Usseglio, Davide (2025) Analytical Approaches for the Relativistic Two-Body Scattering. [Tesi di dottorato]

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Abstract

Detection of scattering of binary black holes is incredibly hard, hence the main focus of the community has been for several years the characterization of gravitational wave signals from binaries in circular or eccentric motion. However by leveraging the relation that is known under the name of bound-to-unbound mapping, it should be in principle possible to get information about bound observables from the unbound ones at very high order in perturbation theory, potentially informing numerical simulations and/or waveform generators. This is (one of) the main motivations behind the interest in studying the scattering of binary black holes. Scattering can be investigated analytically through the post-Minkwoskian (PM) approximation: a weak field approximation that computes corrections around the flat, Minkowski, spacetime. This is the natural ground for modeling an hyperbolic encounter that, in this approximation, could be seen as deviation from the straight line motion in presence of a black hole. An alternative (and independent) expansion, the post-Newtonian (PN) approximation, has been used historically to investigate bound orbits. It can be naively defined by restricting the attention to binary systems, where the relative velocity is small relative to the speed of light. Because of the virial theorem in bound systems, a small relative velocity is immediately translated in a large relative distance approximation. This means that, for bound systems, the PM and PN approximations are degenerate while, in a scattering scenario, the two expansions can be performed independently. Together with these approaches, a third perturbative scheme can be used to obtain analytical results: Self Force. This approximation contains information that are exact in both PN and PM sense, and it has been used in the past to get analytical expressions up to incredibly high orders in PN for bound orbits. In this Thesis, it will be discussed how to obtain analytically scattering observables by using Self Force techniques, generalizing the usual procedure to the case of a generic, equatorial orbit. These methodologies will be applied to study the scattering of a scalar charge off a Schwarzschild black hole. This methodology will also be applied to compute the fluxes and the waveform for a gravitational perturbation. A parallel project, focused on scattering, will also be discussed at the end of this Thesis: it will be presented how to compute the Quasi-Keplerian parameters for parametrizing the hyperbolic orbit in Scalar Tensor theories.

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