Mathematical tools and signal processing algorithms for the study of gravitational waves polarization
This thesis is an interdisciplinary project aiming at proposing new methodologies and algorithms to characterize the polarization of non-stationary polarized signals and to apply these new tools to the context of gravitational wave astronomy. The direct observation of gravitational waves made possible by the advanced LIGO and Virgo detectors constitutes a paradigm shift for the study of compact astrophysical objects such as black holes and neutron stars. Analysis of the large volume of data from these detectors has so far focused on the morphology of the recorded waveform, from which information about the nature of the source can be extracted. The polarization of the waves has received less attention because the number of detectors was insufficient to draw accurate conclusions. However, polarization information is of interest for some astrophysical sources. For example, for mergers of compact binary star systems, the precession of the orbital plane results in a specific evolution of the polarization pattern. This thesis starts from the fundamental theoretical aspects of the representation and characterization of polarized signals to develop analysis and synthesis tools adapted to the context of the considered application. The results presented are of three kinds. First, the different representations of amplitude, frequency and polarization modulated signals are reviewed, evaluating their usefulness for the analysis and synthesis of these signals. This review highlights the problems caused by the degeneracy of certain representations, specifies the conditions of its occurrence and proposes ways to remedy it. On the basis of this study, generative machine learning models are built, and applied to the fast computation of gravitational waveforms, thus allowing the acceleration of the inference of the source parameters. This generative model is proposed both for binary black hole sources without precession and with precession, and its accuracy is evaluated in each case. Finally, new regularization principles based on polarization a priori are introduced to improve the reconstruction of the two signal components from observational data. The method is evaluated on realistic synthetic data. It allows to target the analysis on certain source categories associated with a particular polarization.