An improved detector for non Gaussian stochastic background of gravitational waves.

Testo completo

(1)dipartimento di fisica dell’università di pisa Tesi di Laurea Magistrale. AN IMPROVED DETECTOR FOR NON GAUSSIAN STOCHASTIC B A C K G R O U N D O F G R AV I TAT I O N A L WAV E S. Candidato Riccardo Buscicchio. Relatore. Controrelatori. Dr. Giancarlo Cella. Dr. Isidoro Ferrante Dr. Walter Del Pozzo. Anno accademico 2015-2016.

(2) Riccardo Buscicchio: An improved detector for non Gaussian stochastic background of gravitational waves, Tesi di Laurea Magistrale, © November 2016.

(3) Le sfide sono la nostra cocaina. — A.L. Sansò. He who works with the door open gets all kind of interruptions, but he also occasionally gets clues as to what the world is.. — R. Hammings.

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(5) ABSTRACT. The thesis purpose is mainly to show how some common techniques of quantum field theories could be applied to the design of better detection algorithms, in the context of stochastic background of gravitational waves. The thesis is structured as follows: in Chapter 1 we introduce the framework of propagating wave solutions of Einstein equations. We outline the main features of interferometers, including their coupling to gravitational wave radiation. Then, we summarize the features of our reference cosmological model, a Friedmann-Robertson-WalkerLemaitre universe. We show the performance of single and multiple interferometers detection. In Chapter 2 we overview two interesting sources for our study: cosmic strings cusps and binary black hole coalescences. For the former, we outline the spontaneous symmetry breaking mechanisms, and the formation of topological linear defects. We quantify the associated spectrum of gravitational wave, and characterize in time domain the average signal overlap, as seen from a single interferometer. For the latter, we describe two fiducial cosmological population, and define three different regimes for the background, afflicting deeply the best detection strategy. In Chapter 3 we introduce the basics of stochastic processes, in particular of Campbell processes. We overview two equivalent probabilistic structures behind them. We show how to evaluate their mean values via a generalized concept of probability distributions. We carry out relevant analogies with field theory correlation functions. The reader will be provided here with the necessary tools to manipulate the procedure explained Chapter 5. In Chapter 4 we review the general framework of decision theory. We describe a standard approach to signal detection, and the tools needed to characterize a detector performances. We formulate the criterion for an optimal detection strategy, the Neyman-Pearson lemma. We use some toy signals (both deterministic and stochastic) to split the different aspects of the detection problem into smaller parts. We derive analytically the standard optimal detector for a Gaussian background and study its performances. Then, we switch our attention to the detection of a non Gaussian stochastic background. We show a simple perturbative expansion of the likelihood, assuming the correlators to be hierarchically ordered by a scaling parameter. We show how to construct the relevant statistics of the data. We draw out some considerations on the computational issues of such algorithm.. v.

(6) With all the necessary mathematics in hand, in Chapter 5 we tackle the detection strategy by employing a re-summation technique borrowed from re-normalization group theory. We explain the technique in detail with a simpler example of a single variable, then we extend the results for multiple detector timeseries. They will find best confirmations with both numerical and analytic validation to appear in a proposed scientific paper. Throughout the whole text we will underline the analogies and differences with respect to quantum field theories probabilistic structure, considered in many passages as a compass for right direction. Finally, in Chapter 6 we summarize the results, and draw out some consideration on future developments of the proposed approach.. vi.

(7) SOMMARIO. Scopo principale della tesi è di mostrare come alcune comuni tecniche delle teorie di campo quantistiche possano essere utilizzate per la progettazione di algoritmi migliorati di detection, con particolare attenzione ai fondi stocastici di onde gravitazionali. Nel capitolo 1 descriviamo le soluzioni propaganti delle equazioni di Einstein. Riassiumiamo le principali caratteristiche degli interferometri, compreso il loro accoppiamento alle onde gravitazionali. Poi, descriviamo le caratteristiche del nostro modello cosmologico di riferimento, l’universo di Friedmann-Robertson-Walker-Lemaitre. Introduciamo e descriviamo le performance della detection tramite singoli interferometri, o tramite un network di essi. Nel capitolo 2 descriviamo due sorgenti interessanti come oggetto del nostro studio: cuspidi di stringhe cosmiche e coalescenze binarie di buchi neri. Per le prime, introduciamo i meccanismi di rottura spontanea di simmetria, e la formazione di difetti topologici lineari. Quantifichiamo lo spettro di onde gravitazionali associato e caratterizziamo nel dominio temporale l’overlap medio del segnale, per come è visto da un singolo interferometro. Per le seconde descriviamo due modelli di riferimento della popolazione cosmologica, e distinguiamo tre differenti regimi di generazione del fondo stocastico. Essi influenzeranno profondamente la scelta dell’algoritmo ottimale. Nel capitolo 3 introduciamo le basi matematiche per trattare i processi stocastici, fornendo due equivalenti strutture probabilistiche sottostanti. Mostriamo come valutare i valori medi e manipolare le distribuzioni di probabilità associate a questi processi. Rendiamo esplicite le analogie con le funzioni di correlazione in teoria di campo. Il lettore avrà a disposizione a questo punto gli strumenti necessari per manipolare la strategia di detection proposta nel capitolo 5. Nel capitolo 4 descriviamo il framework generale della teoria della decisione. Descriviamo l’approccio standard alla detection, e gli strumenti necessari a caratterizzare le prestazioni di un detector. Formuliamo il criterio per costruire la strategia di detection ottimale, il lemma di Neyman-Pearson. Presentiamo dei modelli giocattolo (sia deterministici che stocastici) per frammentare le diverse difficoltà che sorgeranno nel caso completo, e risolvere separamente. Deriviamo analiticamente il detector standard ottimale per un fondo stocastico Gaussiano, e descriviamo le sue performance. Poi, rivolgiamo la nostra attenzione alla rivelazione di fondi stocastici non gaussiani. Costruiamo prima una semplice espansione perturbativa della likelihood. Per farlo assumeremo che le correlazioni coinvolte soddisfinino una opportuna legge di scala, sotto forma di potenza di un parametro (piccolo). Dopo aver costruito le statistiche. vii.

(8) dei dati associate, discutiamo le difficoltà computazionali dell’ algoritmo. Con la matematica descritta a disposizione, nel capitolo 5 affrontiamo la strategia di detection impiegando una tecnica di risommazione presa in prestito dalla teoria del gruppo di rinormalizzazione. Descriviamo questa tecnica in dettaglio con un esempio semplice di singola variabile, poi estendiamo i risultati alle serie temporale di un network di rivelatori. Tali risultati troveranno conferme tramite validazioni numeriche e analitiche che verranno presentate nella pubblicazione che accompagnerà questa tesi. Nel corso di tutta l’esposizione sottolineeremo le analogie e le differenze con la struttura probabilistica delle teorie di campo quantistiche, che considereremo quando possibile come bussole per la giusta direzione da seguire. Infine, nel Capitolo 6 riassiumiamo i risultati, e accenniamo alcune considerazioni sugli sviluppi future dell’approccio presentato.. viii.

(9) P U B L I C AT I O N S. Some ideas and figures have appeared previously in the following publications:. [1]. R. Buscicchio and G. Cella. “An improved detector for non Gaussian stochastic background of gravitational waves.” To be submitted.. ix.

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(11) ACKNOWLEDGMENTS. To my parents: they allowed me to play as a child, and to keep doing it as an adult. To my brother, as my first physics teacher: I stand on his shoulder, as everyone do with their giants. To my sister: she always loved me as being extraordinary. Finally, she made me believe I am. To my father: he showed me the value of silence and beliefs, as two different ways of staying closer. To my mother: she showed me goodness and art, as different ways of being alive. To my mirror, Irene. To my travel companions: Francesco and Sapiens. To my neurons: they did a great job, together. To physics, which allowed me to accept death, peacefully. And to its great sister, for his encouraging coherence. To my advisor, an excellent teacher of both, a shining and tireless good example. Lastly, to the daily support of my most intimate discovery. RINGRAZIAMENTI. Ai due che mi hanno permesso di vivere, giocare, e continuare a farlo da adulto: i miei genitori. Al mio primo insegnante di fisica, mio fratello: mi reggo sulle sue spalle, come ognuno fa con i suoi giganti. A chi mi ha sempre trovato straordinario, al punto di convincermene: mia sorella. A chi mi ha insegnato le convinzioni e il silenzio, come espressione dello stesso modo di restare uniti, papà. A chi mi ha mostrato la bontà e l’arte, come espressione della stessa voglia di vivere, mamma. Al mio specchio, Irene. Ai miei due compagni di viaggio, Francesco e Sapiens. Ai miei infaticabili neuroni, per l’ottimo lavoro di gruppo. A questa disciplina, che mi ha dato una percezione pacifica della morte. E alla sua sorella maggiore, per la sua ineludibile coerenza. A un eccellente maestro di entrambe, un fulgido e infaticabile buon esempio: il mio relatore. E infine, al sostegno quotidiano della mia più intima scoperta.. xi.

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(13) N O TAT I O N. Throughout the whole thesis the reader will find symbol explanations in the text. Here a little overview of the most frequent rules adopted. We will use greek lowercase indices µ, ν, . . . for spacetime coordinates, with metric gµν and signature diag (1, −1, −1, −1). An exception is made for τ and σ. They will be reserved in Chapter 2 to the timelike and spacelike coordinate on the two dimensional surface of a string worldsheet. In Chapter 3 σ, σ0 , . . . will be used in reference to the random dots occurrence. Latin lowercase i, j, . . . are used for three dimensional vector components, space components of 4−vectors, or for generic summation. Bold symbol v and associated operators will be employed as well whenever clarity requires it. Einstein rule of summation over repeated indices is followed. Generic parametric arrays (no metric structure assumed therein) are denoted by underlined symbols. The occurrence of multiple integration requires us to contract the notation for compactness. We will generally: indicate any continuous arguments (for example t) of a function f with the standard parenthesis · (t). Multiple arguments will be written sistematically or compactified where necessary from f (t1 , . . . , tn ) into f (t) with obvious meaning. The cardinality of the arguments should be clear from the context. Sometimes the integrations themselves will be reduced to index summation, then recovered in the last step of any derivation. This transition will be denoted by the symbol l. As an example Z X f (x) g (x) dx l fi gi = hi l h(x) i. Multiple integrations obey the same equivalences of multiple arguments Z Z dt1 . . . dtn → dt. ∂ Ordinary derivatives will be indicated by ∂µ , ∂x µ . Functional derivaδ tives with δs(t) . An exception is made for single variable functions f, whose derivatives are f0 . Calligraphic symbols (e.g. A) are used to identify stochastic process, as defined in Chapter 3, and for hypothesis, decisions, and sets as defined in Chapter 4. Handwritten script symbols (e.g. A ) are used for algorithm and tests. Fraktur symbols (e.g. A) denotes correlation functions matrices, and uppercase symbols their components (A ≡ {Aa , a = 1, . . . }). Cartesian products of arrays is denoted by ⊗ exponents (A⊗2 = A ⊗ A = Aa Ab ). Three symbols are chosen to denotes average: h·i,hh·ii,J·K, whose meaning is explained in the text.. xiii.

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(15) CONTENTS 1. 2. 3. 4. 5. 6 a. introduction 1 1.1 Linearized gravity 1 1.2 Test masses and interferometers 4 1.3 Gravitational wave background 9 1.4 FRWL universe and cosmological density parameters 12 1.5 Available SNRs and energy densities 15 sources: two examples 17 2.1 Topological defects 17 2.2 The wire approximation 20 2.3 Loops and cusps in flat spacetime 22 2.4 GWs from cusps: Minkowski spacetime 25 2.5 GW from cusps: FRWL universe 27 2.6 “One-scale” string network 28 2.7 Spectrum and effective number of sources 30 2.8 Unresolved binary black hole coalescence 31 2.9 Cosmological population of binary black holes 32 2.10 Duty cycle and detection regimes 33 mathematical theory of stochastic processes 37 3.1 Basic properties of stochastic variables 37 3.2 Campbell processes 41 3.3 Generating functions 42 3.4 The functions fn , gn , and hn 43 3.5 Connected correlation function 44 3.6 Back to the probability 45 3.7 Stochastic processes and field theory 46 signal detection theory 49 4.1 Decision tests 49 4.2 Optimal test: the Neyman-Pearson lemma 51 4.3 Matched filter for single or multiple events 55 4.4 Unresolvable sources: stochastic approach 56 4.5 Gaussian background: matched filtering on correlation 57 4.6 Non Gaussian background: perturbative optimal detector 62 4.7 Feasibility considerations 64 improved detection algorithm 67 5.1 Single variable: Gram-Charlier re-exponentiation 67 5.2 Single variable: likelihood 72 5.3 Multiple detector correlations 72 5.4 Timeseries array: Gram-Charlier re–exponentiation 74 5.5 Timeseries array: likelihood and statistics 75 conclusions 77 appendix: gram-charlier a series 79. xv.

(16) xvi. contents. b appendix: neyman-pearson lemma 81 c appendix: L NG (s) expansion 83 d appendix: coefficient symbolic computation bibliography. 87. 85.

(17) LIST OF FIGURES. Figure 1.1. Figure 1.2. Figure 1.3 Figure 1.4. Figure 1.5. Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5. Figure 2.6. Polarization tensors plus e+ (ˆz) (left) and cross e× (ˆz) (right), depicted via the linear maps r˙ = eA r along the z−axis. 4 The mesh of a transverse traceless gauge (TT) reference, as stretched by the arrival of a gravitational wave (GW) travelling along z−axis. Nodes on the grid mark constant coordinates values. However, proper time is affected as measured by a travelling photon. 6 Angular sensitivity, depicted as D2+ (left) and 7 D2× (right) for ψ = 0 in polar coordinates. Equivalent strain noise spectral density (loglog plot) for Advanced Virgo interferometer (uppermost black solid curve), and some of the main components in the relevant frequency regime. The plot is an adaptation from [5, fig.2]. 8 ˆ v, ˆ the azArrangement of the detector arms u, ˆ and the wave axis n, ˆ p, ˆ q. ˆ Animutal axis w, gles θ and φ are the spherical coordinates of ˆ The angle ψ compares plus polarvector −n. ization axis pˆ to spherical unit vectors eθ and eφ . 10 Vacuum states for exact (sx) and spontaneously broken (dx) symmetries. 18 The construction of a point P out of equilibrium for the field. 20 Tailored coordinate on the worldsheet neighbourhood 21 Kibble-Turok sphere, showing two path defining a loop with cusps. 23 A loop at two different close values of τ. xc , vc and sc give the instantaneous cusp position, speed and spread. 24 Detailed shape of a cusp (sx) and its time development (dx). 25. xvii.

(18) Figure 2.7. Figure 2.8. Figure 2.9. Figure 4.1. Figure 4.2. Figure 4.3 Figure 4.4 Figure 5.1. Figure 5.2. xviii. Gravitational wave amplitude of bursts emitted by cosmic string cusps in the LIGO/VIRGO frequency band, as a function of log α = 50Gµ, ˙ = 1 yr−1 and f = 150 Hz. with fiducial values N The horizontal dashed lines indicate the effective SNR = 1 noise levels (after optimal filtering for |f|−1/3 ) of LIGO 1 (initial detector) and LIGO 2 (advanced configuration). The shortdashed line indicates the confusion noise part of the signal. Figure adapted from [19, Fig.1] 30 Cosmic evolution rate factor e(z) for two different reference models (a cutoff delay time is assumed to 100Myr). Solid line refers to [21], dashed one refers to [37]. Figure adapted from [38]. 33 Energy density parameter Ωgw as a function of observed frequency for the BBH background corresponding to [21] (solid curve) [37] (dashed curve). Solid-dotted line shows the power-law approximation. Figure adapted from [38, Fig.3]. 34 receiver operating characteristic (ROC) curves for Bernoulli-like tests (solid line) and informative tests (dashed curve). The lower left corner is the most relevant part, since we want a high PD for a fixed, low, PFA . A fiducial value of PFA = 10−5 is usually considered. 51 ROC for the matched filter. Arrows point towards increasing iso−PFA curves. Each curve is labeled with its PFA value. 54 Diagrammatic rules for the perturbative expansion of LNG 64 Λk value at α2 order. External legs are removed, combinatorial coefficients are included. 65 A new symbol is introduced as a diagrammatic representations for theΓ1 function. The generalization to n nodes is obvious. 76 An example of differential equation arising from the expansion (5.24). Some terms of dΓ1 /dα are shown for simplicity at α order. Combinatorial coefficients are left implicit, being irrelevant for the outlined procedure. 76.

(19) LISTINGS. Listing D.1. Sympy script for symbolic computation. 85. ACRONYMS. BBH binary black holes CLT. central limit theorem. CMB cosmological microwave background FRWL Friedmann-Robertson-Walker-Lemaître GCE Gram-Charlier expansion GLRT generalized likelihood ratio test GR. general relativity. GW. gravitational wave. NPl. Neyman-Pearson lemma. QFT quantum field theory ROC receiver operating characteristic SFR. star formation rate. SNR signal-to-noise ratio SGWB stochastic gravitational wave background TT. transverse traceless gauge. xix.

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(21) 1. INTRODUCTION. summary In Section 1.1 we introduce the framework of linearized gravity. We describe propagating wave solutions of Einstein equations. In Section 1.2, we describe interferometers and their coupling to GW radiation, including the main noise sources. In Section 1.4 we summarize the (background) signal spectral properties, and in Section 1.5 we connect them to universe energy densities. We show the performance of single and multiple interferometers via their signal-to-noise ratio (SNR) performances over time. We leave a stochastic description of the spectra to Chapter 3, and a summary of relevant sources to Chapter 2. 1.1. linearized gravity. Standard gravitational waves arise, in the context of general relativity (GR), as spacetime perturbation travelling waves. They could be understood, straightforwardly, as a natural consequence of the existence of a speed limit for the propagation of physical influences. However, their geometrical meaning arises explicitly in GR. For this purpose, we start from the Einstein equations for the metric gµν 1 8πG Gµν = Rµν − gµν R = 4 Tµν (1.1) 2 c involving up to second order gµν derivatives (in Ricci tensor Rµν and scalar curvature R) as well as the energy-momentum tensor Tµν , associated to matter and radiation of the system. We will look for a perturbative solution of (1.1) with respect to a flat spacetime metric ηµν gµν = ηµν + hµν |hµν | 1 The GR invariance under diffeomorfism xµ → x0µ = Fµ (x) must be restricted consistently to an appropriate set of reference frames, everyone exhibiting small hµν . The result is the Poincarè group, with the addition of small diffeomorfisms xµ → x0µ = xµ + ξµ (x). 1.

(22) 2. introduction. The dynamical field then becomes hµν , with associated gauge invariance h0µν = hµν − (∂ν ξµ + ∂µ ξν ). (1.2). By making use of the expansion for gµν , we can linearize the differential operator Gµν and transform (1.1) into an equation for hµν . Any index contraction can be carried over with ηµν at leading order. The result is the gauge invariant equation of motion for hµν 16πG Tµν c4 1 = hµν − hα 2 α. h¯ µν + ηµν ∂ρ ∂σ h¯ ρσ − ∂ρ ∂ν h¯ µρ − ∂ρ ∂µ h¯ νρ = − h¯ µν. (1.3). Choosing a suitable gauge (the so-called Lorenz gauge) ∂µ h¯ µν = 0. (1.4). one can cancel out three left members in (1.3) obtaining 16πG Tµν (1.5) c4 with the conservation of the energy-momentum tensor arising as a consistency condition h¯ µν = −. ∂µ Tµν = −. c4 α ∂ ∂α ∂µ h¯ µν = 0 16πG. (1.6). By contrast, in full GR we have a non-conserved energy-momentum tensor, as shown with the introduction of covariant derivatives µ λ λ Dµ Tµν = ∂µ Tµν − Γµν Tλµ + Γµλ Tν. =0 since in general matter and radiation exchange energy and momentum with the gravitational field, too. Thus, the physical meaning of the approximation (1.6) is clearer now: GW sources propagates and interact with each other in a reference spacetime (flat, with ηµν metric in this case); their energy-momentum tensor is conserved, and their self-coupling to the gravitational field is neglected. Test masses, far from the sources, are affected by a metric perturbation gµν = ηµν + hµν satisfying h¯ µν = 0 whose solution are free waves propagating at light speed. The generation of these waves by their sources is described by the integral of the inhomogeneous (1.5) over the source volume V Z 0| |x 4G 1 − x 0 h¯ µν (x, t) = 4 Tµν x , t − d3 x0 c V |x − x0 | c.

(23) 1.1 linearized gravity. whose boundary value fixes the propagating part. We can further simplify the following sections, by removing once and for all the residual gauge freedom. In fact, Lorenz gauge (1.4) remains satisfied by any coordinate transformation xµ → x0µ = xµ + ξµ (x) if ξµ = 0 as shown below. With this condition the trace reversed metric h¯ µν transformation is deduced from (1.2) accordingly h¯ µν → h¯ 0µν = h¯ µν − (∂ν ξµ + ∂µ ξν − ηµν ∂ρ ξρ ) ≡ h¯ µν − Dµνρ ξρ ≡ h¯ µν − Ξµν. (1.7). with Dµνρ = ηνρ ∂µ + ηµρ ∂ν − ηµν ∂ρ then we see that h¯ 0µν satisfies Lorentz gauge as expected ∂µ h¯ 0µν = ∂µ h¯ µν − Dµνρ ξρ = 0 − ∂µ Dµνρ ξρ. = −∂µ (∂ν ξµ + ∂µ ξν − ηµν ∂ρ ξρ ) = −ξν + [∂ν , ∂µ ] ξµ = −ξν = 0 Moreover, being Dµνρ made of differential operators only ξρ = 0 ⇒ Dµνρ ξρ = 0 ⇒ Dµνρ ξρ = 0 ⇒ Ξµν = 0. Then, we can use the four independent functions ξµ to organize the physical content of a gravitational wave into the hµν components, by using (1.7). The most suitable for our purposes is the TT gauge, defined by the set of Lorentz compatible conditions on the associated T linearized perturbation hTµν hT0µT = 0. hT T. i i. =0. ∂j hTijT = 0. However, this is possible only when Tµν = 0. In fact when sources are present h¯ µν 6= 0, and Lorentz gauge constraints the transformation as before ∂µ h¯ 0µν = 0 ⇒ ξµ = 0. ⇒ Ξµν = 0. 3.

(24) 4. introduction. But we cannot set to zero any further component of h¯ µν subtracting Ξµν , since they are solutions of different differential equation (unless Tµν = 0, precisely) hµν = −. 16πG Tµν = 6 0 = Ξµν c4 ⇒ hµν = 6 Ξµν. TT. T , so the linearized vacuum equaBeing a traceless tensor hµν = hTµν tion reads immediately hTijT = 0 (1.8). and being hTijT also symmetric, the most general solution can be cast in the form of tensor plane-waves with wavevector kµ = 2πf c ,k X hij (x, k) = hA (k) exp(ikµ xµ )(eA )ij (1.9) A=+,×. with eA=1,2 corresponding to a basis for TT 3−tensors eij orthogonal to the propagation direction (as imposed by Lorentz gauge condition ki eij (k) = 0). In the particular case of a 3−vector k in the z direction, the two required tensors could be chosen as     1 0 0 0 1 0       e+ ≡ e1 =  e× ≡ e2 =   0 −1 0   1 0 0  0 0 0 0 0 0. Figure 1.1: Polarization tensors plus e+ (ˆz) (left) and cross e× (ˆz) (right), depicted via the linear maps r˙ = eA r along the z−axis.. 1.2. test masses and interferometers. The (1.9) describes a general solution of (1.8) as a plane wave and, as we will see shortly, it suffices to describe the coupling of an interferometer to a GW. We shall keep in mind that • the meaning of a perturbations hij has still to be specified from a physical point of view, since the TT reference is not necessarily meaningful on Earth.

(25) 1.2 test masses and interferometers. • interferometers outcome will be a scalar value h(t) (or, equiv˜ alently, h(f)): so hij ’s degrees of freedom are to be contracted with a tensor Dij , which depends on the relative orientation of the detector with respect to the chosen TT reference, where a precise description of GWs is available. The first point can be understood by looking at the geodesic equation associated to a point-like test mass d2 xµ dxν dxρ µ + Γ (x) =0 νρ dτ2 dτ dτ. (1.10). µ where τ is the proper time, and Γνρ are the Christoffel symbols associated to the linearized metric gµν = ηµν + hµν. 1 µ Γνρ = ηµσ (∂ν hρσ + ∂ρ hνσ − ∂σ hνρ ) 2. (1.11) i. Now, let us consider two test masses, initially at rest ( dx dτ = 0 at τ = 0) µ and marking two point with coordinates xµ and x , A B respectively. In a TT frame, by plugging (1.11) into (1.10) we obtain d2 xiA,B dx0A,B dx0A,B i (x) + Γ =0 00 dτ2 dτ dτ and observing that 1 i (x) = − (∂0 h0i + ∂0 h0i − ∂i h00 ) Γ00 2 =0 we see that test masses spatial coordinates remain unaltered over time. So, we shall take this as the simplest recipe to construct a TT frame in a laboratory: a set of objects initially at rest marking a coordinate grid (like a regularly spaced cloud of points) also during the passage of a GW. However, a physical observable remains available: the proper distance between points (as measured by a photon travelling along it), or equivalently the proper time. In fact, the null-geodesic of a photon satisfies dx20 = dxi δij + hij dxj. 5.

(26) 6. introduction. Figure 1.2: The mesh of a TT reference, as stretched by the arrival of a GW travelling along z−axis. Nodes on the grid mark constant coordinates values. However, proper time is affected as measured by a travelling photon.. and integrating its square root along the segment [0, L]1 we obtain ZL r i dxj dx ∆x0 = δij + hij (x) dl dl dl 0 r ZL dxj dxi = δij + hij (x) dl dl dl 0 ZL r dxi dxj = 1 + hij (x) dl dl dl 0 ZL 1 ˆ ˆ 1 + hij (x) Li Lj dl = 2 0 ZL 1 hij (x) Lˆ i Lˆ j dl = L+ 2 0. Assuming the typical wavelength λgw of hij big (as compared to the interferometer arms, L ∼ 3km), the integral is then easily approximated: 1 ∆x0 ' L 1 + Lˆ i hij (t)Lˆ j (1.12) 2. This is actually what a terrestrial interferometer measures: the interference between two laser beams, whose photons (originated simultaneously) have propagated along two different Ls, and will therefore be affected differently by the passage of a GW. Moreover, the second point is clear now: being s1,2 the two proper distances along the respective detector arms Lˆ 1,2 = LL1,2 , the inter-. 1 In TT gauge the integration domain (the domain of xi ) remains fixed over time, and we parameterize it with a length element dl = |d~x|.

(27) 1.2 test masses and interferometers. Figure 1.3: Angular sensitivity, depicted as D2+ (left) and D2× (right) for ψ = 0 in polar coordinates.. ference of the two beams will be sensitive to s1 − s2 , so hij will be effectively contracted with a detector tensor D (L1 , L2 ) = Lˆ 1 ⊗ Lˆ 1 − Lˆ 2 ⊗ Lˆ 2. Placing interferometer arms along two orthogonal reference axis Lˆ 1 = ˆ Lˆ 2 = y, ˆ for a wave arriving from an arbitrary direction n ˆ = (θ, φ) x, and whose polarization axis are rotated with respect to (eˆ θ , eˆ φ ) by an angle ψ, we will have an angular sensitivity as follows 1 1 + cos2 θ cos 2φ cos 2ψ − cos θ sin 2φ sin 2ψ 2 1 1 + cos2 θ cos 2φ sin 2ψ − cos θ sin 2φ cos 2ψ D× = 2 D+ =. (1.13) (1.14). The detector response to a plane wave will be characterized by a signal (1.9) h(t) = Dij hij (x, k) X ˆ ij = hA (k) exp(ikµ xµ )Dij eA (n). (1.15). A. =. X. ˆ ψ) hA (k) exp(ikµ xµ )FA (n,. A. where the geometric arrangement of the interferometer and the GW are collapsed into the pattern functions F+,× (a time dependence is implicit in Dij , since a variety of effects may change the interferometer orientation with respect to the reference plane wave, over time). Obviously one has to consider many sources of noise, affecting the signal at the different stages of its manipulation (from the laser beam emission to the optical readout). We will denote any noise source as an effective timeseries n(t) (or equivalently, by their Fourier components n(f)) ˜ to be added directly to the signal h(t). In order to assess the detectability of any signal, it will be useful to consider n(t) via the equivalent strain spectral density, namely the one-sided Fourier transform of the autocorrelation function Z +∞ hn(t)n(t + τ)i e−i2πfτ Sn (f) ≡ 2 −∞. 7.

(28) 8. introduction. Quantum. Gravity gradients Suspension thermal Coating brownian Coating Thermo-optic Substrate Brownian Seismic Excess gas OMC thermo-refractive Alignment Magnetic Sum of the plotted reference AdV curve. Frequency (Hz) Figure 1.4: Equivalent strain noise spectral density (log-log plot) for Advanced Virgo interferometer (uppermost black solid curve), and some of the main components in the relevant frequency regime. The plot is an adaptation from [5, fig.2].. From a physical point of view, any noise is generated along different stage of the signal processing (from the laser emission, to the readout system). For a correct description with a single n(f) ˜ one has to go upstream all the transfer functions of the system, obtaining the effective noise to be compared to the signal. In Figure 1.4 on page 8 many of them are shown, together with the full reference noise curve [5]. Among them, there are three main components: seismic noise. Acting typically under 10Hz, it arises from the microseismic activities (human and environmental), generating a displacement of detector components of some micrometers. They are monitored by seismometers and accelerometers, mostly reduced by use of advanced mirror suspensions, as well as active feedback systems. thermal noise. Due to excitation by thermal kinetic energy fluctuations of detector components (mostly on suspension and mirrors). They dominate the noise spectrum for mid-range frequencies 10 − 100Hz . In order to reduce this contribution, components are designed with resonance frequency as far as possible from the best sensitivity band of the detector. shot noise . Above 100Hz this is the dominant part of the spectrum, arising from the discreteness of the laser beam Poisson statistics. The relative fluctuations can be reduced by making use of powerful and stable lasers, up to powers of 100W for the advanced ones..

(29) 1.3 gravitational wave background. 1.3. gravitational wave background. Now we are in the condition to consider a generic stochastic gravitational wave background (SGWB) signal. We treat it as a linear superposition of (1.9), for both polarization mode +, × and for any wavevector ˆ. kµ = 2πf 1, kc . In TT gauge we have hij (r, t) =. X Z +∞. A=×,+ −∞. Z. ˆ ˆ h˜ A (f, n) ˆ eA ˆ −i2πf(t−nr/c) df d2 n (1.16) ij (n)e. The most general set of data available for the subsequent analysis will be a vector of timeseries sa (t), each coming from one of d different detectors. It is made by the superposition of noises and GW signals sa (t) = na (t) + ha (t). a = 1, . . . , d. (1.17). The signal are conveniently described as in (1.15) by a set of d detector tensors, contracted over the same GW tensor field evaluated at d different locations ha (t) = (Da )ij hij (t, ra ) X Z +∞ Z ˆ a /c) ˆ h˜ A (f, n) ˆ Da eA ˆ −i2πf(t−nr = df d2 n ij (n)e A=×,+ −∞. =. X Z. A=×,+. Z ˆ a /c) ˆ h˜ A (f, n) ˆ FA ˆ ψ)e−i2πf(t−nr df d2 n a (n,. Or equivalently, switching to Fourier amplitudes X Z ˆ a ˜ha (f) = ˆ h˜ A (f, n) ˆ FA ˆ ψ)ei2πfnr d2 n a (n,. (1.18). (1.19). A=×,+. In most of the GW detection contexts, we are usually interested in a peculiar waveform, or equivalently in a particular timeseries, originated directly by the sources we are looking for. Such waveform is usually related to the production mechanism via a deterministic differential equation. Now, we switch our perspective to the received GW signal as a whole process, whose original deterministic time structure (if present) has been buried by the incoherent superposition of sources. As well as random walk can be thought as a stochastic process emerging from a deterministic small-scale dynamics, we will consider now h˜ a (f) to be a stochastic process, emerging from small-amplitude individual GW contributions. Then, any model we make will have to be compared statistically to the received signals, assuming the latter capture information of the stochastic model via their averages. As we will see in Chapter 3, the most general average we shall consider is hsa1 (t1 ) . . . san (tn )i (1.20). 9.

(30) 10. introduction. q p w. u. n v. ˆ v, ˆ the azimutal axis w, ˆ and Figure 1.5: Arrangement of the detector arms u, ˆ p, ˆ q. ˆ Angles θ and φ are the spherical coordithe wave axis n, ˆ The angle ψ compares plus polarization axis nates of vector −n. pˆ to spherical unit vectors eθ and eφ .. computed using the signals recorded by n out of the d available detectors. For conciseness we will denote such averages in two more compact forms: • with a = a1 , . . . an and t = t1 , . . . , tn as substitutes for multiple indices and arguments. ⊗ sa (t) ≡ hsa1 (t1 ) . . . san (tn )i (1.21) • implicitly assuming indices and arguments. ⊗n ⊗ s = sa (t). A sequence of assumptions must be made carefully to proceed in the simplification of the problem. signal-noise uncorrelatedness: any composite correlation function factorizes into two parts, as follows. ⊗n ⊗m ⊗n ⊗m h n = h n Then, we concentrate on the signal part of the correlations. h˜ a1 (f1 ) . . . h˜ an (fn ). We shall use the indices a as a reminder of an important fact: ha (f) is not a detector-independent quantity, since it contains ˆ v, ˆ w ˆ for the the information of two triplets of Cartesian axes (u, ˆ p, ˆ q ˆ for the GW wave), combined contractinterferometer and n, ing the detector tensors over the polarization tensors as shown in Figure 1.5 on page 10 and in (1.18). At this introductory stage we will use as a prototype the two point correlation functions. ˆ h˜ B f0 , n ˆ h˜ A (f, m) and the corresponding ones for hij (t, ra ) or h˜ a (f), depending on the SGWB features we’d like to stress..

(31) 1.3 gravitational wave background. stationarity: being compared to the of a data D typical timescale E ⊗n sample (∼ year), any correlation hij (t, r) does not change over time appreciably. The typical mechanisms acting on the ensemble behavior of (astrophysical and cosmological) SGWB sources vary over longer timescale (t ' 1Gyr). Then a general correlation will be invariant by a uniform time translation. Equivalently, the frequency dependence will be affected by an overall δ(f1 + · · · + fn ). D E ˆ ⊗k ≡ δ (f1 + · · · + fk ) H (f, n) ˆ PA h˜ A (f, n). The latter being the correlation tensor in the polarization space. When plugged into h˜ a , however, some time modulation (e.g. Earth rotation and revolution) may arise. In we. the following. ˜ will ignore this feature, thus assuming the ha (f) correlation function to be stationary as well.. isotropy and non-polarization: for cosmological sources they are to be understood as working hypotheses, as it has been in the past for cosmological microwave background (CMB) radiation. When the sensitivity of the interferometers network will be high enough, we will be able to drop this assumption and to get some precious insight into the early Universe emission anisotropies and polarization fluctuations [7–9, 27]. For an astrophysical background these assumption are more inadequate, since already at a first approximation the expected signals will exhibit correlations with the galactic plane, thus requiring a more detailed treatment [27]. Statistically, we can describe both properties using the two point correlation function. ∗ δ2 (n, ˆ m) ˆ ˆ h˜ B f0 , n ˆ = H(f)δ f − f0 h˜ A (f, m) δAB 4π δ2 (n, ˆ m) ˆ 1 ≡ Sh (f)δ f − f0 δAB 2 4π Plugging this average into the corresponding one for the strain signal (1.19), we shall moreover exclude any correlation between different ψ, introducing a suitable δ (ψ, ψ0 ) function. After inteˆ m, ˆ ψ0 and a summation over polarization indices gration over n, we obtain. ∗ 1 ha (f)hb (f0 ) ∝ Sh (f) δ(f − f0 )γab (f) 2 where the function γab takes into account any geometrical factor of the detector arrangement, as follows " # Z Z ∆ ˆ X A dψ d2 n ˆ ab A c ˆ ψ)Fb (n, ˆ ψ) ei2πfn· Fa (n, Γab (f) = 2π 4π A. Γab (f) γab (f) = max Γab (f). (1.22). 11.

(32) 12. introduction. and the maximum is considered upon all the orientation of one detector with respect to the other. The interferometer relative displacement acts effectively as a frequency cutoff for the correc lation, since for values f > |∆| the integral goes rapidly to zero. For terrestrial interferometers in the worst case |∆| . dEarth = 104 km so one looses sensitivity to background above 104 Hz. gaussianity: mathematically, this means that any correlation function can be factorized into products of two and one point function, according to the Wick-Isserlis theorem [36]. This assumption is rooted directly into the central limit theorem (CLT), and will be the main feature we shall discuss in the following of the thesis. For a cosmological SGWB we can quite easily compare the size of present observable universe with respect to the causally connected portions of it at the emission era: as we will see later in Section 1.4, today N ∼ 1039 GW independent “cosmological sources” are pointed toward our past-cone. It is clear than, in this context, that CLT will hold strongly unless we select among such background signals specific ones characterized, for example, by peculiar time structures. In fact, this will be the case of cosmic strings cusps, as we will see in Section 2.7. For astrophysical sources this assumption has to be checked carefully, since their rate as well as their typical time span might affect significantly the statistics at different redshift. This is the case of our second reference source, described in Section 2.8. 1.4. frwl universe and cosmological density parameters. As a last introductory step, we shall summarize some common properties of our reference Universe: a spatially flat, expanding, FriedmannRobertson-Walker-Lemaître (FRWL) universe. It will be characterized by a metric of the form ds2 = dt2 − a(t)2 dr2 + r2 dΩ2 (1.23). with a(t) a uniform time-dependent scale factor for the spatial physical distances. We use Einstein equations, accommodating a vacuum energy Λ-term 1 Rµν − gµν R = 8πGTµν + Λgµν 2. (1.24). with Tµν the energy-momentum tensor of the fluid (matter or radiation) permeating the Universe. By introducing energy densities and pressures ρ, p for the cosmological homogeneous and isotropic perfect fluid, Tµν reads Tµν = −pgµν + (p + ρ). dxµ dxν dτ dτ.

(33) 1.4 frwl universe and cosmological density parameters. and equation (1.24) becomes 2 a(t) ˙ 8πG Λ H(t) ≡ = ρ+ a(t) 3 3 a(t) ¨ Λ 4πG (ρ + 3p) = − a(t) 3 3 2. . with H(t) the Hubble constant at cosmological time t. We will be denoting with H0 today’s H value H0 ≡ h0 100km · s−1 Mpc−1 ' 67.80 ± 0.77km · s−1 Mpc−1 and the age of our observable universe with t0 = 13.79 ± 0.02 Gyr. By further rescaling the first equation by a factor H20 we obtain ρ Λ + −1 = 0 ρc 3H20 with ρc =. 3H20 8πG. the critical energy density, or equivalently ΩT + ΩΛ = 1. The first term will in general include any contribution from the Universe energy momentum tensor, will it be coming from matter or radiation type fluids. If we would like to include in such a Universe the contributions from GWs, we should define carefully what a gravitational wave is with respect to the background, since it isn’t anymore a flat Minkowski one. In fact, by looking at the metric (1.23) we see that a(t) itself constitutes a time dependent perturbation with respect to a Minkowski spacetime. But, its typical lengthscale of variation is approximately 10 the inverse Hubble constant λB ∼ cH−1 0 ∼ 10 Mpc. If the other perturbations (namely, GWs) arise at much shorter lengthscale λgw , we can decouple them [25] introducing a coarse-grained version of the energy-momentum tensor T¯µν , as well as an averaged metric g¯ µν . We already did this before implicitly, since isotropy and homogeneity (the hypothesis necessary to derive the FRWL solution) hold for large lengthscale only (at least bigger than the typical galaxy cluster size, ∼ 10 Mpc). Small perturbations with respect to g¯ (we will consider them properly gravitational waves) will affect matter dynamics on short lengthscale only, thus decoupling in the averaged g¯ µν description. They still remain however, as energy content of the universe, tµν . 1 8πG R¯ µν − g¯ µν R¯ = 4 T¯µν + tµν 2 c. A careful study of such expansion is available in [25, §1.4 and §1.5], as well as a complete discussion on the decoupling of low-frequency. 13.

(34) 14. introduction. matter dynamics from high frequency gravitational field modes. The remarkable fact is that tµν has a simple expression, if we use the average metric and the associated perturbation gµν = g¯ µν + hµν. c4. tµν = ∂µ hαβ ∂ν hαβ 32πG. and, by using TT reference we obtain the (gauge invariant) energy density c4 ˙ T T ˙ T T ρgw = t00 = h h (1.25) 32πG ij ij with associated cosmological parameter Ωgw =. ρgw ρc. In each frequency band satisfying our approximations, we can describe the spectral content of a cosmological GW source by restricting D E T T T T the average h˙ h˙ to the suitable frequency bin. The result will ij. ij. be the following parameterization of Ωgw Ωgw (f) =. 1 ∆ρgw ρc ∆ ln f. (1.26). By using the plane wave expansion (1.16) we can compute the energy density straightforwardly, under the above assumptions for the SGWB. For unpolarized, isotropic and stationary background we have for the spectral component. 1 ˆ − m) ˆ δ2 (n ˆ h˜ B f0 , n ˆ = δAB h˜ A (f, m) δ f − f0 H f, f0 4π 2. and by plugging it into (1.25) we obtain [25, § 7.8.1] Ωgw (f) =. 4π2 3 f Sh (f) 3H20. (1.27). where we have defined Sh (f) as the one sided spectral density of the signal. As we will explain in more detail in Chapter 3 for the Gaussian case, Sh (f) or the corresponding two-point function H(f, f0 ) embeds all the statistical information of the signal. Since we can always shift the mean to zero , any n−point function will be either a product of a suitable number of Hs, or zero. So a knowledge on Sh (f) suffices in that case for a complete characterization of the signal..

(35) 1.5 available snrs and energy densities. 1.5. available snrs and energy densities. Summarizing what we have described so far, we compute the squared average value of h(t) E. 2 D h (t) = (h+ F+ + h× F× )2 E E D D = (h+ F+ )2 + (h× F× )2. . = F2+ h2+ + F2× h2×. ˆ and polarization angle ψ we leave If we average over the solid angle n unaltered the left member (being the background isotropic) and we group the pattern functions (because their average are equals) Z 2. 2. ˆ dψ 2 2 d n F+ h+ + h2× h = 4π 2π Z q 2y ∞ = 2 F+ dfSh (f) 0. using the brackets J·K for angular averages. Comparing it to the average noise variance (i.e. the squared output in absence of signal) Z∞. 2 n = dfSn (f) 0. we obtain the SNR ratio associated to a single detector in a given frequency bin ∆f centered around f¯. 2 h 2 SNR = hn2 i R q 2 y ∆f dfSh (f) = 2 F+ R ∆f dfSn (f) q 2 y Sh f¯ ' 2 F+ Sn f¯. as a constant quantity. At a fixed threshold SNR, established by detection theory arguments (see Chapter 4) the lowest measurable energy density is therefore given by . Ωgw (f). . min. =. 4π2 3 Sn (f) f q y SNR2 3H20 2 F2+. and the single detector performance does not change over time, except for the frequency resolution one can achieve. A reference value, for space based interferometer is given in [25]  2 1 3 2 f Sn   × Ωgw (f) min = 2.06 × 10−12 − 21 −21 1 mHz 4 × 10 Hz ! 2 2 1 SNR 0.73 √ q y × 2 5 h0 2 5 F+ z. 15.

(36) 16. introduction. Just as a comparison, the performances of a two detector backgroundsearch increase over time, with upper limits for detectable energy density decreasing 2 . Ωgw (f). . min. ∼. 4π2 f3 Sn SNR 3H20 (2T ∆f)1/2 F. (1.28). with a typical improvement factor of 1 (2T ∆f)1/2. ' 1 × 10. −5. . 150 Hz ∆f. 1/2 . 1 yr T. 1/2. We will show such behaviour in Chapter 4, where we will analyze in detail the dependence over integration time of multiple detector SNRs.. 2 we make a slight abuse of notation here. We introduce a fictitious factor F accounting for the relative detectors displacement and orientation, as well as their antenna patterns, to be compared with the averaged F2+ of the single detector case..

(37) 2. SOURCES: TWO EXAMPLES. summary In Section 2.1 we give an overview of spontaneous symmetry breaking in quantum field theory. We describe the formation of topological defects, and we give a common classification of them, based on the homotopy groups of the equilibrium states of the theory. In Section 2.2 we focus our attention on linear defects (strings). We introduce the simplest covariant action describing them, and the associated generally covariant motion equation. In Section 2.3 we show one solution exhibiting cusps, and describe their motion. In Section 2.4 we describe their gravitational wave emission in flat spacetimes, in particular the GW beam emitted by cusps. In Section 2.5 we extend string, as well as their GW individual emissions, to an expanding spacetime. In Section 2.6 we describe qualitatively their properties as a cosmological network. Finally, in Section 2.7, we quantify the associated spectrum, and characterize in time domain the average effective overlap as seen from the detector perspective. Next, in Section 2.8, we switch our attention to unresolved binary black hole coalescence. We first assess the GW emission spectrum. Then, in Section 2.9, we construct a cosmological population of binary black holes (BBH) and give some estimates on the predicted closure energy density. Finally in Section 2.10, as for cosmic strings we describe three different signal regimes we shall consider in the design of a detection algorithm. 2.1. topological defects. Topological defects are predicted from the spontaneous symmetry breaking of a wide variety of elementary particle models. There is a number of mechanisms leading to their formation (phase transitions, quantum tunneling, etc.). The main features can be described by looking at the class of gauge theories L = Dµ φ† Dµ φ − V(φ) − Tr [Fµν Fµν ]. (2.1). Here, φ is a set of scalar complex fields with associated gauge transformation φi → φ0i = Rij (g)φj . The R(g) matrix belongs to the relevant representation of the group G, with generators Ta , elements g, and covariant derivative Dµ = ∂µ − ieAa µ Ta . The gauge fields of the theory are themselves the coefficient of the parallel transport for such derivative. Their kinetic and self-coupling terms arise from the trace. 17.

(38) 18. sources: two examples False vacuum. True vacuum. Figure 2.1: Vacuum states for exact (sx) and spontaneously broken (dx) symmetries.. Tr [Fµν Fµν ]. The V(φ) term contains any higher order power of φ’s. With the choice of a suitable gauge group G we recover many known theories. Some examples: 2 • G = U(1) and V (φ) = 21 α2 φ† φ + 21 β2 φ† φ is the abelian Higgs model 2 • G = SU(2) and V (φ) = 21 α2 φ† φ + 12 β2 φ† φ is the pure non-abelian Higgs model 2 • G = SU(2) × U(1) and V (φ) = 12 α2 φ† φ + 21 β2 φ† φ is the electro-weak standard model The lagrangian is explicitly gauge-invariant if the V(φ) term is symmetric, that is infinitesimally ∂V a T φj = 0 ∂φi ij However, if the minima of V(φ) do not correspond to zero field values, the symmetry is spontaneously broken and a manifold M of vacua arises. Expanding a given theory around one of this vacuum states, we can construct theories with a spectrum of massive particles (without breaking lagrangian invariance with explicit mass terms). This mechanism (the so-called Higgs mechanism) is depicted in Figure 2.1 on page 18 for the abelian Higgs model, where different signs of the α2 -term drive the symmetry from exact (α2 < 0) to spontaneously broken (α2 > 0). The theory just described is oversimplified since it establishes classically the degree of symmetry for equilibrium states. A more precise treatment must encompass thermal and quantum fluctuations, too. A.

(39) 2.1 topological defects. viable approach is to introduce an effective potential Veff (φ), describing quantum and finite temperature corrections to V(φ). As a reference example, for the U(1) abelian Higgs case, the quantum 1−loop correction at T = 0 is given by [15, 35] |φ| 4 V1loop (φ) = A |φ| ln µ2 where µ is the renormalization scale. Thermal corrections at high temperatures are described by VT (φ) = BT 2 |φ|2 + CT 4 thus q changing the scalar field mass term. At temperature T > Tc = 2 − αA the symmetry could be unbroken. The precise behavior of the theory at high temperature (i.e. the onset of first or second order phase transitions, or none of both) depends heavily on the details of the theory. The interested reader might find useful a comprehensive review on the argument [34], and the references therein. Our intent here is to show that symmetry breakings, and the consequent formation of topological defects, might give some insight into earlier stages of our universe thermal histories, when gauge symmetries were thermally restored. Now, still considering the abelian Higgs model, let us analyze the following situation: being M the set of vacuum states for the broken theory, we suppose there exists a closed path γ, along which the fields assume equilibrium values φ = |φ| eiθ(s) . Here θ (s) is a periodic function with T = 2πnL, being the path itself parameterized by s ∈ [0, L]. Let us take a surface Sσ with boundary ∂Sσ = γ and a point Pσ on it, no matter the value φ (Pσ ). For smoothness reasons, collapsing γ to Pσ there has to be a point P on a curve γσ with a non equilibrium value φ(P) = 0 . A neighbourhood on P will have arbitrary small values as well. Then, choosing a non-intersecting sequence of surfaces S’s sharing the same boundary γ, we can construct a path σ of points acquiring zero value (together with its tubular neighbourhood): a cosmic string [10]. Here, we just underline that the mechanism generating topological defects (the Kibble mechanism) has its foundation on the nature of the vacuum manifold M of the simmetry-broken theory: any loop γn cannot be shrunk to a point without leaving M. That is, the first homotopy group π1 (M) is non trivial, and is actually equivalent to Zn . As a comparison the electroweak model, having a doublet of scalar complex fields, has sufficient degrees of freedom to disentangle and collapse any loop to a point without leaving M. In the same fashion, the formation of monopoles and textures defects is strongly related to the triviality of second and third homotopy groups of M, respectively.. 19.

(40) 20. sources: two examples. Figure 2.2: The construction of a point P out of equilibrium for the field.. 2.2. the wire approximation. Being the precise field configuration outside the scope of the thesis, we concentrate our attention on an effective treatment of strings. In fact, if described by means of fundamental fields many strings result having some common properties. As an example we refer to a straight cylindrically symmetric solution for the abelian Higgs case, called the Nielsen-Olesen vortex [10]. It has: • typical diameter equal to the greater inverse mass involved (for 1 the electroweak model they are comparable m1H ' mW,Z ' −16 10 m) • momentum-energy tensor proportional to m2H |φ|2 , describing an object with tension along its longitudinal axis (here taken to be z): Tµν ∝ diag (ε, pr , pθ , pz = −ε) • linear mass density. Z. µ = ε2πrdr If we are not interested in strings self- or mutual- interactions, we can average out the transverse degrees of freedom, considering the string as a curve in space, possessing linear energy density µ. The approximation holds as long as the string has a curvature radius κ−1 much greater than the string typical diameter m−1 H . Its general covariant motion is described as a worldsheet W on a given spacetime, with coordinate Xµ (ζ), ζA=0,1 and curve derivaµ (ζ) tives ∂X ≡ X,µ A . The worldsheet induces a metric γAB on itself, ∂ζA depending on the spacetime metric gµν as well as on the specific parameterization chosen ν A B A B ds2 = gµν dxµ dxν = gµν X,µ A X,B dζ dζ ≡ γAB dζ dζ.

(41) 2.2 the wire approximation. Having W signature (1, 1), we can use the tangent vectors on its points P’s to generate the space . µ TP = tµ A = X,A (ζ(P)), A = 0, 1. and complete it to an orthonormal basis with two spacelike vectors generating. µ NP = nµ A = NA (P) , A = 2, 3. Using these tailored coordinates we can describe a sufficient small spacetime neighbourhood of W as A 0 xµ P0 = Xµ ζA + Nµ χ A ζ P thus giving to any P0 coordinates ζ0 , ζ1 , χ2 , χ3. .. Figure 2.3: Tailored coordinate on the worldsheet neighbourhood. Being the worldsheet itself a solution of the fields equation of motion, we will use its geometrical properties as dynamical effective degrees of freedom. For the U(1) case, the lagrangian density can be rewritten perturbatively [34, § 6.1] L = −µ + ακ + β. κ2 +... µ. and the leading order is the Nambu-Goto action Z 1 S = −µ γ(ζ)− 2 d2 ζ Zp = −µ − det γ (ζ)d2 ζ Zq 2 ν = −µ − det gµν tµ A (ζ) tB (ζ) d ζ Zq 2 ν = −µ − det gµν X,µ A (ζ) X,B (ζ) d ζ. The Euler-Lagrange equations read ∂ ∂L ∂L − =0 ∂ζA ∂Xµ ,A ∂Xµ. (2.2). 21.

(42) 22. sources: two examples. 1 By using the metric determinant and dγ 2 = 12 γ1/2 tr γ−1 dγ one obtains 1 1 BC ∂ 1 1 ∂γBC α α 2 = 0 γ γ gαβ µ (X,B X,C ) − γ 2 γBC 2 ∂X ,A 2 ∂Xµ ,A 1 1 1 β γ 2 γAB gµλ X,λB − γ 2 γAB gαβ,µ X,α A X,B = 0 ,A 2 and defining covariant derivatives along the worldsheet µ µ α β C µ DB tµ A = tA,B + Γαβ tA tB − ΓBA tC. equation (2.2) becomes γAB DA tµ B =0 However, the vectors tives DA , then. tµ A. (2.3). are autoparallel with respect to the derivaν gµν tµ A DA tB = 0. So the (2.3) should be projected (via a suitable tensor qµν ) along the transverse directions, to give some meaningful equations (2.4). ν ρσ qµν γAB X,ν AB = −qµν Γρσ p. The (2.4) is an hyperbolic differential equation, and is solved by two characteristic curves along which Xµ (ζ) propagates. For a given point on the world sheet, the tangent vectors ϑ of the two characteristics are given by the intersection of the local light-cone with the world sheet γAB ϑA ϑB = 0 2.3. loops and cusps in flat spacetime. Let us consider now a flat reference spacetime gµν = ηµν , and a worldsheet W with conformally flat coordinates ζ = (τ, σ) on it, that is γAB = f (σ) diag(1, −1). The motion equation simplifies to [10, 34] (2.5). µ X,µ ττ −X,σσ = 0. and the invariance under parameterization of the world sheet acts as a gauge invariance for the field Xµ (ζ). Thus, (2.5) should be accompanied by gauge consistency conditions (Virasoro conditions) (2.6). ν γ01 = ηµν X,µ τ X,σ = 0. γ00 + γ11 =. ν µν µ ηµν X,µ X,σ X,ν τ X,τ +η σ=. 0. (2.7). A residual gauge freedom in left available by (2.5), for reparametrizations ζ → ζ˜ (ζ) satisfying ˜ 0σ = ζ,1τ ζ,. ˜ 1σ = ζ,0τ ζ,. (2.8).

(43) 2.3 loops and cusps in flat spacetime. We fix it by choosing τ as x0 itself (aligned standard gauge). The solution µ for (2.5) is given by two characteristics curves, Xµ + and X− 1 µ X+ (τ + σ) + Xµ − (τ − σ) 2 1 = τ, (a (τ + σ) + b (τ − σ)) 2. Xµ (τ, σ) =. and (2.6)(2.7) become1 a0. 2. = b0. 2. = 1. A general string solution is therefore specified once we choose on S2 (the boundary of the Kibble-Turok sphere, as shown in Figure 2.4 on page 23) two path describing a0 and b0 respectively.. cusp. a b cusp. Figure 2.4: Kibble-Turok sphere, showing two path defining a loop with cusps.. Considering periodic strings (or loops), we reveal now the most striking feature for our purposes. In fact, being periodic, the following holds for all τ’s Xµ (τ, σ) = Xµ (τ, σ + L) a (τ) + b (τ) = a (τ + L) + b (τ − L) µ and integrating the tangent vectors X,µ τ e X,σ along a period we obtain ZL ZL µ dσX,σ = dσX,µ τ =0 0 0 ZL ZL 0 0 dσ a − b = dσ a0 + b0 = 0 0 0 ZL ZL dσa0 = dσb0 = 0 0. 0. The vector fields a0 , b0 describing the characteristics speeds are zeromean. Than, it is reasonable to assume one of the following scenarios: 1. 0. means derivative with respect to its own argument.. 23.

(44) 24. sources: two examples. • the path are discontinuous at some point, and the loop results having kinks • the path do intersect, and the string loop exhibit cusps The equation (2.6) provides a meaning of 4−momentum to X,µ τ , beµ cause of the orthogonality with respect to X,σ . Now, if we express 0 0 ν X,µ τ X,τ using a and b µ ηµν X,ν τ X,τ = 1 −. =. 2 1 a0 · b0 1 0 a + b0 = − 4 2 2. 1 (1 − cosθ) 2. we notice that near-cusp regions move at ultrarelativistic speed a0 ≈ b0 → ∂τ Xµ ∂τ Xµ ≈ 0. By a suitable (τ, σ)-origin choice, we can describe the worldsheet via a Taylor expansion of the space components X (τ, σ). 0 0 a + b0 a − b0 1 a00 + b00 (τ + σ)2 = xc + τ+ σ+ 2 2 2 2 000 3 000 3 a + b000 τ a − b000 σ τ2 σ 2 + + 2τσ + + 2 6 2 6 2 = xc + vc τ + tc (τ + σ)2 + 3 3 τσ2 τσ2 σ τ + + sc + + jc 6 2 6 2. (2.9). (2.10). cusp loop. Figure 2.5: A loop at two different close values of τ. xc , vc and sc give the instantaneous cusp position, speed and spread.. In Figure 2.6 on page 25 we show the approximate cusp trajectory in the (τ, σ = 0) subdomain X (τ, 0) = xc + vc τ + tc τ2 + jc and its shape in the (τ = 0, σ) subdomain. τ3 6.

(45) 2.4 gws from cusps: minkowski spacetime. X (0, σ) = xc + tc σ2 + sc. σ3 6. ke eli ac p s cusp. timelike. spac e. like. cusp. ke timeli. Figure 2.6: Detailed shape of a cusp (sx) and its time development (dx).. However, it should be kept in mind that an exact cusps is only a mathematical artifact, since the wire approximation doesn’t hold anymore for small values of (τ, σ), when the two segments of the string are close enough (as compared to their thickness). For a consistent treatment of string self-interaction near cusps we shall follow a fundamental field approach. So far we have described some kinetic features of cuspy strings in flat spacetime. Now we compute the associated GW emission. Then we will extend both results to a FRWL reference universe. 2.4. gws from cusps: minkowski spacetime. We first consider the energy-momentum tensor of a string in a covariant form Z µν ν µ ν 4 T (x) = µ (X,µ (2.11) τ X,τ −X,σ X,σ ) δ (x − X (τ, σ)) dσdτ with the following dynamical quantities in Minkowski spacetime: ZL. (2.12). ε = µ dσ 0 ZL. ZL. p = µ dσX,τ (τ, σ) ≡ µ dσX˙ (τ, σ) 0 ZL. (2.13). 0. J = µ dσX (τ, σ) × X˙ (τ, σ) 0. Our aim is to obtain the time structure of the GW signal emitted by ultrarelativistic cusps. In harmonic gauge, the solution for (1.5) reads. hµν (t, x) = −4G. Z. Σ.

(46)

(47) Tµν (t0 , x0 ) − 12 ηµν T (t0 , x0 ) δ(t0 −

(48) x − x0

(49) )dt0 d3 x0 0 |x − x |. 25.

(50) 26. sources: two examples. where Σ ≡ Σ(t, x) is the intersection between the past light-cone of the event (t, x) and the source worldsheet W. Integrating over spacetime and substituting the source momentum-energy tensor (2.11), we obtain Z Tµν (t − |x − y| , y) − 12 ηµν T (t − |x − y| , y) 3 hµν (t, x) = −4G d y |x − y| Σ Z −1 Ψµν (τ, σ) = −4Gµ 1 − n · X˙ (τ, σ) dσ (2.14) |x−X (τ, σ)|. with. β ν µ ν 1 µν α β ηαβ X,α Ψµν (τ, σ) = X,µ τ X,τ −X,σ X,σ , τ X,τ −X,σ X,σ − η 2 x − X (τ, σ) , n= |x − X (τ, σ)| and the retarded time being implicitly defined as τ = t − |x − X (τ, σ)| The [. . . ]−1 factor in (2.14) has an important consequence: if we look exactly along n=X˙ on W, being X˙ (0, σ) ≈ vc ≈ 1 during the cusp event, we notice that GW amplitude integrand diverges. To compute more precisely the waveform emitted, one uses the periodicity (T = L2 ) of the loop to write the traceless part of h¯ µν as a sum 2π over discrete Fourier wavevectors kµ m = T m(1, n) whose coefficients are proportional to the Fourier transform of the momentum-energy tensor ZT Z µ µν ν µ ν −ikX T (k) = dτ dσ (X,µ τ X,τ −X,σ X,σ ) e T 0. We defines in the wave zone the asymptotic part of the GW amplitude propagating along n as ¯hµν (x) ≡ kµν (t − x, n) + O 1 r r2 Z |ω| kµν (ω, n) = dtk (t, n) eiωt 2π. and after some careful calculations for the dominant part of series and integrals involved [19], we obtain the amplitude in the nc ≡ vc direction.

(51) 2.5 gw from cusps: frwl universe. µ ν iωtc ν µ e A A + A A + − − + |ω|1/3 1 µ X+ (τ + σ) + Xµ Xµ (τ, σ) ≡ − (τ − σ) 2 Xµ ± ,ττ ≡ Aµ

(52)

(53) µ ±

(54) X ,ττ

(55) 4/3 ±. kµν (ω, nc ) ' −C. Gµ. 4π(12)4/3 C≡ 2 3Γ 13. (2.15) (2.16) (2.17) (2.18). where tc corresponds to the burst peak arrival time and Γ is the Euler’s gamma function. Then, the cusp waveform has the following spiky behavior kµν (t) ∝ |t − tc |1/3 The actual computation, for small declinations n · nc ≡ cos θc provides a smoothing of the burst peak over timescales τs ∼ θ3c T , or equivalently an exponential decay in frequency domain ∼ e−|ω|τs . An estimate of the asymptotic waveform is provided by [19] k (ω, n) ∼. GµT 2/3 1/3. Θ θ˜ (ω) − θc. |ω| −1/3 ˜θ (ω) ≡ |ω| T 2π. 2.5. . (2.19). gw from cusps: frwl universe. We shall summarize some minor changes needed to describe string dynamics, and GW emission, in a FRWL universe. It is convenient to dt use the conformal time dη = a(t) so the metric introduced in (1.23) takes the form ds2 = a2 (η) dη2 − dx2. We drop the gauge condition (2.7) and set ζ0 ≡ τ = η. Then, by the introducing an auxiliary function 0 X = q 2 1 − X˙ the analogue of (2.12), as well as its conformal-time derivative, are easily derived [32, 34] Z (η) ε=a µ dσ (2.20). ε˙ a˙ = 1 − 2 v2 ε a. (2.21). 27.

(56) 28. sources: two examples. with. 2 v =. R. dσX˙ R dσ. 2. Large horizon-sized strings will therefore gain energy by cosmological stretch or loose it by velocity redshift. Small loop, the main emitters of the GW bursts, will evolve as in Minkowski spacetime. Therefore, we shall use previous results on emission as local wave zone values. A GW burst at cosmological distance is redshifted ω → ωobs = (1 + z)−1 ωem and the physical distance will be stretched by universe expansion accordingly r = a0 η Its polarization tensor is parallel transported along the null propagation geodesic γ (γ). (γ). kµν ≡ kem eµν → kµν ≡ kobs eµν. Using (2.19) we provide a final estimate for a cusps at cosmological distances GµT 2 1+z hcusp (ω) ∼ (2.22) 1/3 ω 3H0 z (1 + z)T 2π. under the assumption of a matter dominated universe, and the hypothesis of an observation angle θ . θmax ≡ 2.6. hω. 2π. (1 + z) T. i−1/3. “one-scale” string network. We have derived so far the GW amplitude emitted by individual cusps at cosmological distances. We shall now consider the contribution from a cosmological network of string loops. Summarizing the results found by [19, § V] we will use a fiducial one-scale model [13, 34]. At a cosmological time t, a horizon-sized volume contains a few dH −sized string and many small loops. We parametrized the loop spacetime density as nl (t) and their typical size as l (t) = 2T (t) with a single parameter α l ∼ αt. nl (t) ∼ α−1 t−3. connected to the loops energy loss by GW back-reaction [13] α ∼ Γ Gµ ≡. 1 dEgw µ dt.

(57) 2.6 “one-scale” string network. We shall give now an estimate of the logarithmic rate density of cusps event at a given redshift z, around frequency f ˙ dN ω ˙ N(f, z) = ,f = d ln z 2π The key components are: 2. (f,z). around vc along which the burst esti• the solid angle ∼ θmax4π mate (2.22) is consistent • the number of cusps event per unit spacetime ν(t) • the average number of cusps per loop c • the number of loops per unit volume at a given redshift nl (t) • the proper volume at a given redshift interval z, z + dz,dV (z) • the matter-radiation equality redshift zeq ' 2.4 × 104 Ω0 h20 • an interpolating function ϕn (z) smoothing over different cosmological era The result for differential rate reads −8/3 ˙ N(f, z) ∼ 102 ct−1 (ft0 )−2/3 ϕn (z) 0 α 11/6 ϕn (z) = z3 (1 + z)−7/6 1 + z/zeq. and its integral, the effective rate of observable events in a frequency bin, is dominated by the greatest redshift value, because of the monotonicity of ϕn (z) Z zmax ˙ = ˙ ˙ (f, zmax ) N d ln zN(f, z) ∼ N (2.23) 0. Then, the GW signal from a cosmological population of cusps burst at ˙ will correspond to events from a redshift shell zmax , zmax + fixed rate N dz obtained by solving (2.23) for zmax . Unfortunately, the functional form of the solution is not simple with respect to zmax . It is easier to write it in close (interpolated) form using two auxiliary functions (see [19] for details) ˙ f) ≡ 10−2 N/c ˙ y(N, t0 α8/3 (ft0 )2/3 11/6. g(x) ≡ x−1/3 (1 + x)(1 + x/zeq )−3/11. by which the log Fourier transformed signal simplifies to 2. 3 ˙ ∼ Gµα 1 Θ 1 − θm αmax , N ˙ max , fmax g y N, ˙ f hcusp (f; N) (2.24) (ft0 ) 3. Figure 2.7 on page 30 compares the value of hcusp to one sigma noise levels for matched filtering burst search by ground-based interferometers, as a function of the model parameter α. It is clear from it that the parameter space might be well constrained by advanced GW detectors, provided we have a robust detection algorithm for the considered signals. 29.

(58) 30. sources: two examples. -21.5 -22.0 -22.5 -23.0 -23.5 -24.0 -24.5 -12. -10. -8. -6. -4. Figure 2.7: Gravitational wave amplitude of bursts emitted by cosmic string cusps in the LIGO/VIRGO frequency band, as a function of ˙ = 1 yr−1 and f = 150 Hz. log α = 50Gµ, with fiducial values N The horizontal dashed lines indicate the effective SNR = 1 noise levels (after optimal filtering for |f|−1/3 ) of LIGO 1 (initial detector) and LIGO 2 (advanced configuration). The short-dashed line indicates the confusion noise part of the signal. Figure adapted from [19, Fig.1]. 2.7. spectrum and effective number of sources. We now switch our attention to the signal as a stochastic ensemble of bursts X h (t) = h (t − tσ , zσ , pσ ) (2.25) σ. where we have anticipated the notation of Chapter 3 with a (provisionally ill-defined) sum over the burst event of interest, with associated random arrival time tσ of the cusps, redshift of their production zσ and any other random parameter flagged by pσ . We will denote in from now on random quantities via the subscript σ, whose meaning will be clearer in the next Chapter. The log-Fourier transform of the GW amplitude is X h(f) ≡ |f| h˜ (f) = e2πiftσ h (f, zσ , pσ ) (2.26) σ. with usual spectral density . ∗ h˜ (f) h˜ f0 ≡ Sh (f) δ f − f0. (2.27). In addition, we define the analogue for h (f) as a dimensionless rms value. ∗ h (f) h f0 ≡ δ f − f0 |f| h2rms (f).