Data analysis pipeline for the spherical gravitational wave antenna MiniGRAIL

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Thesis

Reference

Data analysis pipeline for the spherical gravitational wave antenna MiniGRAIL

DA SILVA COSTA, Carlos

Abstract

MiniGRAIL est un détecteur d'ondes gravitationnelles composé d'une d'une masse résonante sphérique de 68cm de diamètre avec une fréquence de résonance de ~2850kHz et une bande passante de ~240Hz. Sa sensibilité sera de 10^{-21} Hz^{-1/2} pour une température effective de 40mK. Nous avons testé par simulation une stratégie d'analyse des données alliant un algorithme "waveburst" redéveloppe pour la sphère suivi d'un filtre de Wiener. Le premier permet d'obtenir les "trigger" rapidement et indépendamment de la direction des ondes. Le filtre de Wiener permet une meilleure caractérisation de la fréquence centrale, de l'amplitude et du temps d'arrivée des triggers. La détermination de la direction des ondes se fait par trois méthodes indépendantes. Ces trois déterminations servent de vetos permettant de réduire le taux de faux trigger à ~10^{-3}Hz. La résolution de la direction sur toute la sphère varie de 16 à 70mrad. La polarisation linéaire des ondes fut testée pour des signaux de forte amplitude.

DA SILVA COSTA, Carlos. Data analysis pipeline for the spherical gravitational wave antenna MiniGRAIL. Thèse de doctorat : Univ. Genève, 2010, no. Sc. 4210

URN : urn:nbn:ch:unige-68913

DOI : 10.13097/archive-ouverte/unige:6891

Available at:

http://archive-ouverte.unige.ch/unige:6891

Disclaimer: layout of this document may differ from the published version.

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S´ECTION DE PHYSIQUE Professeur Martin Pohl Professeur Michele Maggiore

Data Analysis Pipeline For The Spherical Gravitational Wave Antenna MiniGRAIL

TH`ESE

présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès Sciences, mention Physique

par

Carlos Filipe Da Silva Costa de

Anadia, Portugal

Th`ese N˚4210 Gen`eve, Avril 2010

Atelier d’impression ReproMail, Uni Mail

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ulfivr3silTE

DT CTNTY:

FÂCULTI pË5 SCtXlleË5

Doclorof ès sciences mentîon physîque

Thèse ae A/lonsieur Carlos DA SILVA COSTA

i n t i t u l é e :

"Dqlo Anolysis Pipeline For The Sphericol Grovilqlionol Wove Anlenno MiniGRAl["

Lo Foculté des sciences, sur le préovis de Messieurs M. POHL, professeur ordinoire el directeur de thèse (Déportement de physique nucléoire et corpusculoire), M. MAGGIORE, professeur ordinoire (Déportement de physique théorique), G. FROSSATI, professeur (Leiden Cryogenics - Leiden, The Netherlonds), et Modome V. FAFONE, professeure (Universitù di Romo <Tor Vergotol - Diportimenio di Fisico - Romq, ltolio), outorise I'impression de lo présente thèse, sons exprimer d'opinion sur les propositions quiy sont énoncées.

G e n è v e , l e 5 m o i 2 0 1 0

Thèse - 4210 -

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Lo thèse doit porter dons les "lnformotions

lo déclorotion orécédente et remplir relotives oux thèses de doclorot è

les conditions énumérées l'Université de Genève".

N . B .

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ii

Remerciements

La quˆete du Graal ne se fait pas seul. Autour de la table ronde, il y a plusieurs chevaliers sans lesquels je n’aurai rien pu faire, que je m’empresse de remercier :

– Giorgio Frossati, MiniGRAIL c’est son petit, il a lancé la quête et mérite donc le titre de roi Arthur.

– Puis Merlin (Michele Maggiore), il a écrit un grimoire sur les ondes gravitationnelles et lorsque j’ai écrit mon petit grimoire il m’a aidé à déjouer bien des pièges.

– Florian Dubath a été la première personne avec qui j’ai découvert les ondes gravitationnelles.

– Ceux qui m’ont suivi tous les jours sur les routes de la quˆete et qui m’ont bien souvent indiqu´e le chemin, Stefano Foffa et Ricardo Sturani.

– Arlette de Waard, plus que des informations, m’a fourni plusieurs cartes de route.

– Sacha Usenko qui ne fut pas seulement un compagnon de route mais un ami ! – Emlyn Corrin, qui a déchiffré mes écrits en anglais et m’a appris plusieurs outils

de programmation et surtout pour sa sympathie et sa patience.

– Urs S., un ami qui m’a fourni de pr´ecieux conseils sur la structure de programmation.

– Peggy Argentin, Catherine Blanchard et Nathalie Chaduiron, qui m’ont dépa- touillé dans les méandres obscures de l’administration de l’uni.

– Yann Meunier pour ses conseils sur des scripts, le monde informatique et les chocolats.

– A Umberto mon coll`egue de Donjon.

– A Leiden, il faudrait encore remercier tous les autres occupants des ´ecuries (lab) qui ont m’ont fait une place parmi eux.

– Tous les autres du DPT, du DPNC et DPMC (Estela, Rikarrd, Philippe Jaquet, Christophe, Karina, Anna-Sabina, Sonia, Mercedes, Vivian, Cyril Petitjean, Hil- lary Sanctuary, Simon Nigg, Jose Garcia, Andreas Malaspinas,..) pour des conver- sations sur d’autres sujets que la méta-physique et plusieurs bons moments qui font qu’une quête se fait dans des conditions très agréables.

– Les professeurs C. Leluc, D. Rappin et M. Bourquin pour des conseilles.

– Ma famille, mon fr`ere S´ergio.

– Mes amis (Olga, P-E H., Victoria, ..., car le Filipe est parfois difficile quand il est stress´e).

Pour terminer, le chevalier Martin Pohl. C’est lui qui m’a parlé de la quête du Graal et m’a présenté au Roi Arthur et Merlin. Sans lui, je ne serais jamais moi-même devenu chevalier.

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R´ esum´ e

La periode actuelle est une p´eriode prometteuse pour la recherche des ondes gravitationnelles.

L’étude du système binaire d’étoile à neutron dans lequel se trouve le pulsar 1619+13 découvert par J. Taylor Jr. and R. Hulse a permis de prouver l’existence des ondes gravitationnelles grâce à l’accord entre la variation du périastre dû à la perte d’énergie par ondes gravitationnelles prédit par la relativité générale et la variation observée.

De nos jours, nous disposons de deux syst`emes relativistes observables en plus, dont un form´e de deux pulsars.

Le but n’est plus de prouver l’existence des ondes gravitationnelles mais d’exploiter leur potentiel pour l’astronomie, c’est `a dire l’information transmise par les ondes sur le coeur et la structure des ´etoiles et autres objets massifs ayant un moment quadripolaire.

Il existe deux types de détecteur, les interféromètres et les masses résonnantes (barres et sphères). La sensibilité des interféromètres (LIGO et Virgo) a progressé de telle sorte que la détection des ondes gravitationnelles est attendue dans les dix prochaines années. Actuellement, les résultats obtenus par ces détecteurs, les taux et amplitudes maximum des ondes, permettent déjà de donner des informations perti- nentes sur les sources.

Depuis les barres de Weber en 1965, les barres ont été les instruments pionniers dans la recherche des ondes gravitationnelles. Elles ont largement contribué aux connais- sances actuelles en donnant les premières limites sur les taux d’événements.

Les barres interagissent avec les ondes dans une bande de fréquence d’une centaine de hertz autour de leur fréquence de résonance. Les ondes provoquent des oscillations cohérentes des surfaces, qui sont amplifiées par des transducteurs, oscillateurs capacitifs de plus petite masse accouplés à la barre. Les signaux électriques sont amplifiés par des SQUIDs.

Les détecteurs sphériques sont basés sur le même principe de détection que les barres. Les ondes gravitationnelles interagissent uniquement avec les modes quadrupolaires de la sphère. La forme sphérique permet une détection isotrope et un système multi-canaux mesurant les déformations quadrupolaires permet la détermination de la direction d’arrivée des ondes gravitationnelles.

MiniGRAIL est situé à Leiden, en Hollande. Ce détecteur est composé d’une sphère de CuAL de 68cm de diamètre avec une fréquence de résonance de ∼ 2850kHz. La bande passante est de∼240Hz. Sa sensibilité sera de 10⁻²¹Hz⁻^1/2 pour une tempéra- ture effective de 40mK. Les sources potentielles dans cette bande de fréquences sont les “burst” et les périodiques. Les systèmes binaires coalescents ne sont détectés que sous la forme des “bursts”.

Le but de cette thèse fut de développer une chaˆıne d’analyse des données compre- nant le transfert automatique des données, le formatage pour l’échange des données avec les autres expériences, la recherche des “trigger” et leurs caractérisations.

Dans un premier temps, notre groupe à Genève a développé un modèle du détecteur

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simulant un signal de sortie le plus réaliste possible. Ce signal comprend plusieurs sources de bruit, des transducteurs non identiques et différents modes d’oscillation de la sphère en plus des modes quadrupolaires. Ce modèle inclut aussi la fonction de transfert qui transforme les courants mesurés en amplitudes des oscillations des modes quadrupolaires, ce que nous avons nommé les canaux des modes quadrupolaires.

Nous avons testé par simulation une stratégie d’analyse alliant un algorithme“waveburst”re-développé pour la sphère suivi d’un filtre de Wiener. L’algorithme“waveburst”

appliqué à la sphère permet d’obtenir les “trigger” rapidement et indépendamment de la direction des ondes. Deux méthodes indépendantes (une likelihood et une géomé- trique) de détermination de la direction des ondes sont incorporées à cet algorithme. Le filtre dépend inversement de la direction car la forme du signal dépend de la direction d’arrivée des ondes. C’est pourquoi nous appliquons d’abord l’algorithme “waveburst”

pour limiter le champ des directions à tester. Par contre, le filtre permet une meilleure caractérisation de l’ordre de grandeur de la fréquence centrale, de l’amplitude et du temps d’arrivée des triggers. Avec le filtre, nous construisons une troisième détermi- nation indépendante de la direction d’arrivée des ondes, basée sur la maximisation du SNR en fonction de la direction.

Les comparaisons entre les trois déterminations des directions des ondes gravitationnelles sont utilisées comme vetos sur les “triggers”. Cela permet de réduire le taux de faux trigger à 1 ou 3·10⁻³Hz en fonction de l’amplitude du signal injecté hrrs={3−10} ×10⁻²¹Hz^−1/2.

Après avoir démontré l’efficacité de notre chaˆıne, nous avons optimisé la technique de détermination de la direction. Cette méthode facilite la détermination de la réso- lution de la direction sur toute la sphère. On constate que la résolution varie de 16

à 70mrad (pour un signal de h_rss = 10⁻²⁰ Hz^−1/2) en fonction de la direction, de la polarisation et de l’angle (θ ou φ). Le SNR lui aussi varie du simple au double en fonction de la direction. Ces deux phénomènes trouvent leur source dans l’anisotropie introduite par la suspension de la sphère et la fixation des transducteurs sur la sphère.

Pour terminer, cette m´ethode nous permet aussi une mesure de la polarisation lin´eaire.

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Chapter 1 Introduction

A promising period for gravitational-wave research

During the last decades we have accumulated important knowledge on gravitational waves (GWs) from both the experimental and the theoretical point of view.

The studies on the sources have stimulated the research for new analysis methods and experiment evolution. Vice versa, the new experimental limits have fixed new goals for the comprehension of the source mechanisms.

Nowadays (2010) experiments have reached a sensitivity at which the probability of detection becomes substantial. Moreover, as we will discuss in more detail below, these experiments are still improving their sensilbilities and will have major upgrades in the next years.

From this cumulated progress on both sides, and in particular the detector im- provement, a direct detection can be expected in the next decade.

GWs, a window on the universe still to be opened

The final goal is not just to detect GWs nor to confirm Einstein’s theory. Their existence has already been proved since Joseph H. Taylor Jr. and Russell A. Hulse discovered the pulsar (PSR 1913+16) in a Neutron Star (NS) binary system [1] for which they were awarded with the Nobel Prize in 1993 [2].

A pulsar and its companion, both follow elliptical trajectories around their common center of mass. They are on opposite sides of a line passing through the center of mass.

The relative coordinate between bothr=rp−rc, where indexpandcstand for pulsar and companion, describes also an ellipse of eccentricity eand semimajor axis a. The angle formed between the orbiting plane and the plane perpendicular to the line of sight is noted ι. The periastron is the point where the two stars are closest together.

We denote the position of the periastron with an angle ω from the intersection of the orbital plane and the plane perpendicular to the line of sight. The period of the binary system is noted by P_b and we define a time of reference T0 when the system passes through the periastron. All these parameters:

P_b, ω, e, xand T₀, where x=asin(ι)/c, are the Keplerian parameters.

We can also measure the post-keplerian parameters< ω >˙ and γ (Einstein parameter). From these two parameters we can obtain precisely the masses of the system

1

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2 CHAPTER 1. INTRODUCTION m_p and m_c. In General Relativity, having the the masses and the Keplerian parameters, all the post-Keplerian parameters can be fixed. In particular the orbital period derivative ˙P_b serves as a prediction of the system evolution. The system loss of energy by GW emission induces the change of the orbit and period.

The Hulse-Taylor binary system has a periodPb less than 8h which indicates that the system is highly relativistic with a speedv∼10⁻³c. This system has been followed for almost 30 years, see Figure 1.1. The predicted gravitational orbital period derivative is ˙P_b,GR = 2.40242±0.00002×10⁻¹²s/s and the value observed¹ by Weisberg and Taylor is ˙P_b= 2.4056±0.00051×10⁻¹²s/s. Then the ratio is :

P˙_b

P˙_b,GR = 1.0013±0.0021

The concordance between the prediction and the measurements of the energy radiated by GW is impressive, even more because no free parameter are used.

Presently, we have at our disposal in total four NS binary systems with one pulsar [3]. From two (the Hulse-Taylor binary + the PSR B1534+12) it has been possible to measure the decrease of the orbital period derivative due to GW finding agreement with General Relativity. We have also one more remarkable system for many reasons.

It is composed of two pulsars beaming in our direction [5]: PSR J0737-3039 A and B. It is a high relativistic NS-NS system with a period of 2.4hr and we can observe its periastron decay in a few days. Its coalescing time is the shortest among binary NS system. Furthermore it has a sharp pulse with a large flux allowing a high time precision on the measures. All these characteristics indicate that a precision 0.02% on the orbital period derivative can be achieved.

So the existence of the GWs has already been demonstrated. Nowadays the aim is to do GW astronomy which would be complementary to conventional astronomy, for the following reasons [6, 7]:

– GWs are produced by coherent displacement of masses while EM radiation are a superposition incoherent sources. For instance, they can give information on the structure of the stars.

– As they have a very small cross section even the denser matter distribution is transparent to GWs, so for instance, they can give information on the core of massive objects like neutron stars.

– Gravitational detectors are all-sky sensitive. It means also that we have diffi- culties on the localization but on the other hand we can detect many sources.

For example, NS binaries which are not beaming in our direction are potentially detectable by GWs. Current EM telescopes have to be focused on the source and have narrow angle of view.

GWs offer a huge potential to be exploited, as each time that we opened a new window to space (radio astronomy, X-rays and γ-rays) we have had unexpected discoveries.

Two generations of detectors

We have two classes of detectors that correspond to two generations, first the resonant mass detectors then the interferometers.

1. Note that the value ofPbcould not be directly measured. The observations are corrected from many effects: motion of system solar, time delay etc, see [3].

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Figure 1.1: Comments and figure from [4], Orbital decay: the points are the observed change in the periastron and the line is the prediction from GR.

The history of resonant bars and in general of gravitational wave detectors started in 1965 with the first resonant bar of Joseph Weber [8, 9].

After this pioneering work many bars have been built, in particular in the nineties:

ALLEGRO, AURIGA, EXPLORER, NAUTILUS and NIOBE. They are all cryogenic and their sensitivity improved with respect to the original Weber’s bars by five orders of magnitude in energy resolution. They have been working for many years, and they also performed observations as a network. They are medium scale experiments run by collaborations of a few to a few tens of people.

The interferometers are experiments requiring considerable manpower (hundreds of people) and financial resources. Due to their complexity, they started taking data after the year 2001 even though their studies of feasibility start in the seventies. The LIGO and Virgo interferometers are remarkable instruments with a large bandwidth and a promising sensitivity. GEO600 is smaller than LIGO and Virgo. It also con-

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4 CHAPTER 1. INTRODUCTION tributes to GW research and tests future technologies which are going to be applied to the upgrades Advanced LIGO and Advanced Virgo. The two advanced detectors are planned to take data by 2015. Both collaborations are since a few years working together. The first GW detection on Earth is expected from these detectors or their upgrades.

Present situation

The three interferometers of LIGO have finished in 2007 their fifth run (called S5), a long run of more than one year. At the same time Virgo was performing its first scientific run SRV1. Data from both experiments have been analyzed, separately or jointly. Even if no detection was claimed, the strong sensitivity of the interferometers allows to do relevant physical interpretation from the results, for instance:

– The study of the periodic GWs from Crab Pulsar shows that this pulsar loose not more than 4% of the spin-down energy available [10].

– The search for GWs bursts from soft gamma ray repeaters (SGR) gives an upper limit consistent with theoretical model predictions [11].

– The gamma ray burst GRB 070201, which is located in the Andromeda galaxy, could have been produced by the merging of neutron stars or a neutron star and black hole. The results have excluded such a possibility [12].

– The stochastic background of gravitational waves should transport information from the earliest epoch of our universe. GWs placed upper limits that improved the nucleosynthesis bound in the frequency band around 100Hz, see the article on Nature [13].

Recently, in July 2009 LIGO has started its sixth scientific run.

MiniGRAIL

The focus of this thesis is on MiniGRAIL, a spherical resonant mass detector which is being commissioned in Leiden, Netherlands. Compared to LIGO and Virgo and even to resonant bars, MiniGRAIL is a small scale experiment. The budget is much lower and it requires limited manpower, 10 people in rush ours.

MiniGRAIL share the same detection principles as the resonant bars, but it is a spherical resonant mass and has a higher cross section for the same mass. Compared to all other experiments, as we will see in this thesis, it has a good isotropy. It can give the GW direction thanks to its multimodes.

Despite the fact that its sensitivity is not comparable to interferometers, it could be a useful addition to the existing network of GW detectors because of these characteristics.

The fact that MiniGRAIL is a small scale experiment, was a chance for me. I had the possibility to work in different domains. Even if my contribution was minor in the hardware, I have collaborated directly on it. For example, I prepared the transducers, made superconducting cables, installed the magnetic shielding on the dewars and other small activities. The main purpose of my thesis was however to develop the data analysis pipeline. In this context, I also had the responsibility for the data transfer

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1.1. THESIS OUTLINE 5 between our universities². Then the core of my work, the pipeline, was to test different analysis options with trigger search and characterization. The goal of our group was to develop a data analysis pipeline that takes complete benefit of the advantages of a spherical detector. I have also studied the LIGO-Virgo data frame format to prepare our data for a possible exchange.

In a few words, I made the bridge between hardware and analysis, focusing myself on data analysis. In this thesis I will explain these different features from the hardware to the analysis with a sphere.

I have worked in collaboration with MiniGRAIL, but the results presented here could be applied with appropriate changes, to other spherical detectors such as the Mario Schenberg detector being developed in Brazil.

1.1 Thesis outline

In Chapter 2, we will see how we derive the GWs in the linearized theory of General Relativity. We will discuss the effects of GWs on free particles, and this will lead us to understand the principles of detection. The linearized theory assumes certain approximations, we will see the limits of such approximations when applying the developed theory in the laboratory frame.

In Chapter 3, I describe the landscape of the actual detectors, the bars, the interferometers and future detectors. We will discuss the working and detection principles of the bars and interferometers. Because MiniGRAIL is a resonant mass detector and shares common principles with the bars, I will go into more details about the bar description.

Chapter 4 is devoted to spherical detector principles of detection. We will see that GWs couple only with quadrupolar modes of the sphere. Then I will introduce the quadrupolar mode channels which are used in our analysis. The following chapter gives an overview how MiniGRAIL and how the signal is amplified.

Once the acquisition of the data done, the data are formatted. The complete soft- ware treatment of the signal and the complete features of our data analysis pipeline are described in Chapter 7.

Our group developed the Coherent Waveburst for the Sphere, a method able to detect triggers without information about the GWs arrival direction. The characterization of the triggers is improved using a Matched Filter as follow up. The results of this analysis strategy are given in Chapter 8. With the Matched Filter we develop a technique, based on SNR maximization, to improve the GW arrival direction determ- ination. This method suffers from limitation due to the variation of the resolution over the sphere.

In the last Chapter, I present another method that does not suffer from this limitation. This new method allows us also to characterize the SNR variation and resolution over the sphere. It gives also the possibility to study the linear polarization of the GWs.

2. MiniGRAIL in Leiden, NL and our groupe in Geneva, CH.

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6 CHAPTER 1. INTRODUCTION

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Chapter 2 Effect Of Gravitational Waves On Matter.

2.1 The nature of Gravitational Waves

In this chapter, I will briefly recall how the gravitational waves are derived from the Einstein equations and under which conditions. I will introduce the TT gauge, which is of importance for our analysis. To understand how GWs act on one resonant mass, I present briefly their effect on the test particles in our lab as reference frame. For more information and details see Reference [3] from where the following demonstrations are taken.

2.1.1 Weak gravitational fields

First we have to define weak gravitational fields. In the absence of matter the metric is g_µν = η_µν, where the Minkowsky metric η_µν = diag(−1,1,1,1). A weak gravitaional field is a “nearly” flat space time where the metric, in suitable coordinate system, is described by:

g_µν(x) =η_µν+h_µν(x), (2.1)

where |h_µν(x)| 1. The perturbation h_µν(x) depends on the space-time position x.

To simplify the notations, I will omit the dependence on x except when the develop- ments ask to explicitly represent it.

There are two kind of coordinate transformations, the Lorentz transformations and the Gauge transformations, which preserve the properties of the metric above.

If we apply a Lorentz transformation with a boost with v 1 (using c = 1, so γ = (1−v²)⁻^1/2)):

g_αβ⁰ = Λ^µ_αΛ^ν_βg^µν (2.2)

and substituting g_µν with Equation (2.1) we get:

g_αβ⁰ = Λ^µ_αΛ^ν_βη_µν+ Λ^µ_αΛ^ν_βh_µν. (2.3) It follows, using the facts that the Minkowski metric is same in any Lorentz frame, that:

g_αβ⁰ =η_αβ+h⁰_αβ. (2.4)

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8 CHAPTER 2. EFFECT OF GRAVITATIONAL WAVES ON MATTER.

Then under a Lorentz boost, h_µν transforms as a (0,2) tensor defined on a flat space time¹; and for different observers the space time looks nearly flat.

From the point of view of a nearly flat space-time (at the first order inh), we can raise index with η^µν, so we have:

h^µ_ν =η^µαh_αν, h^µν =η^µαη^νβh_αβ and∂^µ=η^µρ∂_ρ. 2.1.2 Einstein equations in a weak gravitational field.

In a weak gravitational field, i.e. for a nearly flat space-time, we can simplify the Einstein equations. The goal is to show that h_µν obeys wave equations.

The Einstein equations can be written in various forms. For our purpose we take the form:

Rµν =−8πG

c⁴ Tµν−gµνT^ρ_ρ

, (2.5)

where R_µν = R^α_αµν is the Ricci tensor, which is a contraction of the Riemann- Christoffel tensor. The Ricci tensor is symmetric and the Riemann-Christoffel tensor is invariant under gauge transformations. The tensor Tµν is the energy-momentum tensor which takes into account matter and electromagnetic energies-momentum.

The interest of this form is to have on left-hand side only the Ricci tensor R_µν. For the future derivation, we have to express the Einstein equation far from matter so we need to describe the Ricci tensor in a weak field. First, we define g^µν in the approximation of a weak field. We know that g^µνg_µν = 1 so in the first order in h:

g^µν = (ηµν+hµν)⁻¹ =η^µν−h^µν. (2.6) The Ricci tensor is defined by:

R_µν = 1 2g^αρ

∂_αρ² g_µν+∂_µν² g_αρ−∂_αν² g_µρ−∂_µρ² g_αν

+g^ραg_ησ

Γ^η_αρΓ^σ_µν−Γ^η_ανΓ^σ_µρ

. (2.7) To express it in powers ofh we substitute g^αβ in the above expression with (2.1) and (2.6) as follows:

R⁽¹⁾_µν = 1 2

∂_µν² h^ρ_ρ+∂_ρ^ρh_µν−∂_µρ² h^ρ_ν −∂_ρν² h^ρ_µ

(2.8) R⁽²⁾_µν = −1

2h^αβ

∂²_µνh^α_β+∂_αβ² h_µν−∂_µβ² h_αν−∂_αν² h_βµ + η^αβη_ρσh

Γ^ρ_αβΓ^σ_µν−Γ^ρ_ανΓ^σ_µβi

(2.9)

R⁽¹⁾ = ∂_µ^µh^ρ_ρ−∂²_µρh^µρ (2.10)

R⁽²⁾ = η^µνR⁽²⁾_µν−h^µνR⁽¹⁾_µν (2.11)

where we note by (1) the first order in h and (2) the second order. We stop at the second order, superior orders are negligible because|h_µν| 1.

1. The newh⁰ is|h⁰_αβ| 1 because the speedv1 and|hαβ| 1 for allαβ.

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2.1. THE NATURE OF GRAVITATIONAL WAVES 9 2.1.3 Gauge transformations

We have already seen that the metric transforms as a tensor under Lorentz transformations. Now let us see what happens for gauge transformations. We start by performing a very small change in our coordinate system:

x^0α=x^α+ξ^α(x^β), (2.12)

taking the derivative:

∂x^0α

∂x^β =δ_β^α+∂ξ^α

∂x^β , (2.13)

and we define:

x^α=x⁰^α−ξ^α(x^β), (2.14)

and its derivative:

∂x^α

∂x^0γ =δ_γ^α−∂ξ^α

∂x^β

∂x^0γ 'δ_γ^α− ∂ξ^α

∂x^γ . (2.15)

We get the right-hand part by substituting _∂x^∂x₀^βγ in the same way we are doing with

∂x^α

∂x⁰^γ, this is an iteration. Then considering|^∂ξ_∂x^α^γ| 1 for allα, β, we can neglect terms of second and superior order. We have also used the fact that the Kronecker delta is the same in any coordinate system.

Now, we would like to verify if the primed system is also defined in a nearly flat space-time with the form Equation (2.1). We start with the general definition:

g_αβ⁰ = ∂x^µ

∂x^0α

∂x^ν

∂x^0βgµν, (2.16)

then substituting with (2.1) and (2.15), we get:

g⁰_αβ =

δ_α^µδ^ν_β− ∂ξ^µ

∂x^αδ^ν_β− ∂ξ^ν

∂x^βδ^µ_α

η_νµ+δ_α^µδ_β^νh_µν, (2.17) we can simplify it using the fact that we raise and lower index in nearly flat space-time withη_µν:

g⁰_αβ =η_αβ +h_αβ − ∂ξ_β

∂x^α − ∂ξ_α

∂x^β . (2.18)

Now we get what we call the gauge transformation rewriting the above equation:

h⁰_αβ =h_αβ− ∂ξ_β

∂x^α − ∂ξα

∂x^β , (2.19)

then we have the same form as (2.1):

g_αβ⁰ =η_αβ+h⁰_αβ. (2.20)

We can also note that, if|^∂ξ_∂x^α^γ| 1 then |_∂x^∂ξ^α^γ| 1 andh⁰_αβ is also small.

We conclude that once we have identified a coordinate system for which we have (2.1), we can apply a coordinate transformation (2.12), that corresponds at the first order to the gauge transformation (2.19), without altering our assumption that the space-time is nearly flat (providing that ξ^α is small).

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2.1.4 Linearization of the Einstein equations

At the first order inh_µν and using the Ricci tensor (2.8) defined before, the Einstein equations can be defined as:

R⁽¹⁾_µν =−8πG c⁴

T_µν−1 2η_µνT_ρ^ρ

. (2.21)

We take T_µν at the lowest order in g_µν and we use the property that it satisfies the continuity equation² :

∂_µT_µν = 0. (2.22)

This equation can be simplified by introducing the symetrique tensor:

Sµν =Tµν− 1

2ηµνT_ρ^ρ, (2.23)

which has the properties:

S^ρ_ρ=−T^ρ_ρ, (2.24)

and from the continuity equation, we get:

∂_µS^µ_ν = 1

2∂_νS^ρ_ρ. (2.25)

Then we rewrite the Einstein equations in weak gravitational field:

∂^ρ∂_ρh_µν+∂_µν² h^ρ_ρ−∂_µρ² h^ρ_ν−∂_νρ² h^ρ_µ=−16πG

c⁴ S_µν. (2.26)

To simplify it further we find a gauge transformation as define in Equation (2.19) that will eliminate the last three terms of the left-hand side of the equation. So we substitute in the Equation (2.26):

h⁰_µν =hµν−∂µξν−∂νξµ, (2.27) We already know that this transformation keeps the properties of the nearly flat space- time. But we need to verify that ifhµν is a solution of the Einstein equations thenh⁰_µν is also solution of the Einstein equations:

∂^ρ∂ρh⁰_µν+∂_µν² h⁰_ρ^ρ−∂_µρ² h⁰_ν^ρ−∂_νρ² h⁰_µ^ρ (2.28)

= ∂^ρ∂_ρh_µν+∂_µν² h^ρ_ρ−∂_µρ² h^ρ_ν −∂_νρ² h^ρ_µ (2.29)

−∂^ρ∂_ρ(∂µξ_ν+∂_νξ_µ)−2∂²_µν∂_ρξ^ρ (2.30) +∂_µρ² (∂^ρξ_ν +∂_νξ^ρ) +∂²_νρ(∂^ρξ_µ+∂_µξ^ρ) (2.31) The addition of the two last lines (2.30) and (2.31) is equal to zero. It confirms that our gauge does not change the Einstein equations.

2. In General Relativity we should apply the covariant derivative DµT^µν = 0. In linearized approximation the covariant could be approximate by conventional derivative.

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2.1. THE NATURE OF GRAVITATIONAL WAVES 11 Then it remains to eliminate the three last terms of line (2.29), the gauge that corresponds to the change of variable (2.27) is the Lorenz gauge³ which is defined by:

∂_µh^µ_λ = 1

2∂_λh^µ_µ. (2.32)

The demonstration of this part is in Appendix B.1. We can rewrite this gauge grouping the expressions ofh^µ_λ in the form:

h¯^µ_λ =h^µ_λ−1 2η^µ_λ h^ρ_ρ and the gauge is written:

∂_µh^µ_λ−1

2∂_λh^µ_µ=∂_µ¯h^µ_λ = 0 (2.33) which has the same form as the Lorenz gauge used in electrodynamics ∂_µA^µ= 0.

To conclude, in a weak gravitational field using our gauge and boost, we can always write the Einstein equations in the fully covariant form:

∂^ρ∂_ρh_µν =−16πG

c⁴ S_µν. (2.34)

2.1.5 Solution of the Einstein equations in an empty space and Grav- itational Waves

Far from masses, so in a free space, we take the energy momentum tensorT_µν equal to zero and the Einstein’s equations are written:

∂^ρ∂_ρh_µν(x) = 0, (2.35)

with the gauge conditions defined in Equation (2.32). This equation is the equation of waves (∇²−∂_t²)hµν = 0. The general solution of these equations is:

hµν(x) =eµνe^ik^ρ^x^ρ+e^∗_µνe⁻^ik^ρ^x^ρ, (2.36) with the conditions:

k^µk_µ= 0 and k^µeµν = 1

2k^νe^µ_µ,

wheree_µν is the polarization tensor andk^µ= (ω, ~k) with c= 1 and~k the wave vector.

With this definition of k^µ, we have the following dispersion relation:

ω² =|~k|². (2.37)

At this stage we have shown that small perturbation on the flat metric represented by h_µν are waves propagating with the speed of light.

3. The Lorenz condition is named after Ludvig Lorenz but it is often confused with Hendrik Lorentz (of the boost transfromation) and written Lorentz gauge [14].

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2.1.6 The transverse-traceless gauge

From now we will always consider the case of an empty space where T_µν = 0. We have started with hµν which contains 16 components. Using the symmetry of hµν

it remains 10 free components. We have also the conditions imposed by the Lorenz gauge: ∂_µh^µ_λ = ¹₂∂_λh^µ_µ, then deriving the solution (2.36) we get k^µe_µν = ¹₂k^νe^µ_µ and using the compact form (2.33) we get:

¯k^µe_µν = 0. (2.38)

where ¯k is the wave vector defined with the compact form of ¯h. This means that the direction of propagation is orthogonal to the tensor of polarization. Another point, is that the relation (2.38) imposes four more equations. So we pass from 10 to 6 degrees of freedom. We still have some possibilities to simplify the equations. The Lorenz gauge conditions do not fix all the degree of freedom. We can add another change of variablesx⁰⁰^µ=x⁰^µ+^ν(x⁰^ν) which gives the gaugeh⁰⁰_µν =h⁰_µν−∂µν−∂νµ. This change of variables does not change our Lorenz gauge if:

∂^ν∂_ν_µ= 0. (2.39)

Now we can rewrite the general solution of the Einstein equation in an empty space using the last change of variables.

h⁰⁰(x)µν =e⁰⁰_µνe^ik^ρ^x^ρ+e^00∗_µνe^−ik^ρ^x^ρ, (2.40) where

e⁰⁰_µν =e_µν+k_µ_ν+k_ν_µ. (2.41) We examine this equation in one particular direction of propagation, let us choose z, then k^µrevrites:

k^µ= (k,0,0, k). The Lorenz gauge condition imposes that:

k(e_0ν +e_3ν) = 1 2k_ve^ρ_ρ, which gives:

e₀₁=−e₃₁, e₀₁=−e₃₁,

e₁₁=−e₂₂, e₀₃=−¹₂(e00+e₃₃). (2.42) With the relation (2.41), we use our last gauge choosing

₀= e₀₀

2k, ₁ =−e₁₃

k , ₂ =−e₂₃

k , ₃=−e₃₃

k . (2.43)

It gives

e⁰₁₁=e₁₁, e⁰₁₂=e₁₂,

e⁰₂₁=e₂₁, e⁰₂₂=e₂₂=−e₁₁, (2.44) and all other e⁰_µν = 0.

In conclusion, we have used our last four degrees of freedom by imposing (2.43) and we remain with two free components. The Einstein equations in an empty space can be rewritten omitting the prime:

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2.2. INTERPRETATION OF FRAMES AND EFFECTS OF GRAVITATIONAL WAVES13

hµν(x) =eµνe^ik^ρ^x^ρ+e^∗_µνe⁻^ik^ρ^x^ρ, (2.45) with

e_µν =







0 0 0 0

0 e11 e12 0 0 e₁₂ −e₁₁ 0

0 0 0 0







. (2.46)

The matrix of polarization e_µν is transverse and traceless, so we call the last gauge (2.39)the Transverse Traceless gauge, abbreviated the TT gauge.

To finish, we redefine the temporal part of 2.45 using only the real part and positive propagation:

h₁₁(x) =h₁₁cos(k⁰x⁰−k³x³) andh₁₂(x) =h₁₂cos(k⁰x⁰−k³x³),

then we take into acount that k⁰=k³ and giving explicitly the coordinates, we get:

h_xx(t,−→x) =h_xxcos [k(t−z)] (2.47) and

h_xy(t,−→x) =h_xycos [k(t−z)]. (2.48)

2.2 Interpretation of frames and effects of gravitational waves

2.2.1 Interpretation of the TT frame

We will consider a GW arriving on two free test particles. It means that we are considering the effect of the perturbation hµν on the flat metric and its effect on the test particles. We consider particles at rest. This means that, at time t= 0:

x^µ(0) =x^µ and dx^µ

dτ =cδ₀^µ,

where τ is the proper time. The trajectory of a particle is described by the geodesic equation:

d²x^µ

dτ² + Γ^µ_νρdx^ν dτ

dx^ρ

dτ = 0. (2.49)

If we substitute with the linear approximation of the Christoffel tensor:

Γ^µ_νρ= 1

2η^µα[∂ρhαν+∂νhαρ−∂αhρν], (2.50) we get in particular,

Γ^µ₀₀= 1

2η^µα[2∂0hα0−∂αh00] = 0, (2.51) then, in TT gauge the hα0 and h00 are equal to zero everywhere. All the components of the Christoffel tensor are equal to zero Γ^µνρ= 0 for any point P. Then the geodesic equation becomes:

d²x_µ dτ²

P

= 0. (2.52)

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Hence in the TT gauge particles at rest remain at rest indefinitely. This kind of system is called a “freely falling frame”.

At this stage one could think that nothing happens. However we have examined the evolution of the coordinates. In general relativity the coordinates are not physical quantities. We have to examine the proper distance. For example, if we consider two particles, one at the origin and the second one xⁱ = (δ,0,0). The proper distance is given by:

∆l=Z _δ

0

√ds² =Z _δ

0 |g_αβdx^αdx^β|^1/2 'δ|g_xx(0)|^1/2. (2.53) We evaluated the metric at the origin. As h_xx 1 in our approximation of a nearly flat space, we can do the following approximation:

∆l≈δ[1 + 1

2h_xx(x= 0)]. (2.54)

Now we see, sincehxxis not constant, that the proper distance changes as the gravitational waves pass. It is this change in the distance that detectors attempt to measure.

The coordinates in the TT frame stretch in such a way that the positions marked by the coordinates are always the same.

2.2.2 The TT frame and polarization

To understand what is the effect of the polarizations found in Section 2.1.6, we examine the geodesic deviations on a ring of free particles, see Figure 2.1. The ring lies on thexy plane and the GW travels on the zdirection.

The acceleration of the geodesic deviation is given by:

D²ζ^µ

Dτ² =R^µ_αβγv^αv^βζ^γ, (2.55) whereζ is distance of separation of two geodesics marked by two test particles,τ is the proper time, x^µ are the coordinates of one test particle andx^µ+ζ^µthe coordinates of the other one and the speed v^α= ^dx_dτ^α. The covariant derivative is defined by:

DA^µ

Dτ = dA^µ

dτ + Γ^µ_αβA^αdx^β dτ , where A^µ is a general vector field.

In a weak gravitational field we can take the proper time equal to the time: τ ≈t, then Equation (2.55) rewrites:

∂²ζ^µ

∂t² =R^µ_αβγU^αU^βζ^γ, (2.56) whereU^α= ^dx_dt^α. As before we consider particles initially at rest, one at the origin and the other one at a distanceon the circle as shown in Figure 2.1. ThenU^µ= (1,0,0,0) and ζ^µ= (0, cosθ, sinθ,0) and we get :

∂²ζ^µ

∂t² =R^µttxcosθ+R^µttysinθ . (2.57)

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2.2. INTERPRETATION OF FRAMES AND EFFECTS OF GRAVITATIONAL WAVES15

Figure 2.1: Illustration of the gravitational effects on a ring of free particles.

We substitute R^µ_ttα by its linearized expression of the first order inh_µν: R_αβγδ= 1

2(∂β∂_γh_αδ+∂_α∂_δh_βγ−∂_α∂_γh_βδ −∂_β∂_δh_αγ) , (2.58) and finally we get two expressions withh defined in TT gauge:

∂²ζ^x

∂t² = 1 2

cosθ∂²

∂t²h_xx+ sinθ∂²

∂t²h_xy

, (2.59)

and ∂²ζ^y

∂t² = 1 2

cosθ∂²

∂t²h_xy−sinθ∂²

∂t²h_xx

. (2.60)

Then we substitute with the definitions (2.47) and (2.48), and we can also set z = 0 since the behavior of the ring does not depend on the positionz, we get:

ζ^x =cosθ−1 2

ω² (cosθh_xxcos(ωt) + sinθh_xycos(ωt)), (2.61) and

ζ^y =sinθ−1 2

ω² (cosθh_xycos(ωt)−sinθh_xxcos(ωt)). (2.62) Now we can see what is the effect of the two polarizations. First we look at the case when h_xx 6= 0 and h_xy = 0.

ζ^x = cosθ−1

2cosθh_xxcos(ωt), (2.63) ζ^y = sinθ+1

2sinθhxxcos(ωt), (2.64)

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If we examine the oscillation for all angles between 0 and 2π, we have contraction and dilatation of the proper distances on the x and y axes. This situation corresponds to the left scheme in Figure 2.1. The amplitudeh_xx is noted h₊, where + represents the direction of its effects on the particles.

When hxx = 0 and hxy 6= 0, we get:

ζ^x = cosθ−1

2sinθh_xycos(ωt) (2.65)

ζ^y = sinθ−1

2cosθh_xycos(ωt) (2.66)

Now we have contraction and dilatation of the proper distances on the diagonals, as represented in the right scheme of Figure 2.1. The amplitude hxy is noted h_×, also where ×represents its effects on the particles.

To conclude this section:

The plus and cross notation corresponds to the orientations of these dilations and contractions. Then we write usually the matrix of the amplitudes of h by:





h₊ h_× 0 h_× −h₊ 0

0 0 0



 . (2.67)

If we proceed to rotation of the coordinate system, we can show that we can pass from the set of Equation (2.63) and (2.64) to the set (2.65) and (2.66) by a rotation of π/4.

2.3 The laboratory referential

Until now we developed our equations in a free falling frame. But on Earth, in our laboratory, we do not use free falling frame. The frame of our laboratory takes into account the gravitation of Earth, its motion in space (self rotation of Earth, rotation around the sun and we can include galactic displacement, see [15]) and finally the gravitational waves.

How can we detect the effects of gravitational waves between other effects? Ac- tually, the GWs that we are looking for are expected to have frequencies around kHz and Earth effects are of the order of Hz. So our detectors work in a frequency band where we can beat the noise, eliminate other gravitational effects and expect certain GW signals. For instance, we need to go in space to observe frequencies below 1kHz.

At high frequencies our detectors are also limited by other sources.

Now we will see that the effects of gravitational waves can be translated into effects of a force in the laboratory frame. This is done starting with the geodesic deviation Equations 2.56. This equation could be simplified as shown in [3]:

∂²ζ^µ

∂t² =−c²Rⁱ_0j0(U⁰)²ζ^j. This equation is valid under a few conditions:

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2.3. THE LABORATORY REFERENTIAL 17 – The detector does not have a relativistic motion so dxⁱ/dτ is negligible with

respect to dx⁰/dτ.

– We have also used the approximation already discussed for weak gravitational fields dτ 'dt.

– Using pure time-like or proper distance geodesics in TT frame, it can be shown that Γ^µνρ(P) = 0 at P the point where we develop the geodesic deviation.

– We use the linearized Riemann tensor R_0j0ⁱ which is frame invariant. The non linearized form is rather covariant.

The last point gives a hint how to compute the Riemann tensor. We use its form computed in TT frame (2.58). We get then:

∂²ζ^µ

∂t² = 1

2h¨^{T T}_ij ζ^j.

I recall thatζ is the distance between two test mass, so ^∂_∂t²^ζ2^µ is the relative acceleration between them. Now we can write it in a more useful way, in terms of forces:

F_i= m

2¨h^{T T}_ij ζ^j, (2.68)

where m is the mass of one test mass. This force is starting point to understand how works resonant mass gravitational waves detectors that we will see in the next chapter.

2.3.1 Limit of the approximations

In the Equation (2.56), when we take the difference between two geodesics, we evaluated Γ^µ_αβ(x) and Γ^µ_αβ(x+ζ) at two different points. We could approximate that they are equal, expanding them to first order inζ. This approximation is only possible if we take |ζ|smaller than the scale on which changes the gravitational field.

So on which scale could we make our approximation? One definition is given taking the reduced wavelength [3] and the size of our experiment L:

Lλ/2π . (2.69)

In the case of bars and miniGRAIL, all our approximations fulfill perfectly this condition. For example, the frequency of expected gravitational wave for miniGRAIL are around 3kHz and the experiment is ∼0.5m so we have:

λ/2π '15915mL'0.5m.

To clarify the ideas, if the experiment were bigger than the length scale of the GW, then, for example, one bar will experiment forces going in one direction and in the opposite direction. All the variations over the lenght of bar will cancel and we could not measure gravitational wave. If the experiment is of the order of GW length then the force is not constant on all the length or the perturbation h is not constant. For other experiments this condition is not so clearly fulfilled.

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Data analysis pipeline for the spherical gravitational wave antenna MiniGRAIL

Thesis

Reference

Data analysis pipeline for the spherical gravitational wave antenna MiniGRAIL

Data Analysis Pipeline For The Spherical Gravitational Wave Antenna MiniGRAIL

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i n t i t u l é e :

"Dqlo Anolysis Pipeline For The Sphericol Grovilqlionol Wove Anlenno MiniGRAl["

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Le

Remerciements

R´ esum´ e

Contents

Chapter 1

Introduction

1.1 Thesis outline

Chapter 2

Effect Of Gravitational Waves On Matter.

2.1 The nature of Gravitational Waves

2.2 Interpretation of frames and effects of gravitational waves

2.3 The laboratory referential