Rethinking the basics of the unmodelled analysis of gravitational-wave transients in a modern (and future) perspective

This Thesis covers some important topics concerning gravitational-wave data analysis. Particularly, it focuses on unmodelled analysis, which does not assume any waveform template for the signal, and reconstructs it by evaluating only the coherence of data among different detectors. The core tool of unmodelled analysis is the time-frequency transform used to represent data. Therefore, following the initial chapters in which I introduce gravitational waves from both a theoretical and experimental point of view, and in which I explain the generalities of unmodelled analysis, the Thesis covers the topic of time-frequency transforms, with particular reference to wavelets. In this framework, I focus on the Q-transform, a time-frequency transform based on Morlet wavelets whose Gaussian envelope is parameterized by the Q parameter: this parameter allows to balance time and frequency resolution given by the Gabor-Heisenberg uncertainty principle, whose lower bound which is attained by the Morlet wavelets, making the Q-transform an optimal tool for time-frequency representation. Despite this desirable property, the Q-transform until recently lacked an exact and effective inversion formula. In this Thesis I apply an alternative inversion formula to Morlet wavelets, including their Q-parameterized version (the "wavelet Q-transform"): above all, from the inversion formula I demonstrate an exact and effective time-frequency denoising formula, which allows to disentangle signal and noise according to their spread in the time-frequency plane. I also extend the wavelet Q-transform by introducing a new parameter p, which introduces a frequency-dependent frequency modulation: I call this extension the "wavelet Qp-transform", finding that it is equipped with the same inversion and denoising formula presented above. The Qp-transform is better suited to gravitational-wave signals rapidly evolving in frequency, and can represent those signals in a more compact way compared to the wavelet Q-transform. I test the wavelet Q- / Qp-transform and their inversion formula showing that these are well-suited for gravitational-wave unmodelled analysis. The second part of the Thesis is focused on the interferometer response: indeed, the current detector response implemented in gravitational-wave data analysis (both modelled and unmodelled) is obtained assuming that the wavelength of the detected gravitational waves is much larger than the interferometer arm, which is the so-called long-wavelength approximation. This approximation works well with the current detectors, but that could change for the next generation of detectors, such as Cosmic Explorer and Einstein Telescope, which are planned to have far longer arms compared to the current detectors. Therefore, starting from the current literature which considers that problem mainly in the frequency domain, I obtain the time-domain interferometer response beyond the long-wavelength approximation. Then, after extending the unmodelled-analysis formalism to the detector response without the long-wavelength approximation, I perform some simulations involving the next generation of detectors to understand when and how much the long-wavelength approximation can bias unmodelled analysis for such detectors, particularly from the point of view of waveform reconstruction. Finally, I analyse the case of the (possible) triangular design for Einstein Telescope, with a detector made by three 60° interferometers making an equilateral triangle. First, I evaluate its detector response beyond the long-wavelength approximation. Then, I analyse one of the main properties of Einstein Telescope triangular design, i.e., the fact that by summing the responses of the three interferometers the signal cancels, showing that this no more holds at those frequencies (more or less above 1000 Hz) for which we need to drop the long-wavelength approximation.