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Probing (beyond) general relativity with compact binaries and gravitational waves

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General relativity enjoys phenomenal success in agreeing with experiments and observations, but it must break down at some point. Astrophysics can give guidance for what type of theory may correct general relativity, if we know which phenomenology to look for. I will discuss the possible corrections to the structure of compact objects, the binary problem, and observations with pulsar timing and gravitational wave detection. These corrections are computed in specific effective theories, such as Einstein-dilaton-Gauss-Bonnet (EdGB) and dynamical Chern-Simons (dCS) gravity, and we may further generalize the common aspects of these theories to build a parametrized framework which captures a large number of effective field theories.