suggesting that their detection is challenging but possibly feasible with next-generation gravitational-wave detectors; (iii) neutron stars with quadrupolar deformations, where the latter is induced by an anisotropic crust (modeled by a thin shell), finding indications that a more complete of study of elastic properties of neutron stars may be required to understand the gravitational-wave signatures of coalescing neutron-star binaries. We also considered a particular model of an exotic compact object with self-gravitating anisotropic matter, where we developed the first fully covariant study for this type of object. We find that anisotropic stars have interesting phenomenological signatures, such as tidal Love numbers and gravitational echoes, and can evade some of the most pressing challenges that other exotic compact objects face when attempting to mimic the black hole model. The fully covariant framework suggest that these objects may be used to study numerically the behavior of ultracompact objects