The present-day interest in gravitational waves, justified by the recent direct detections made by LIGO, is opening the exciting possibility to answer many questions regarding General Relativity in extreme situations. One of these questions is whether black hole are – indeed – described totally by their mass, charge and angular momentum or whether they can have additional long-range hair. This project is concerned with this question. We aim at studying the influence of additional structure on the black hole horizon in the form of long-range hair by studying linearized Einstein equation the solutions when perturbed. More precisely, we will study the Schwarzschild solution, pierced by an infinitely long and thin cosmic string such that the space-time possesses a global deficit angle. Quasi-normal modes are believed to dominate the gravitational wave emission during the ring down phase of an excited black hole that would e.g. be the result of a merger of two ultra-compact objects, therefore linearized perturbations can be considered. With the advent of gravitational wave astronomy the proposed study will be very important when reconstructing the source of the detected gravitational wave signals. / O atual interesse em ondas gravitacionais, justificado pelas detecções diretas feitas pela colaboração LIGO recentemente, está abrindo a excitante possibilidade de responder várias questões a respeito da Relatividade Geral em condições estremas. Uma dessas questões é se buracos negros são – realmente – totalmente discritos apenas por sua massa, carga e momento angular ou se eles podem ter os chamados cabelos de longo alcance adicionais. Nosso projeto se preocupa em responder esta pergunta. Nosso objetivo está em estudar a influência de uma estrutura adicional no horizonte de eventos de um buraco negro através do comportamento da equação linearizada de Einstein quando a solução é perturbada. Mais precisamente, nós estudaremos a solução de Schwarzschild atravessada por uma corda cósmica infinitamente fina, tal corda faz com que o espaço-tempo tenha um hiato angular em seu plano equatorial. Acredita-se que modos quasi-normais dominem a emissão de ondas gravitacionais durante a fase de ringing down de buracos negros excitados que podem, por exemplo, se originar da colisão de objetos ultra compactos, portanto perturbações lineares podem ser consideradas. Com o advento da astronomia através de ondas gravitacionais o estudo proposto será importante para que se possa reconstruir a origem de sinais detectados.

Les ondes gravitationnelles ont été prédites par Einstein dans sa théorie de laRelativité Générale. Elles sont des perturbations de l’espace-temps que lon essaie de mettre en évidence parinterférométrie laser. Plus précisément les détecteurs sont des interféromètres de Michelson de plusieurs kmde long combinés avec des cavités Fabry-Perot afin d'augmenter la sensibilité de linstrument. La premièregénération de détecteurs (Virgo, LIGO, GEO) n’a pas permis d’obtenir une détection directe malgré plusieursphases d’observations en coïncidence à la sensibilité prévue. Une seconde g´enération de détecteurs estactuellement en préparation avec notamment le projet européen Advanced Virgo. Ce détecteur devraitavoir une sensibilité améliorée d’un ordre de grandeur par rapport à linterféromètre Virgo. Les miroirs del’interféromètre jouent un rôle primordial dans la sensibilité d’Advanced Virgo puisque celle-ci est limitéeà dans les fréquences médianes par le bruit thermique des miroirs et aux hautes fréquences par la quantitéde photons que lon arrive à collecter dans les cavités de linterféromètre. La haute puissance contenue dansles cavités Fabry-Perot induit des effets de lentille thermique importants. Cette thèse s’intéresse dans unpremier temps aux effets de lentille thermique dans linterféromètre pour différentes configurations optiques.Par la suite, nous nous intéresserons aux miroirs qui composent les cavités Fabry-Perot depuis la définitiondes besoins en termes de planéité à la réalisation de cette planéité et à sa mesure. La planéité de ces miroirsdoit être sub-nanométrique de faon à limiter les pertes optiques dans les cavités Fabry-Perot et ainsi r´eduireles effets du bruit de photons et de la lumière diffusée. Nous verrons la réalisation de la correction de laplanéité des substrats par la technique dite du traitement correctif. Nous étudierons aussi l’uniformité dudépôt des couches minces diélectriques nécessaires à l’obtention de surface hautement réfléchissante avec enparticulier l’étude du mouvement planétaire des substrats dans la machine de dépôts. / Gravitational waves have been predicted by Einstein in his General Relativity theory. Theyare perturbation of the space-time metric and we try to reveal them by laser interferometry. More precisely,gravitational wave detectors are km long Michelson interferometers combined with Fabry-Perot cavities.The network of first generation detectors (Virgo, LIGO, GEO) did not permit a direct detection afterseveral observational runs in coincidence at the nominal sensitivity. A second generation of detectors is inpreparation with in particular the European project Advanced Virgo. This detector should have a sensitivityincreased by an order of magnitude compared to Virgo. The interferometer mirrors play a crucial role inthe Advanced Virgo sensitivity as it is limited by the mirror thermal noise in the mid-frequency regionand by the amount of photons collected in the interferometer cavities at high frequencies. The high powercirculating in the Fabry-Perot cavities induces important thermal lensing effect. This thesis is interestedfirst in the thermal lensing effect in the interferometer for different optical configurations. Then we areinterested in the mirrors composing the Fabry-Perot arm cavity from the calculation of the requirements interms of flatness to the realization of the mirrors flatness and its measurement. The mirror flatness shouldbe sub-nanometric in order to limit the optical losses in the Fabry-Perot cavities to reduce the effect of theshot noise and of the diffused light. We will see the correction of the substrates flatness by the so-calledcorrective coating technique. Finally, we study the uniformity of the dielectric multilayer coating depositionnecessary to obtained high-reflective mirrors. We study in particular the planetary motion of the substratesin the coating machine.

Classical theory cannot provide a satisfying scenario for a unitary thermodynamic evolution of black holes. To preserve information one requires quantum mechanical effects on scales reaching beyond the traditional horizon radius. Therefore, common to many of the theories attempting to resolve the paradox is the existence of exotic horizon structures. The recent advent of gravitational wave astronomy provides a possible means for detecting the existence of such structures through gravitational wave emission in the ringdown phase of binary black hole mergers. Such emission is described by quasinormal modes (QNMs) in which the gravitational waves originates outside the black hole, in the vicinity of the photon spheres. Requiring reflective properties of the horizon structure results in the existence of gravitational echoes that may be detected by facilities such as LIGO etc.. This thesis studies geodesic motion of such echoes in the equatorial plane of a rotating black hole. Depending on the extent of the horizon structure, and the particular mode of emission, one can expect different timescales for the echoes. For a horizon structure extending <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5CLambda%20r%20=" /> <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?10%5E%7B-12%7D" /> <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?M" /> outside the traditional horizon of a  <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?M%20=%2022.6%20M_%7B%5Codot%7D" />, <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?a%20=%200.74%20M" /> black hole one would ideally find echoes appearing as integer multiples of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5CDelta%20t_%7Becho%7D" /><img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?=%200.0204%20s" /> after the primary signal. The time delay is expected to increase by at least an order of magnitude if one lets <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5CDelta%20r%20%5Csim%2010%5E%7B-80%7D%20M" />. The expected echo timescales for gravitational waves emitted from any point around the black hole, in arbitrary modes, is an interesting further study.

In der vorliegenden Dissertation wird ein neues experimentelles Konzept zur Durchführung von Gravitationswellendetektion und Spektrometrie mit einem Hong-Ou-Mandel (HOM) Interferometer im Weltraum untersucht. Dabei wird das Rauschbudget des Instruments bewertet. Die grundlegenden experimentellen Anforderungen werden berechnet. Es wird gezeigt, dass die Leistung und Wellenlänge der verschränkten Photonenquelle, zusammen mit der Winkelgenauigkeit der Messung der Photonenpolarisationsdrehung, die Haupteinschränkungen bilden, um die Art der Gravitationswellenquellen zu bestimmen, die das Ziel von HOM - Gravitationswellenspektrometern sein würden. Die derzeit verfügbaren pW-Leistungen mit typischen Photonenfrequenzen in der Größenordnung von 1014 Hz (sichtbarer - UV-Anteil des optischen Spektrums) sind völlig ungeeignet, was die erforderliche Detektionszeit für eine der Gravitationswellenquellen betrifft, die derzeit von bodenund raumgestützten Gravitationswellendetektoren anvisiert werden. Der Betrieb des HOM-Interferometers als GW-Spektrometer wird mit einem numerischen Modell veranschaulicht, das die von LIGO am 14 September 2015 (GW150914-Ereignis) aufgezeichneten Gravitationswellendehnungsdaten verwendet. Unter der Annahme einer Winkelgenauigkeit von μrad für die Messung der Polarisationsdrehung von Photonen könnten diese Messungen nur mit Armlängen des HOM-Interferometers in der Größenordnung von 10.000 km (nur im Weltraum erreichbar, wenn wir keine optischen Kavitäten verwenden) und unter Verwendung von verschwänkten Photonenquellen von etwa 1 W Leistung durchgeführt werden, die verschränkte Photonen mit Wellenlängen im Radiowellenbereich des elektromagnetischen Spektrumserzeugen, (10 MHz), und unter Verwendung von Photodetektoren mit minimaler Detektionszeit für einzelne Photonen und minimaler detektierbarer Leistung, die weit von den Möglichkeiten der gegenwärtigen Photodetektortechnologie entfernt sind. Auch die erforderliche Präzision der Uhrensynchronisation, um die Koinzidenz- Zählgeschichte zu erfassen, ist noch ange nicht erreicht. Obwohl die Technologie zur Herstellung der erforderlichen verschränkten Photonenquellen, Photodetektoren und Uhrensynchronisationsgenauigkeit derzeit nicht verfügbar sind, diskutieren wir verschiedene Missionsszenarien zur Implementierung eines großarmigen HOMInterferometers.:Contents 1 Introduction 1 2 Gravitational waves and their measurement 7 2.1 Theory of general relativity in a nutshell . . . . . . . . . . . . . . . . 8 2.2 On the physical nature of gravitational waves . . . . . . . . . . . . . 13 2.2.1 Effect of gravitational waves on the test masses of a detector . 15 2.2.2 Estimation of gravitational wave’s amplitude . . . . . . . . . . 18 2.2.3 Gravitational radiation luminosity and cross section of the Hydrogen atom to GWs . . . . . . . . . . . . . . . . . . . . . 21 2.3 Measuring cosmic distances with GW astronomy . . . . . . . . . . . . 25 2.4 Influence of gravitational waves on photon’s polarization . . . . . . . 28 2.4.1 Effect of gravitational waves on the parallel transport of photon’s polarization four-vector - revisited . . . . . . . . . . . . 29 2.4.2 Effect of primordial gravitational waves on the polarization of the cosmic microwave background . . . . . . . . . . . . . . . 36 2.4.3 Gravitomagnetic Faraday effect . . . . . . . . . . . . . . . . . 40 2.5 Michelson type gravitational wave antennas . . . . . . . . . . . . . . 41 2.6 Rough estimation of the sensitivity and cross section of Michelson type detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3 Interaction of gravitational waves with Hong-Ou-Mandel interferometers 47 3.1 Fundamental nature of quantum entanglement in brief . . . . . . . . 47 3.2 Why a HOM interferometer to detect GWs? . . . . . . . . . . . . . . 50 3.3 Quantum mechanics of Hong-Ou-Mandel interferometers . . . . . . . 54 3.4 Principle of gravitational waves detection with a Hong-Ou-Mandel interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.5 Instrument noise budget . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.6 Basic experimental requirements for HOM based gravitational waves detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4 Gravitational waves spectrometry with a Hong Ou Mandel interferometer in space 77 4.1 Principles of gravitational waves spectrometry with a Hong-Ou-Mandel interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2 Hong-Ou-Mandel spectrometer in geostationary orbit . . . . . . . . . 93 4.3 Hong-Ou-Mandel spectrometer scanner in space . . . . . . . . . . . . 95 5 HOMER mission scenarios for gravitational waves spectrometry - basic design requirements 97 5.1 HOMER mission design analysis . . . . . . . . . . . . . . . . . . . . . 98 5.1.1 HOMER GEO mission . . . . . . . . . . . . . . . . . . . . . . 99 5.1.2 HOMER ground-GEO mission . . . . . . . . . . . . . . . . . . 105 5.1.3 HOMER scanner mission . . . . . . . . . . . . . . . . . . . . . 106 5.2 Influence of earth gravitomagnetism on photon polarization . . . . . 106 5.3 Payload design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.4 Spacecraft design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.5 Summary of HOMER mission requirements . . . . . . . . . . . . . . 120 6 Outlook and conclusions 129 6.1 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.1.1 HOM gravitational wave detector with optical cavities . . . . 129 6.1.2 Bright entangled heralded photon sources . . . . . . . . . . . 130 6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7 ANNEX: Detailed derivation of gravitational waves and gravitoelectric and gravitomagnetic fields 137 7.1 Weak gravitational fields . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.2 General relativity for the practical physicist . . . . . . . . . . . . . . 138 7.3 Gravitational wave equation . . . . . . . . . . . . . . . . . . . . . . . 144 7.4 Gravitoelectromagnetic split of spacetime . . . . . . . . . . . . . . . 145 7.4.1 Gravitational scalar potential . . . . . . . . . . . . . . . . . . 147 7.4.2 Gravitomagnetic vector potential . . . . . . . . . . . . . . . . 147 7.4.3 Space curvature . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.5 Maxwell-type gravitational equations . . . . . . . . . . . . . . . . . . 151 7.6 Gravitomagnetic waves . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.7 The equations of motion in the weak field approximation . . . . . . . 156 7.8 Production of gravitational radiation . . . . . . . . . . . . . . . . . . 160 Bibliography 163

No description available.

LISA, the Laser Interferometry Space Antenna, is the 3rd large mission (L3) of the ESA program Cosmic Vision with a junior partnership from NASA planned to be launched around 2034. Space-based gravitational wave observatories such as LISA have been developed for observation of sources that produce gravitational wave (GW) signals with frequencies in the mHz regime. The frequency band is achievable by having a longer-baseline interferometer compared to ground-based detectors. In addition, the significant size of the LISA arms-length guarantees the detection of many astrophysical sources. The absence of Newtonian noise in space, which is the dominant source of noise below few hertz for ground-based detectors, allows LISA to be sensitive to lower frequency compared to the former. Thus, going to space allows studying different sources with respect to the ones of interest for ground-based detectors such as supermassive black holes. Although having very long baselines between the satellites generally increases the sensitivity to gravitational waves, it also implies many technical challenges, such that a balance must be found between scientific performance and technical feasibility.In the actual proposal LISA is designed to be a constellation of three identical spacecraft in a triangular formation with six active laser links connecting the three spacecraft, which are separated by 2.5 million km. To fulfil the observatory program every spacecraft has a minimum requirement of two free-falling test masses, two telescopes, and two lasers. The detector’s center-of-mass follows a circular, heliocentric trajectory, trailing 20 degrees behind the Earth and the plane of the detector is tilted by 60 degrees with respect to the ecliptic.The goal of LISA is to detect GWs which manifest themselves as a tiny fluctuation in the frequency of the laser beam measured at the phase-meter. Thus, to detect GW you need to compete with many sources of disturbance that simulate the effect of a GW frequency modulation. Laser noise is an example of those. Therefore, one key element in the LISA data production chain is the post-processing technique called Time Delay Interferometry aimed at suppressing the intense laser frequency noise that would completely cover the astrophysical signal. Data from the six independent inter-satellite links, connecting the three spacecraft, are properly time-shifted and combined to form the final scientific signal. This post-processing technique circumvents the impossibility of physically building in space an equal arm interferometer, which would intrinsically beat the frequency noise by comparing light generated at the same time.The following work is focused on revisiting the Time-Delay-Interferometry (TDI) for LISA and studying the usage of all the possible TDI combinations we can build for the LISA instrument characterisation and science extraction. Many possible TDI combinations that suppress the frequency noise have been identified in the past and this thesis revisits the TDI technique focusing on the physical interpretation of it, that is a virtual interference of photons that have been travelling through the constellation via different paths but performing the same total distance. We illustrate all possible TDI configurations that suppress the laser noise contribution to the level required by the mission to understand how TDI channels can be best used for the diagnostic of the instrument and LISA science. With this philosophy, we develop an algorithm to search for all possible combinations that suppress laser noise at the same level as the classical TDI X, Y, and Z combinations presented in the TDI literature. This algorithm finds new combinations that fulfill the noise suppression requirement as accurately as X, Y, and Z.The LISA mission has been also advertised to probe the early Universe by detecting a stochastic GW background. Once the laser frequency noise has been subtracted, the stochastic signal, both cosmological and astrophysical, is itself going to contribute to the noise curve. Therefore it is necessary to have a good estimate of the noise of the instrument to discriminate between the stochastic background signal and the LISA noise.The strategy that has been suggested in the literature is to use the TDI T, insensitive (up to a certain order) to GW signals to estimate the pure instrumental noise in order to distinguish between the LISA background noise and the GW stochastic signal. Following this idea, as instrument noise is expected to have multiple, independent sources, this thesis explores combinations that could allow discriminating among those sources of noise, and between them and the GW signal, with the purpose of understanding how we can characterise our instrument using TDI. We illustrate special TDI combination signals in LISA, in addition to TDI T, that we call null-channels, which are ideally insensitive to gravitational waves and only carry information about instrumental noise. Studying the noise properties that can be extracted by monitoring these interferometric signals, we state that individual acceleration noise parameters are not well constrained. All null-channels behave as an ideal Sagnac interferometer, sensitive just to a particular linear combination of the six test masses acceleration that resembles a rotational acceleration signal of the entire constellation. Moreover, all null-channels show approximately the same signal to noise ratio remarkably suppressed relative to that of the TDI X. In support and application of our theoretical studies, we also give an introduction on calibrating the LISA instrument by injecting spurious signals in a LISA link and see how these propagates through a TDI channel. Indeed, this will be useful to calibrate the instrument during operations and also to build the basis for the data analysis to discriminate spurious signals from gravitational waves. My contribution to the results we present in this thesis can be summarised as the following. I supported the studies and the realisation of the search TDI algorithm whose results are published in the article. In particular, I took care of cataloging the new TDI combinations and consolidating the results we found. I have updated the TDI combinations reported in the above-mentioned work, the final version of it is reported in this thesis. I worked on the characterisation of these combinations concerning secondary noises such as clock noise, readout noise, residual laser frequency noise, and acceleration noise. In particular, I studied how these noises are transferred through the various TDI and I derive the correspondent analytical models. I then realize a software with Wolfram Mathematica, design to load and combines phase data produced by an external simulator to build the final TDI outputs, besides I also did the noise models’ validation. The basis of this program was then used to implement these TDI combinations in LISANode. Finally, I developed the algorithm to study how disturbances in force, such as glitches, and simple GW signals, such as monochromatic GW binaries, propagate through TDI and null-channels. Moreover, I tested through simulations the validity of these TDI and null-channels to distinguish instrumental artefact from GW signals and to characterise the instrumental noise.

No description available.

Die Erforschung von Gravitationswellen hat unsere Sicht auf das Universum revolutioniert. Mit dem bevorstehenden Start von LISA, einem Weltraum-Gravitationswellendetektor, und neuen Berichten über Hinweise auf einen Gravitationswellenhintergrund im Nanohertz-Bereich aus Pulsar Timing Array (PTA)-Experimenten, eröffnen sich neue Möglichkeiten und Herausforderungen. Diese Dissertation entwickelt innovative Datenanalysetechniken und Wellenformmodelle, um Erkenntnisse aus diesen Beobachtungen zu gewinnen. Ein Schwerpunkt liegt auf der Untersuchung von Extreme Mass Ratio Inspirals (EMRIs) durch LISA. Diese Quellen bestehen aus kleinen, kompakten Objekten, die sich um ein zentrales Schwarzes Loch bewegen. Die Wellenformen von EMRIs bieten die Möglichkeit präziser Parametermessungen, sind jedoch aufgrund ihrer langen Signaldauer und harmonischen Komplexität schwer zu berechnen. Wir präsentieren die Implementierung einsatzbereiter EMRI-Wellenformen im Frequenzbereich für Grafikprozessoren (GPUs) und zentrale Recheneinheiten (CPUs). Zudem untersuchen wir das wissenschaftliche Potenzial von EMRIs innerhalb von Akkretionsscheiben, erforschen den Einfluss von Umwelteffekten mittels bayesianischer Methoden und bewerten die Multimessenger-Aussichten dieser Systeme. Im PTA-Bereich entwickeln wir Methoden zur Optimierung der Datenkombinationen für PTA-Analysen und tragen zum European Pulsar Timing Array bei, indem wir alternative Sampling-Pipelines für die Analyse von Gravitationswellenhintergründen und individuellen Quellen implementieren. Mit transdimensionalen Sampling-Methoden suchen wir nach einzelnen supermassiven Schwarzen Löchern und bewerten deren Signifikanz. Diese Dissertation trägt zur Weiterentwicklung der Gravitationswellenastronomie bei, indem sie neue Methoden und Modelle entwickelt, die tiefere Einblicke in die kosmischen Phänomene ermöglichen, die von LISA- und PTA-Beobachtungen erfasst werden. / Gravitational wave astronomy has reshaped our understanding of the cosmos. As we look towards the future launch of LISA, a space-based gravitational wave detector, and analyze recent evidence of a nanohertz gravitational wave background from Pulsar Timing Array (PTA) experiments, new opportunities and challenges emerge. This thesis delves into developing novel data analysis techniques and waveform models to extract information from these observations. Focusing on LISA, we delve into Extreme Mass Ratio Inspirals (EMRIs). These sources consist of small compact objects spiralling into massive black holes at the centres of galaxies. Their observations are expected to provide precise parameter measurements for these systems. However, EMRI waveform generation poses challenges due to the long signal duration and large harmonic content. For the first time, we provide a fast implementation of EMRI waveforms in the frequency domain, suitable for both graphics processing units (GPUs) and central processing units (CPUs). In addition, we explore the scientific potential of EMRIs embedded in accretion disks. Employing Bayesian inference, we investigate the measurability of environmental effects and explore these systems' multimessenger prospects. Transitioning to PTA, we develop methods to optimize data combinations for PTA analyses. We present our contributions to the second data release of the European Pulsar Timing Array collaboration, which consists of implementing alternative sampling pipelines for gravitational wave background and individual source analyses. Using trans-dimensional sampling methods, we search for individual supermassive black hole binaries and assess their significance. The burgeoning field of gravitational wave astronomy has the potential to transform our understanding of the Universe. The work in this thesis develops new approaches that will facilitate the delivery of the best possible scientific results from current and future gravitational wave observations.

FundaÃÃo Cearense de Apoio ao Desenvolvimento Cientifico e TecnolÃgico / Wormholes sÃo estruturas exÃticas que podem ligar regiÃes longÃnquas do espaÃo e por isso sÃo bastante visados como objeto de pesquisa. Pode-se mostrar que buracos negros possuem modos de vibraÃÃo amortecidos, os chamados de modos quase-normais. Esses modos sÃo responsÃveis por irradiar as ondas gravitacionais para espaÃo. Isso se deve ao fato de pequenas perturbaÃÃes na mÃtrica poderem ser descritas pelas equaÃÃes de Einstein, em forma de equaÃÃes de onda. Esse comportamento à semelhante para wormholes, onde devemos encontrar um potencial de Regge-Wheeler associado a equaÃÃo de onda. Nosso objetivo nesse trabalho serà tentar ver se à possÃvel encontrarmos um potencial do tipo Regge-Wheeler para um wormhole, que se supÃem ser sustentado pela matÃria escura na regiÃo dos halos galÃcticos. E por consequÃncia, verificarmos se à possÃvel determinar modos quase-normais usando o mÃtodo WKB, para esse wormhole em particular. / Wormholes are exotic structures that can connect distant regions of space and therefore are highly targeted as a research object. It can be shown that black holes have damped vibration modes, The so-called quasi-normal modes. These modes are responsible for radiating gravitational waves to space. This is due to the fact that small disturbances in the metric can be described by Einsteinâs equations, in the form of wave equations. This behavior is similar for wormholes, where we must find a Regge-Wheeler potential associated with the wave equation. Our goal in this work will be to see if it is possible to find Regge-Wheeler potential for a wormhole, Which are supposed to be sustained by dark matter in the region of galactic halos. And consequently, Verify if it is possible to determine quasi-normal modes using the WKB method, for that particular wormhole.

Les signaux bivariés apparaissent dans de nombreuses applications (optique, sismologie, océanographie, EEG, etc.) dès lors que l'analyse jointe de deux signaux réels est nécessaire. Les signaux bivariés simples ont une interprétation naturelle sous la forme d'une ellipse dont les propriétés (taille, forme, orientation) peuvent évoluer dans le temps. Cette propriété géométrique correspondant à la notion de polarisation en physique est fondamentale pour la compréhension et l'analyse des signaux bivariés. Les approches existantes n'apportent cependant pas de description directe des signaux bivariés ou des opérations de filtrage en termes de polarisation. Cette thèse répond à cette limitation par l'introduction d'une nouvelle approche générique pour l'analyse et le filtrage des signaux bivariés. Celle-ci repose sur deux ingrédients essentiels : (i) le plongement naturel des signaux bivariés -- vus comme signaux à valeurs complexes -- dans le corps des quaternions H et (ii) la définition d'une transformée de Fourier quaternionique associée pour une représentation spectrale interprétable de ces signaux. L'approche proposée permet de définir les outils de traitement de signal usuels tels que la notion de densité spectrale, de filtrage linéaire ou encore de spectrogramme ayant une interprétation directe en termes d'attributs de polarisation. Nous montrons la validité de l'approche grâce à des garanties mathématiques et une implémentation numériquement efficace des outils proposés. Diverses expériences numériques illustrent l'approche. En particulier, nous démontrons son potentiel pour la caractérisation de la polarisation des ondes gravitationnelles. / Bivariate signals appear in a broad range of applications (optics, seismology, oceanography, EEG, etc.) where the joint analysis of two real-valued signals is required. Simple bivariate signals take the form of an ellipse, whose properties (size, shape, orientation) may evolve with time. This geometric feature of bivariate signals has a natural physical interpretation called polarization. This notion is fundamental to the analysis and understanding of bivariate signals. However, existing approaches do not provide straightforward descriptions of bivariate signals or filtering operations in terms of polarization or ellipse properties. To this purpose, this thesis introduces a new and generic approach for the analysis and filtering of bivariate signals. It essentially relies on two key ingredients: (i) the natural embedding of bivariate signals -- viewed as complex-valued signals -- into the set of quaternions H and (ii) the definition of a dedicated quaternion Fourier transform to enable a meaningful spectral representation of bivariate signals. The proposed approach features the definition of standard signal processing quantities such as spectral densities, linear time-invariant filters or spectrograms that are directly interpretable in terms of polarization attributes. More importantly, the framework does not sacrifice any mathematical guarantee and the newly introduced tools admit computationally fast implementations. Numerical experiments support throughout our theoretical developments. We also demonstrate the potential of the approach for the nonparametric characterization of the polarization of gravitational waves.