This Thesis covers some important topics concerning gravitationalwave 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 timefrequency 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 timefrequency transforms, with particular reference to wavelets. In this framework, I focus on the Qtransform, a timefrequency 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 GaborHeisenberg uncertainty principle, whose lower bound which is attained by the Morlet wavelets, making the Qtransform an optimal tool for timefrequency representation. Despite this desirable property, the Qtransform until recently lacked an exact and effective inversion formula. In this Thesis I apply an alternative inversion formula to Morlet wavelets, including their Qparameterized version (the "wavelet Qtransform"): above all, from the inversion formula I demonstrate an exact and effective timefrequency denoising formula, which allows to disentangle signal and noise according to their spread in the timefrequency plane. I also extend the wavelet Qtransform by introducing a new parameter p, which introduces a frequencydependent frequency modulation: I call this extension the "wavelet Qptransform", finding that it is equipped with the same inversion and denoising formula presented above. The Qptransform is better suited to gravitationalwave signals rapidly evolving in frequency, and can represent those signals in a more compact way compared to the wavelet Qtransform. I test the wavelet Q / Qptransform and their inversion formula showing that these are wellsuited for gravitationalwave unmodelled analysis. The second part of the Thesis is focused on the interferometer response: indeed, the current detector response implemented in gravitationalwave 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 socalled longwavelength 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 timedomain interferometer response beyond the longwavelength approximation. Then, after extending the unmodelledanalysis formalism to the detector response without the longwavelength approximation, I perform some simulations involving the next generation of detectors to understand when and how much the longwavelength 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 longwavelength 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 longwavelength approximation.
This Thesis covers some important topics concerning gravitationalwave 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 timefrequency 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 timefrequency transforms, with particular reference to wavelets. In this framework, I focus on the Qtransform, a timefrequency 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 GaborHeisenberg uncertainty principle, whose lower bound which is attained by the Morlet wavelets, making the Qtransform an optimal tool for timefrequency representation. Despite this desirable property, the Qtransform until recently lacked an exact and effective inversion formula. In this Thesis I apply an alternative inversion formula to Morlet wavelets, including their Qparameterized version (the "wavelet Qtransform"): above all, from the inversion formula I demonstrate an exact and effective timefrequency denoising formula, which allows to disentangle signal and noise according to their spread in the timefrequency plane. I also extend the wavelet Qtransform by introducing a new parameter p, which introduces a frequencydependent frequency modulation: I call this extension the "wavelet Qptransform", finding that it is equipped with the same inversion and denoising formula presented above. The Qptransform is better suited to gravitationalwave signals rapidly evolving in frequency, and can represent those signals in a more compact way compared to the wavelet Qtransform. I test the wavelet Q / Qptransform and their inversion formula showing that these are wellsuited for gravitationalwave unmodelled analysis. The second part of the Thesis is focused on the interferometer response: indeed, the current detector response implemented in gravitationalwave 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 socalled longwavelength 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 timedomain interferometer response beyond the longwavelength approximation. Then, after extending the unmodelledanalysis formalism to the detector response without the longwavelength approximation, I perform some simulations involving the next generation of detectors to understand when and how much the longwavelength 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 longwavelength 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 longwavelength approximation.
Rethinking the basics of the unmodelled analysis of gravitationalwave transients in a modern (and future) perspective / Virtuoso, Andrea.  (2024 May 17).
Rethinking the basics of the unmodelled analysis of gravitationalwave transients in a modern (and future) perspective
VIRTUOSO, ANDREA
20240517
Abstract
This Thesis covers some important topics concerning gravitationalwave 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 timefrequency 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 timefrequency transforms, with particular reference to wavelets. In this framework, I focus on the Qtransform, a timefrequency 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 GaborHeisenberg uncertainty principle, whose lower bound which is attained by the Morlet wavelets, making the Qtransform an optimal tool for timefrequency representation. Despite this desirable property, the Qtransform until recently lacked an exact and effective inversion formula. In this Thesis I apply an alternative inversion formula to Morlet wavelets, including their Qparameterized version (the "wavelet Qtransform"): above all, from the inversion formula I demonstrate an exact and effective timefrequency denoising formula, which allows to disentangle signal and noise according to their spread in the timefrequency plane. I also extend the wavelet Qtransform by introducing a new parameter p, which introduces a frequencydependent frequency modulation: I call this extension the "wavelet Qptransform", finding that it is equipped with the same inversion and denoising formula presented above. The Qptransform is better suited to gravitationalwave signals rapidly evolving in frequency, and can represent those signals in a more compact way compared to the wavelet Qtransform. I test the wavelet Q / Qptransform and their inversion formula showing that these are wellsuited for gravitationalwave unmodelled analysis. The second part of the Thesis is focused on the interferometer response: indeed, the current detector response implemented in gravitationalwave 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 socalled longwavelength 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 timedomain interferometer response beyond the longwavelength approximation. Then, after extending the unmodelledanalysis formalism to the detector response without the longwavelength approximation, I perform some simulations involving the next generation of detectors to understand when and how much the longwavelength 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 longwavelength 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 longwavelength approximation.File  Dimensione  Formato  

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