• black holes
• effective field theory
• general relativity
• Horndeski theory
• modied theories of gravity
• no-hair theorems
• odd sector
• perturbations
• quasi-normal modes
• Regge-Wheeler equation
• Schwarzschild spacetime
• static and spherically symmetric background
• symmetry breaking
• unitary gauge

The recent observation of gravitational waves has opened a new window for the investigation of our universe, also providing the first direct detection of black holes (BH). Gravitational waves, indeed, can provide information about the dynamics of gravitational systems in regimes in which general relativity (GR) has never been directly observed before, thus leading to new possible ways to distinguish between GR and modified theories of gravity. A noticeable example of such phenomena is provided by the quasi-normal modes (QNM), which are emitted, for example, in the ringdown phase of BH mergers. This work focuses on the analysis of the alternative theories of gravity in which a single real scalar field is added to the Einstein-Hilbert action; the approach chosen is model independent and the formalism developed can be used to describe a variety of theories as, for example, the Horndeski theory and its extensions, which provide a general setup, including various models as sub-cases such as: quintessence, P(X) theories, Brans-Dicke theory and Galileons. The aim of the thesis is to provide an effective description of how those modified gravitational theories introduce novel features which could be observed in the QNM spectrum. Thus, it provides the theoretical low energy description of the perturbations in gravitational theories where an extra real scalar degree of freedom is introduced, under the assumption that they allow for static and spherically symmetric background configurations. In order to make optimal use of the isometries of the physical system, a (3+1) spacetime decomposition is performed and the construction of the effective action describing the low energy regimes of perturbations of the aforementioned modified theories of gravity over static and spherically symmetric background solutions is illustrated. The similarities and differences with the effective field theory of inflation are extensively discussed, together with the motivation behind the use of such techniques. In a second part, the odd sector of the QNM spectrum in modified theories of gravity is analysed using the effective approach. In particular, in the thesis, it is shown that the new effects caused by the scalar field dynamics can be traced into two different contributions. One is the modification of the Schwarzschild background, while the other is the consequence of the more complex structure of the linearised equations compared to the GR one. While the general setup is provided, the technique is mainly applied to a hairy, static and spherically symmetric black hole solution known in the literature. The results show that in the odd sector the QNM spectrum is different from the one predicted by GR.