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Gravitational Waves Signal Analysis: Matched Filtering, Typical Analyses and Beyond

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https://lib.uliege.be https://matheo.uliege.be Gravitational waves signal analysis: Matched filtering, typical analyses and beyond Auteur : Janquart, Justin Promoteur(s) : Cudell, Jean-Rene Faculté : Faculté des Sciences Diplôme : Master en sciences spatiales, à finalité approfondie Année académique : 2019-2020 URI/URL : https://github.com/lemnis12?tab=repositories; http://hdl.handle.net/2268.2/9211 Avertissement à l'attention des usagers : Tous les documents placés en accès ouvert sur le site le site MatheO sont protégés par le droit d'auteur. Conformément aux principes énoncés par la "Budapest Open Access Initiative"(BOAI, 2002), l'utilisateur du site peut lire, télécharger, copier, transmettre, imprimer, chercher ou faire un lien vers le texte intégral de ces documents, les disséquer pour les indexer, s'en servir de données pour un logiciel, ou s'en servir à toute autre fin légale (ou prévue par la réglementation relative au droit d'auteur). Toute utilisation du document à des fins commerciales est strictement interdite. Par ailleurs, l'utilisateur s'engage à respecter les droits moraux de l'auteur, principalement le droit à l'intégrité de l'oeuvre et le droit de paternité et ce dans toute utilisation que l'utilisateur entreprend. Ainsi, à titre d'exemple, lorsqu'il reproduira un document par extrait ou dans son intégralité, l'utilisateur citera de manière complète les sources telles que mentionnées ci-dessus. Toute utilisation non explicitement autorisée ci-avant (telle que par exemple, la modification du document ou son résumé) nécessite l'autorisation préalable et expresse des auteurs ou de leurs ayants droit. Université de Liège Master thesis Gravitational waves signal analysis: Matched filtering, typical analyses and beyond JANQUART Justin In Partial Fulfilment of the Requirements for the Degree of Master in Space Sciences Under the supervision of: CUDELL Jean-René Academic year 2019-2020 Student is not a container you have to fill but a torch you have to light up. —Albert Einstein Ideas and Opinions, 1954 Acknowledgements I would like to express my thankfulness to all the persons that have helped me and supported me during the completion of this work. In particular, I want to thank Professor Jean-René Cudell for giving me the opportunity to enter the world of gravitational waves data analysis. I am very grateful for his counsels and the time he has dedicated to discussions about the subject. I also thanks Gregory Baltus for the discussions we had about this work, the interpretations of some results and the investigation ideas. Additionally, I would also thank Dr. Atri Bhattacharya for the numerical support brought during this year. My acknowledgements also go to the review panel: Dr. Maxime Fays, Pr. Dominique Sluse and Pr. Eric Gosset to take the time to read this (rather long) manuscript. Finally, I want to thank my family and friends for their support. Especially Amélie, Emmy-Lee, Agnès, Patrick and Sabine for their patience and encouragement in the fulfilments of my studies. I am forever grateful to Leo Ocquet to have given me the love and curiosity for sciences, as well as the counsels to find my way. Without them, this work would never have been possible. Moreover, I thank Mister Thomas Michel for his support and good mood during these years. III Contents Introduction 1 1 Theory of matched filtering 10 1.1 Importance of this technique in the detection of gravitational waves....... 10 1.2 Basic concepts of the filtering process........................ 10 1.3 Characteristic functions............................... 13 1.4 Khinchin’s theorem.................................. 13 1.5 Detection of a signal with a known form...................... 16 2 Visualization of the event present in the data 21 2.1 Introduction...................................... 21 2.2 Raw strain....................................... 22 2.2.1 Acquiring the data.............................. 22 2.2.2 Sources of noise................................ 24 2.3 First manipulation of the data............................ 25 2.4 Power spectral density................................ 27 2.4.1 Mathematical tools for the computation of the PSD: FFT, window func- tions and Welch’s method.......................... 27 2.4.2 Practical computation of the PSD...................... 32 2.5 Whitening the data.................................. 34 IV 2.6 The Q-transform................................... 37 2.6.1 Mathematical details of the Q-transform.................. 37 2.6.2 Practical computation............................ 39 3 Significance of the event 41 3.1 Preconditioning the data............................... 42 3.2 Making a template to fit the signal......................... 42 3.3 Signal to noise ratio and matched filtering..................... 46 3.4 Chi-square test.................................... 50 3.4.1 Mathematical description of the chi-square test.............. 50 3.4.2 Practical computation of the chi-square test................ 53 3.5 Down-weighting of the SNR............................. 55 3.6 Calculation of the background and significance................... 57 3.7 The false-alarm rate................................. 60 4 Finding the best fit parameters for the event 62 4.1 One detector: LIGO-Livingston........................... 62 4.2 Several detectors................................... 67 4.3 The error bars of the estimated parameters..................... 74 5 Investigation of effects leading to variations in the analysis 75 5.1 Value of the time slice in the PSD computation.................. 75 5.1.1 Shape of the PSD............................... 75 5.1.2 Value of the SNR............................... 78 5.2 Influence of the window used in Welch’s method.................. 82 V 5.2.1 Modification of shape of the PSD depending on the window used in Welch’s method................................ 83 5.2.2 Modification of the signal-to-noise ratio depending on the window taken in order to compute the PSD........................ 85 6 Dependence of the results on the signal and noise characteristics 87 6.1 Evolution of the SNR as a function of the fraction of the signal taken into account 87 6.1.1 The SNR of GW170814 as a function of the fraction of time taken into account..................................... 88 6.2 Evolution of the SNR as a function of the noise amplitude and characteristics.. 90 6.2.1 Signal swamped in gaussian white noise................... 90 6.2.2 Evolution of the SNR as a function of the amplitude of the noise present in the detectors and as a function of the amplitude of the signal..... 93 7 Neutron stars overview 96 7.1 Interest of these objects and difference with black holes.............. 96 7.2 Matched filtering for the GW170817 event..................... 99 7.3 Possibility to detect the merger early on...................... 101 7.3.1 Method of search............................... 101 7.3.2 Results..................................... 104 8 Overlapping signals 107 8.1 Injection of the signal: black hole mergers..................... 108 8.2 Overlap of two black-hole mergers.......................... 111 8.3 Overlap of a binary black hole merger and a neutron star merger......... 114 8.4 Overlap of two neutron star signals......................... 115 8.5 Remark about the overlaps.............................. 117 VI Conclusions 118 VII Introduction The theory of general relativity as it is known today has not been written in one day. It needed the evolution of ideas on a long time scale. It is also the case for gravitational waves theory. The first correct theory of gravity was proposed by Newton in his work Philosophiae naturalis principia mathematica, published in 1687 [1]. In this theory, gravitation is depicted as an attractive force acting between two massive objects. The biggest problem, already raised by Newton, is that it acts instantaneously without taking into account any distance effect between the two bodies. Nevertheless, at the time, it worked well for the description of the motion of celestial objects and it was kept for the next centuries. A first theory of relativity was proposed by the French physicist Henri Poincaré in 1900, and in 1905 he proposed gravity to be mediated by waves, which he already called gravitational waves. However, it is not the theory that remained famous over the years. Nevertheless, it was probably a source of inspiration for Einstein. The latter proposed in 1905 his theory of special relativity, which led to the surprising effect of time dilation and length contraction. He completed this theory ten years later to formulate his famous theory of General Relativity, still seen today as the best theory to describe gravitation. This theory describes gravity geometrically, as the manifestation of the space-time manifold. The curvature arises when a massive object is present, as well as when it has energy or momentum. Basically, this is described by Einstein’s equations which make the link between the local curvature of space and time and the energy and the momentum present in the same region of space. In more technical terms, it gives the link between the Ricci tensor, the Ricci scalar and the metric on one side and the energy-momentum tensor on the other. This equation cannot, in fact, be solved analytically in many cases and has often to be solved numerically [2]. After his theory was completed, Einstein found the possibility to have the formation of grav- itational waves. Those can be compared with the waves from the theory of electromagnetism (proposed by JC Maxwell in 1864 and published in 1865 [3]), where the acceleration of electric charges can produce electromagnetic waves. However, this analogy is not perfect. Indeed, for the latter theory, a dipolar emission is enough to obtain the wave. On the other hand, for gravity, the emission has to come from a higher order, the quadrupolar order. This result was not directly obtained by Einstein. Indeed, he first realized that because there are no positive and negative masses, the dipolar emission is not possible. In the case of electromagnetism, there are positive and negative charges. A dipole is then made of one positive and one negative charge and the emission arises from the relative motion of both charges.
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