Diagrammatic Effective Field Theory Approach to Coalescing Binary Systems in General Relativity and Gravitational Waves Phenomenology

In this thesis we study the dynamics of a gravitationally bound binary system composed of two spinless compact objects, which could be black holes or neutron stars, in the post-Newtonian (PN) approximation scheme of general relativity. The predictions obtained within this scheme have already been fundamental for the observation of gravitational waves by the LIGO-Virgo-KAGRA collaboration, yet an improvement of their accuracy will be of uttermost importance to match the precision of future gravitational wave observatories, such as Einstein Telescope, Cosmic Explorer and LISA. Specifically in this work we employ an effective field theory approach to the gravitational dynamics, applying modern diagrammatic techniques to address the computation of the post-Newtonian corrections: these techniques have been first developed in the context of quantum field theory for the evaluation of elementary particles scattering amplitudes, yet recently they have been successfully applied also to the study of coalescing binary systems in general relativity. Using these techniques we thoroughly derive the corrections to the Lagrangian of the binary system up to the 2.5PN order (v^5), i.e. at next-to-next-to-leading order in the conservative sector and at leading order in the dissipative sector. The former sector includes corrections to the binding energy of the binary system, whereas the latter encodes radiation-reaction effects. From these results then we analytically compute the observable gravitational wave. To evaluate the conservative diagrams we have also developed a Mathematica code, which we apply as well to evaluate some selected conservative diagrams first contributing at 7PN order (v^14), so N^7LO corrections to the Newtonian potential. Finally, we perform a Fisher matrix forecast on the precision with which the future space-based LISA gravitational wave observatory will be able to constrain possible deviations from general relativity during the early inspiral phase of compact binary systems. In particular we introduce a parametric deformation of the post-Newtonian expression for the phase of the emitted gravitational waves, finding that it may be possible to constrain relative deviations from the post-Newtonian coefficients ranging from O(0.1) for the 2PN coefficients to O(0.001) for the leading order one. Throughout this thesis we review many of the needed topics and explicitly evaluate most of the necessary results, with the aim of presenting an accessible and self-contained exposition, spanning from the derivation of the post-Newtonian corrections to their application in a phenomenological analysis. The approach presented in this thesis could possibly be extended to modified theories of gravity as well.