Softwareandtheoreticalmethodsforgravitational-wavedataanalysis PhDthesis

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Dipartimento di Fisica E. Fermi

Corso di Dottorato in Fisica

PhD thesis

Software and theoretical methods for

gravitational-wave data analysis

Candidate:

Supervisor:

Giulia Pagano

Prof. Walter Del Pozzo

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Abstract

The direct detection of gravitational waves has opened the era of gravitational-wave astronomy. The LIGO and Virgo detectors observed dozens of gravitational-wave sig-nals from the coalescence of binary black holes and one confirmed binary neutron star merger hitherto. Moreover, as the detector sensitivity increases, gravitational-wave ob-servations will become an everyday reality.

The analysis of such signals through parameter estimation software programs allowed the LIGO-Virgo Collaboration to shed light on the properties of the compact sources and to test general relativity in the strong-field regime for the first time. Therefore, the importance to develop robust software for parameter estimation of gravitational waves cannot be understated. However, currently, only few software packages are available for gravitational-wave data analysis.

In this thesis we present G W MO D E L, a software package that we developed for pa-rameter estimation of gravitational waves from compact binaries. By analysing a set of simulated events as well as public LIGO-Virgo data, we demonstrate that G W MO D E L can infer the parameters of the compact sources observed by LIGO and Virgo, thus be-ing a valid tool for the analysis of current and future ground-based detector data. Moreover, we investigate a specific aspect of gravitational waves: gravitational lensing. Gravitational lensing of gravitational waves is predicted by general relativity and might be first observed by ground-based detectors in the coming years. Besides magnifying the signal amplitude, lensing distorts the gravitational waveform by inducing charac-teristic signatures that might allow us to study the lens properties. Moreover, if not accounted for, the lensing magnification will bias the inferred source characteristics: the binaries will appear to be more massive and closer than they actually are.

Hence, it is fundamental to develop software and data analysis techniques for gravita-tional-wave gravitational lensing. However, only few lensing systems can be solved analytically and a limited number of lensing configurations has been worked out in the context of lensing of gravitational waves.

In this thesis we also present L E N S I N GG W, a software package that we developed to model lensing of gravitational waves from arbitrary lensing configurations. We demon-strate that L E N S I N GG W is an efficient tool to predict lensed gravitational waves and show that it can be used to study prospects of detection and the properties of such sig-nals. In this respect, we find that LIGO-Virgo might in principle be able to distinguish lensing signatures at design sensitivity.

We further provide theoretical methods to determine lensing magnifications and asses or rule out lensing for binary neutron star mergers. We show their application to simulated

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highly magnified gravitational waves and to the more massive coalescence consistent with a binary neutron star merger observed by LIGO-Virgo hitherto. We find no evi-dence in support of lensing for this detection.

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Chapter 1 Introduction

More than one hundred years after its formulation, general relativity (GR) still stands as the description of gravitational interactions. Multiple observations [5] have shown stunning agreement with the predictions of the theory over time, corroborating its fundamental role in the description of our universe.

Amongst its predictions, GR has foreseen the existence of gravitational waves (GWs) [6, 7]: ripples in the space-time fabric that propagate at the speed of light. In-deed, in analogy with the electromagnetic (EM) interaction, where accelerated charges emit EM waves, the motion of accelerated masses produces gravitational radiation.

Two characteristics imply that GWs produce observable effects: they carry energy and deform the objects on their path according to specific patterns. However, the grav-itational interaction being among the weakest forces, the effects induced by GWs are in turn weak. Because of that and because the gravitational interaction of an object is related to its mass, compact objects in accelerated motion such as binary black holes (BBHs) and binary neutron stars (BNSs) are amongst the most promising sources for GW detection.

Over the years, both indirect and direct methodologies have been proposed to de-tect GWs. The former compare the observed energy loss of the emitting source to GR predictions. The latter, instead, measure GW-induced deformations. Only direct measurements, however, allow to reconstruct the gravitational waveform.

The first indirect detection of GWs was made in 1981, through the observation of the decay of the orbital period of the Hulse-Taylor binary pulsar [8]. Direct searches, instead, have focused on measuring deformations induced by GWs in ground-based interferometers (IFOs).

Since GWs carry the imprints of the source properties, direct measurements of GWs allow to investigate the objects that generated them. Therefore, GWs from compact binary coalescences (CBCs), such as BBH mergers or BNS coalescences, offer a unique channel to investigate the strong-field regime of GR, which will never be accessible in laboratories.

Typical GW amplitudes, however, are buried in the detectors’ noise. Thus, the first direct detection required incredibly sensitive IFOs and the development of ad hoc search methodologies for the identification of coincident events in the instruments [9].

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On September 14, 2015 at 09:50:45 UTC, the Advanced LIGO detectors [10, 11] made the first direct observation of GWs. At the beginning of their first observing run (O1), they detected GW150914 [12], a GW from the coalescence of a binary of stellar-mass black holes1_.

Indeed, parameter estimation (PE) studies on GW150914 allowed the LIGO-Virgo Collaboration (LVC) to infer the properties of the compact binary system, such as the component masses, and to perform pioneering tests of GR [13, 14]. From the morphology of the GW, PE analysis also demonstrated that the coalescence dynamics was consistent with the predictions of GR.

Therefore, not only the first direct detection proved that stellar-mass black holes form binary systems, but also that they evolve and merge according to the predictions of Einstein’s theory. In addition, PE studies on GW150914 allowed to investigate for-mation models, place bounds on the BBH merger rate and on the related stochastic background [15, 16, 17].

The LIGO detectors unveiled two additional GW signals from BBH coalescences during O1. The second observing campaign (O2), instead, took place in 2017 and included the Advanced Virgo detector [18] in the global network.

PE analysis of the GWs detected during O1 and O2 allowed the LVC to produce the first catalogue of compact binary merges (GWTC-1) [19], which includes the properties inferred for ten BBH coalescences and one BNS merger.

In particular, BNSs are expected to emit in the EM band during the merger [20]. Thus, EM counterparts allow us to study BNS coalescences both in the GW and in the EM channel (multi-messenger studies) [21].

PE studies on the BNS signal detected during O2 and the identification of the related EM counterpart [22, 23, 24, 21, 25, 26, 27, 28, 29, 30] led to the first GW standard siren measurement of the Hubble constant [31], whereas the combined analysis of the events of GWTC-1 provided more precise tests of GR and population studies [32, 33], as well as a joint measurement of the Hubble constant [34].

Thanks to the increased detector sensitivity, the LIGO-Virgo observatories detected dozens of events during the latest observing run, O3 [35]. For the first time, a BBH coalescence with asymmetric mass ratio has been detected [36]. In such signals, where the mass of one black hole is much lighter than the one of the other compact object, higher harmonics are excited. Therefore, PE analysis could measure the contribution of such subdominant modes in a GW for the first time. More recently, the detection of GW190814 [37], a BBH merger with even more asymmetric masses, offered a unique opportunity to test even higher harmonics in the GW signal.

In addition to that, LIGO-Virgo detected the BBH merger GW190521 [38]. PE of such signal has shown that the mass of the remnant object formed by the coalescence is larger than 100M [39], where Mindicates one solar mass. This makes it strong

observational evidence of a black hole in the mass range 102 _{− 10}3_M

and the first

evidence that such black holes can form from BBH mergers [39].

Thus, PE analysis was fundamental to uncover the physics of the GWTC-1 [19] and of the recently unveiled GWTC-2 [35, 40, 41] detected sources and it will play a

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crucial role in the next observing runs as well.

In the coming years the additional LIGO India detector will join the global net-work [42]. The recently-constructed KAGRA IFO will reach better sensitivity [43, 44, 45, 46] and the LIGO and Virgo detectors will improve their performance thanks to the planned instrumental upgrades [47, 48]. Moreover, the planned Einstein Telescope and Cosmic Explorer, two third-generation ground-based IFOs, will further expand the global network [49, 50, 51, 52, 53]. Therefore, a growing number of detections is expected in the next observing runs [41, 35, 16, 19].

With a broader statistics at hand, PE will produce a clearer picture of the population of compact binaries (CBs) in the universe, as well as allow for more stringent tests of GR and alternative gravity theories.

PE of GWs relies on GW templates predicted by GR. These templates depend on 15 parameters in the case of BBHs, plus two tidal parameters in the case of BNSs. The source characteristics are inferred through dedicated software packages that compare (match, in technical terms) such templates to the observed data [54].

PE software packages must then investigate such 15-dimensional parameter space to find the source parameters that give the largest match to the data, thus unveiling the source properties. However, different sets of parameters can produce the same GW signal due to correlations, which complicates the inference of the true source characteristics.

Besides providing accurate science, PE software packages should also support ex-tensions and developments that might be required in the coming years. Indeed, the ability to produce better science is also driven, for example, by advancements in the-oretical waveform modelling. Accurate signal models will result in better parameter inference, as they offer an improved representation of the observed data. Moreover, new waveform templates might offer novel tests of the nature of the detected sources. For these reasons, a fundamental characteristic of software packages for PE of GWs is to ease the implementation of new features and the customisation of the analysis.

As the physics extracted from GW observations is finding an increasing number of applications, it is therefore essential to develop reliable software for PE of GWs. LIGO-Virgo PE analyses have been performed with L A L SU I T E (LAL) [55], the LVC algorithm library suite. LAL is a complete and reliable software library for GW data analysis and is the product of years of contributions by hundreds of scientists. Thus, it has reached a certain level of complexity, which makes it non-trivial to extend or customise the inference. More recently, PYC B C IN F E R E N C E[56] and BI L B Y [57] have been presented, which are a PY T H O N-based toolkit for PE of CBCs and a multi-purpose inference library, respectively.

After reviewing the theory of GWs and of GW data analysis, in the first part of this thesis we present G W MO D E L: a functional software package that we have developed for PE of CBCs.

G W MO D E L can infer the properties of GW signals from both real and simulated ground-based detector data. Indeed, we equipped it with simulation tools that allow us to generate synthetic GWs. Such signals can then be analysed by the software package.

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synthetic signals, G W MO D E L is a complete software program for CBC data analysis. Our code supports any waveform model implemented in LAL, which includes the signal templates employed by the LVC. Therefore, any waveform development in LAL will benefit G W MO D E L. Moreover, we designed G W MO D E L to allow simplified usage for beginners, but also to facilitate the customisation of any step of the analysis and the addition of new features from expert users.

After introducing the software package, we present the analysis that we performed with G W MO D E L. In this respect, we have simulated and analysed a set of 100 mock events, to test the performance of our software package on a statistical basis. Further-more, we have analysed the public LIGO-Virgo data for the ten BBH detections of GWTC-1, reconstructing the source properties for each event. We have then compared our results to the measurements published by LIGO-Virgo.

With this analysis we show that G W MO D E L can reasonably infer the parameters of simulated signals. Moreover, we demonstrate that G W MO D E L correctly recovers the parameters of the BBH mergers detected during the first and second LIGO-Virgo observing runs. Thus, our software package is a valid tool to analyse both simulated and real ground-based detector data.

With the prospects of an increasing number of detections and better detector sensitiv-ity [46], novel observations might be unveiled in the coming years. One example of such an observation is gravitationally lensed GWs, which might be first observed in the next future: from predictions of the number of GW sources and of the lensing probability, Refs. [58, 59] estimated that LIGO-Virgo could observe ∼ O(1) lensed events per year at design sensitivity. Lensed GWs could provide exciting applications in fundamental physics, astrophysics and cosmology [60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 2, 70].

Just as in the lensing of EM waves, GW lensing can occur when GWs propagate near massive astrophysical objects, such as galaxies or galaxy clusters. The gravity of the galaxy can deflect the GW into multiple paths, producing multiple images of the same source. The trajectories of such images are curved with respect to the original wave and focused towards focal lines. The focusing will change the amplitudes of the images and different images will reach the same detector at different times. Indeed, they will have travelled different trajectories at the same speed, hence the arrival time differences [71, 72, 73, 74, 75, 76, 77, 78, 79, 80].

Because the amplitude of the images can be magnified by the focusing of the de-flector, this phenomenon is called lensing and the astrophysical objects that produce it are referred to as lenses. Consequently, one typically indicates the original GW as unlensedwave and the images as lensed waves.

While lensing of EM waves can be assessed through the angular direction of the lensed photons, detection of GW lensing requires different methodologies. Indeed, current GW detectors differ from optical telescopes in multiple ways. One is that they identify GW signals from their time of arrival at the instrument, rather than from the sky localisation of the incoming waves.

Therefore, current IFOs must be able to resolve the image arrival time differences to identify different lensed images of the same source. In contrast to optical detectors, that are limited by their angular resolution, LIGO and Virgo will thus be limited by

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their time resolution.

The time resolution of ground-based GW detectors is of the order of the inverse of the observable frequencies. Since LIGO and Virgo are sensitive to the frequency range between a few Hz and a few kHz, their typical time resolution is of the order of milliseconds.

Several methodologies have been proposed to distinguish lensing signatures in LIGO-Virgo data. For instance, one can search for multiple images of the same com-pact source [81, 62, 82, 83, 84, 85, 86], or look for populations of CBs which appear to have high masses because of the lensing magnification [87, 79, 78]. Both these method-ologies have been recently applied to the detections of the first and second observing runs, finding no robust evidence of lensing signatures [88].

Just like for unlensed waves, then, LIGO-Virgo will require templates of lensed waveforms to reconstruct the lensed GWs embedded in the data and study the related compact source.

To predict the lensed GWs, one needs to model the gravitational potential of the lens. Indeed, the latter determines the lensing signatures on the waves. However, currently, only the lensing effects of isolated lenses described by very specific mathematical models can be worked out analytically (see, for instance, Ref. [89]). On the other hand, there was no numerical software that targets lensing of GWs until the work presented in this thesis. It is therefore crucial to develop a framework that is not limited to the few lens models that we can solve analytically to study lensing effects on GWs.

In the second part of this thesis, we present our work on GW lensing, aimed at addressing the lensing of GWs both by a numerical and a theoretical perspective.

When the lenses are galaxies or galaxy clusters (strong lensing), the typical time delays of the images span from minutes to years and the image angular separations on the sky are_{. arcseconds (arcsec) [89, 90, 91, 92, 64, 93]. Therefore, the arrival} time differences of strongly lensed GWs are such that those signals will be detected as individual events in the IFOs. Distinguishing such waves as lensed, however, is non-trivial.

Indeed, in the geometrical optics limit, which holds when the wavelength of the wave is smaller than the lens characteristic size, the effect of strong lensing is that of magnifying the signal amplitude, without changing the signal morphology 2 [96, 87]. Such magnification is degenerate with the luminosity distance of the binary and, normally, it is not possible to break this degeneracy. Therefore, the analyses of the LVC do not take possible magnifications into account and assume unlensed signals.

However, the magnification-distance degeneracy is such that, if the source is lensed, it will appear to be closer than it is [87, 58, 78, 79] and, as a result, the redshift measure-ment will be biased towards lower values. Since such estimate is used to reconstruct the intrinsic (source-frame) masses of the binary, the latter will also be biased. In particu-lar, the compact objects will appear to be more massive than they are. Therefore, not accounting for strong lensing of GWs will result in biased inference of the properties of the population of CBs.

2_{See, however, Refs. [94, 95], that suggest that certain images could produce minor changes in the}

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As opposed to strong lensing, the lensing effects induced by masses of roughly 10−6 _{. M/M} . 106 are referred to as microlensing [89]. Typically, the arrival time

differences induced by low-mass (. 100M) isolated microlenses are such that the

GW microimages will overlap in ground-based detectors and their time delays will fall below the IFO time resolution [3]. Therefore, LIGO and Virgo may not be able to distinguish such incoming GWs as the superposition of lensed microimages.

However, Diego et al. [3, 97] recently proved that, if the microlenses are embedded in the gravitational field of a strong lens, their interaction with the gravitational potential of the galaxy or galaxy cluster can enhance the microlensing effect. Indeed, they showed that extragalactic strongly lensed GWs that propagate across the gravitational field of a microlens or of a microlens population are more likely to be microlensed than in the isolated microlens case [3]. Moreover, the enhancing of the microlensing effect due to the interaction with the strong lens can result in higher time delays than for isolated microlenses [3].

In such cases, microlensing can produce interference that might be observable at LIGO-Virgo frequencies in the lensed signal that reaches the IFOs [3]. Such interference originates from the superposition of the microimages of the original strongly lensed GW and might be used to infer the properties of the microlenses, such as their mass (see Refs. [62] and [82] for studies of intermediate-mass and stellar-mass black hole and star microlenses). Therefore, it is crucial to study the microlensing of strongly lensed GWs.

In order to study such signals, one needs to model the lensed GWs that will reach the detectors. For this purpose, it is necessary to predict the images and the image properties induced by arbitrary lens distributions, such as microlenses embedded within galaxies or galaxy clusters. Indeed, one needs to be able to predict the strongly lensed images produced by galaxies and galaxy clusters and determine the microimages of such strongly lensed GWs induced by the microlenses.

When microlenses are embedded in galaxies or galaxy clusters, microimages sep-arated by scales as small as microarcsec can form from the original strongly lensed wave [3]. For arbitrary lens distributions, it is not possible to predict analytically the number and the properties of the images of a given source. The latter can instead be found through numerical procedures. Given the lens distribution, such algorithms iden-tify how many images of the same source form and their angular positions on the sky. From these, it is then possible to retrieve the image properties and reconstruct the lensed signal.

Software packages for gravitational lensing, however, target light lensing instead of lensing of GWs [98, 99]. Therefore, they do not model lensed GWs. Since microlensing is not directly resolvable in optical [89], then, such software programs focus on strongly lensed images as opposed to microimages.

In the second part of this thesis we present L E N S I N GG W, a software package that we developed to model lensed GWs from arbitrary CBCs and generic lens distributions, in the geometrical optics limit.

We equipped L E N S I N GG W with a new algorithm that we have designed to model both strongly lensed images and microimages simultaneously. Thus, L E N S I N GG W

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can predict both the strong lensing of GWs produced by galaxies and galaxy clusters and the microlensing induced by microlenses embedded within strong lenses, while retaining fast performance. Indeed, we designed L E N S I N GG W to handle arbitrary numbers of microlenses embedded in multicomponent strong lenses, such as galaxies formed by bulges, disks and halos.

Besides identifying the images of the compact source and the image properties resulting from the examined lens model, L E N S I N GG W also simulates the related lensed and unlensed GWs.

The modelling of lensed and unlensed GWs is necessary for assessing the detectabil-ity of lensed CBCs and for investigating the relevant parameter space for microlensing. Indeed, the distinguishability of lensed signals is typically assessed by evaluating the dephasing induced by lensing in the lensed GW, with respect to the unlensed wave.

Thus, L E N S I N GG W will contribute to establishing a complete framework for the study of lensed GWs, a field that is receiving increasing attention from both astronomy and modelling sides [100, 90, 101, 3, 102, 94, 87, 62, 82, 63, 81, 103].

After presenting L E N S I N GG W, we demonstrate that the software package can correctly recover the images of microlensed systems studied in the literature. We then present the analysis that we have performed withL E N S I N GG W: first, we have studied how the CBC and the microlens properties affect the detectability of lensed GWs, in the case of a microlens embedded within a galaxy. We found that LIGO-Virgo might in principle be able to distinguish lensing signatures at design sensitivity. Then, we studied how the properties of the strong lens affect the lensed GWs, now considering the more realistic scenario of a halo of hundreds of microlenses embedded within a galaxy.

In the second part of this thesis, we also present our work on strong lensing of GWs. In this respect, we have developed mathematical methods to determine strong lensing magnifications from individual GW detections of BNS mergers. These magnification estimates might be used to infer the true redshift of the source, thus correcting the biased parameters.

BNSs undergo tidal deformations during the coalescence due to the gravitational field of the companion [104, 105]. The GW signal allows us to infer these tidal effects through the measurement of the phase evolution of the GW, which is unaffected by strong lensing. Thus, such estimates are unbiased.

However, tidal effects can be independently estimated from the measurement of the intrinsic component masses and an assumed equation of state (EOS), which describes the neutron star (NS) internal structure [105]. These latter estimates are therefore biased by strong lensing and result in lower tidal deformabilities inferred for lensed systems.

Thus, BNSs tidal effects measured from the GW phase represent unbiased param-eters to which one can compare the magnification-corrected (and thus, bias-corrected) measurements derived from the objects’ masses. Through the agreement between the two, one can then determine or rule out strong lensing for BNS systems.

Determining if a BNS is lensed or not is particularly important for measurements of the Hubble constant, multi-messenger studies and tests of GR that rely on strongly lensed GWs with EM counterparts [61, 106, 67, 68, 69], as well as for tests of GR that

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benefit from multi-messenger observations of CBCs [107].

We show the application of our methods to simulated lensed and unlensed BNS mergers and show how they can be used to confirm or rule out strong lensing for these systems. Furthermore, we show the results of the application to GW190425 [108], the more massive CBC consistent with a BNS merger detected by LIGO and Virgo so far: the intrinsic mass of this system is higher than what expected from the known galactic double neutron star (DNS) population [109, 110, 108]. Such application does not find evidence in support of strong lensing for this signal.

The thesis is organized as follows: in chapter 2 we cover the basics of GWs from CBCs and of GW lensing. We review the fundamentals of GW data analysis and discuss selected GW observations. In chapter 3, chapter 4 and chapter 5 we present our original contributions to data analysis of GWs. In chapter 3 we present the software package G W MO D E L, together with the analysis of simulated signals and of the ten BBH events of GWTC-1. In chapter 4 we present L E N S I N GG W. We illustrate the code validations and the analysis of simulated microlensed systems. In chapter 5 we present the theoretical methods to infer strong lensing magnifications of BNS mergers and their application to assess or rule out strong lensing on simulated BNS coalescences. Moreover, we show the application to GW190425. We conclude in chapter 6.

1.1 Conventions and notation

We indicate space-time points with the standard four-vector notation, xα. We use Greek indices α, β, ... to indicate space-time indices, 0, 1, 2, 3. We denote the temporal component of four-vectors with the index α = 0, and the spatial components with the indices α = 1, 2, 3, so that x1_{, x}2 _{and x}3 _{are the four-vector spatial components along}

the x, y and z directions, respectively. Latin indices i, j, ... stand for the spatial indices 1, 2, 3. Moreover, we adopt the convention of summation over repeated indices, unless otherwise mentioned.

The metric of a flat space-time is:

ηµν = diag (−1, 1, 1, 1) ,

while we indicate a generic space-time with the metric gµν and its determinant with

g = det(gµν).

We define the Riemann tensor as:

Rµ_νρσ = ∂ρΓµνσ − ∂σΓµνρ+ Γ µ αρΓ α νσ − Γ µ ασΓ α νρ,

where the Christoffel symbols Γα µν are

Γα_µν = 1 2g

ασ_[∂

µgσν+ ∂νgσµ− ∂σgµν] .

The Ricci tensor is R_µν = Rα

µαν and the scalar curvature is R = gµνRµν.

We indicate the four-derivative with respect to xα_{as ∂}

α, where: ∂α = ∂ ∂xα = ₁ c∂0, ∂i ,

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1.1. Conventions and notation

and c is the speed of light. The covariant derivative of a rank-two tensor Vαβ instead is:

DµVαβ = ∂µVαβ − ΓρµαVρβ− ΓρµβVαρ. (1.1)

Finally, the energy-momentum tensor of the matter and energy in the universe, Tµν,

is defined through the variation of the action of the non-gravitational fields, SNG, with

respect to the metric:

δSNG ≡

1 2c

Z

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Chapter 2 Gravitational waves: theory and

observations

One of the pillars of the modern description of fundamental interactions is field theory. In this context, particles are represented by fields: dynamical degrees of freedom that describe the particle at each space-time point, xα_{. Their interactions are determined}

by field equations, that specify the evolution of the fields under their mutual influence. GR is a metric theory of gravity: it describes the gravitational interactions of fields through their coupling with the metric tensor, gµν(xα). The metric characterizes the

geometry and the curvature of the space-time at each point xα and is itself a field governed by evolution equations. Therefore, not only it provides a unified description of space and time, but it also classifies the space-time as a dynamical entity which affects, and is affected by, the matter and energy in it through its coupling with the matter and energy fields.

In analogy to the EM interactions, where EM waves arise from the perturbations of the EM field, GWs arise from the perturbations of the metric tensor gµν. In section 2.1

and section 2.2, we cover the theory of such perturbations. In section 2.3, we describe the effects of GWs on test masses. We discuss the GWs produced by generic matter distributions in section 2.4 and we focus on CBCs in section 2.5. We discuss lensing of GWs from a theoretical perspective in section 2.6.

In section 2.7, we describe current detectors of GWs. In section 2.8 we illustrate the standard framework employed to reconstruct the source properties from the ob-served GW signals. We conclude this chapter by presenting selected GW observations in section 2.9.

Throughout this chapter, we follow the derivation of Refs. [111, 77, 89, 112].

2.1 Gravitational waves as solutions of Einstein’s field

equations

Within GR, the evolution of the space-time metric is determined by the Einstein’s field equations:

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Rµν−

1

2gµνR = 8πG

c4 Tµν, (2.1)

where Rµν and R are the Ricci tensor and scalar respectively, c denotes the speed of

light and G is the Newtonian gravitational constant. In the above and in what follows, tensor quantities are intended to be calculated at a generic space-time point xα.

The left-hand side of Eq. (2.1) depends only on the metric and thus on the space-time geometry. The energy-momentum tensor Tµν, instead, describes the energetic

balance of the matter and energy fields in the universe. Therefore, the curvature of the space-time is influenced by the presence of matter and energy through Tµν, and in turn

the distribution of matter and energy, hence the motion of masses, is determined by the space-time geometry gµν.

GWs arise as solutions of the Einstein’s equations for the metric gµν. The latter

constitute a system of ten, non-linear coupled equations, which is non-trivial to work out. Thus, one usually solves them in the linearised regime: when the perturbations of the metric tensor are small enough, one can consider only the linear order in its fluctu-ations, which simplifies the equations. To illustrate it, we first consider the geometry predicted by GR for an empty universe and then work out the evolution equations of the fluctuations produced by matter and energy in the linearised theory.

• Empty universe

In an empty universe, neither matter nor energy are present and Tµν = 0 at

each space-time point. The solution of the Einstein’s equations, then, is the Minkowskian metric ηµν, where

ηµν =      −1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1      . (2.2)

The Minkowskian metric describes a flat space-time: indeed, the metric restricted to the spatial directions, ηij, is Euclidean.

• Universe with matter and energy

If the universe is not empty, Tµν 6= 0 and the flat metric does not describe the

gravitational field any more, i.e. it is not a solution to the gravitational field equations.

However, if the observer is sufficiently far away from the matter distribution (as it is the case for the astrophysical objects that we observe from Earth), one can nonetheless assume the space-time to be asymptotically flat and expand the metric gµν around the flat space-time. Namely,

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2.1. Gravitational waves as solutions of Einstein’s field equations

where hµνis the perturbation to the flat geometry, which is assumed to be small1.

The above expansion is therefore meaningful when |hµν| 1 for each

compo-nent of the metric perturbations, hµν.

Physically, this condition implies that we are only allowed to choose reference frames in which the smallness of the perturbations with respect to the flat space-time holds. Indeed, the covariant formulation of GR allows us to change the co-ordinate system without altering the description of the physics, which is known as gauge freedom. By assuming the smallness of the perturbations, we have restricted ourselves to coordinate systems that respect the above-mentioned con-dition.

Since the Minkowskian metric is constant, the part of the gravitational field which evolves under the influence of matter and energy is the perturbation hµν. To find

its evolution equations, we consider the so-called trace-reversed perturbation, ¯ hµν = hµν− 1 2ηµνh, h = η µν hµν, (2.4)

which preserves the smallness condition and amounts to changing the sign of the trace of the perturbation, ¯h = ηµν¯hµν = −ηµνhµν = −h, hence the name. This

change of variable allows us to simplify the Einstein’s equations.

Since we are interested in the linearised theory, we raise and lower indices in the Einstein’s equations with the Minkowskian metric ηµν, so that for a generic

tensor Bµ, one has that Bµ = ηµνBν. Indeed, considering the full metric gµν

would produce higher order terms. Moreover, from the explicit expression of the Minkowskian metric, Eq. (2.2), it holds that ηµν _{= η}

µν.

The equations can be further simplified by using the gauge freedom to impose the condition ∂µh¯µν = 0. This coordinate system is also known as the Lorentz

gaugeand always exists (see, for instance, Ref. [111]).

By expanding the Einstein’s equation to the linear order in ¯hµν and by means of

the Lorentz gauge, one gets [111]: 2¯hµν = −

16πG

c4 Tµν, ∂

µ_¯_h

µν = 0 , (2.5)

where 2 ≡ ∂µ∂µ = −(1/c2)∂2/∂t2 + ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2 is the flat

space-time d’Alembertian operator with speed of propagation equal to c.

Therefore, in the linearised regime, the metric perturbations hµν obey a wave

equation and such gravitational waves propagate at the speed of light. They can be thought of as ripples in the fabric of the space-time, which perturb the flatness of the Minkowskian geometry as they propagate.

The first one of Eqs. (2.5) also reveals that the source of GWs is the matter and energy in the universe, through the energy-momentum tensor Tµν. However, GWs

1_{We recall here that the metric tensor is dimensionless, as it describes the space-time geometry. Thus,}

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are suppressed by a factor G/c4 _{∼ 8 · 10}−45_s2_{/kg · m, which anticipates that their}

effects are weak.

2.2 Propagation of gravitational waves

The propagation of GWs can be studied by assuming that the perturbations were created by a certain matter distribution Tµν, but that the observer is detecting them

outside of the source, where Tµν = 0. In this case, their propagation in the linearised

theory is determined by the wave equation:

2¯hµν = 0, ∂µ¯hµν = 0 . (2.6)

In vacuum, additional gauge transformations can be performed to fix the redundant degrees of freedom of the GW ¯hµν. Indeed, since the metric is a symmetric tensor, the

metric perturbation is itself symmetric, which implies that only 10 out of its 16 space-time components are independent. Moreover, the Lorentz gauge fixes 4 more degrees of freedom through the four conditions ∂µ¯_h

µν = 0. Therefore, in the Lorentz gauge the

GW ¯hµν has 6 independent components. We exploit the remaining gauge freedom to

work out the independent degrees of freedom of ¯hµν in what follows.

As a matter of fact, one can perform a further coordinate transformation that respects the Lorentz gauge and imposes the following 4 additional constraints on the GW [111]:

¯

h = 0, h0i = 0 . (2.7)

The first one of Eqs. (2.7) implies that ¯hµν = hµν, as one can verify from Eq. (2.4).

Thus, the wave equation and the gauge conditions derived for the trace-reversed pertur-bations apply in fact to the actual fluctuations hµν.

By using the second of Eqs. (2.7) and by making explicit the summation over repeated indices, the Lorentz gauge condition with ν = 0 becomes:

∂µhµ0 = ∂0h00+ ∂ihi0

= ∂0h00

= 0 .

(2.8)

Therefore, the temporal component of the metric perturbation is constant.

However, GWs are defined as the time-dependent (propagating) part of the fluctua-tions. Thus, the constant temporal component is a contribution to the static gravitational field, which implies that the GW temporal component is trivial.

Therefore, the total gauge conditions imposed on GWs amount to:

h0µ = 0, h = 0, ∂ihij = 0 , (2.9)

where we used the fact that h00 = 0 for the “true” GW. The gauge conditions above

include both the Lorentz gauge and the four additional constraints of Eqs. (2.7) and are called the transverse-traceless gauge, or TT gauge.

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2.3. Effects on test masses

Indeed, the metric perturbation hµν is traceless in this coordinate system and is

non-trivial only in its spatial components µ ≡ i, ν ≡ j. Thus, we refer to the GWs in this particular gauge as hT T_ij in what follows. The gauge is transverse in the sense that the only non-trivial components of the waves are in the transverse spatial directions with respect to the direction of propagation.

As a matter of fact, the general solution of Eqs. (2.6) is a superposition of plane waves of the form hT T

ij = Aij(k)eik

β_x

β_{, where k}β _{= (ω/c, k) is the four-dimensional}

wave-vector of the given plane wave, ω = c|k| is the wave frequency and Aij(k) is the

wave amplitude. The direction of propagation of the wave is determined by the direction of the spatial wave-vector and is defined as ˆn = k/|k|. The Lorentz gauge condition then implies that for each plane wave, ∂jhT T_ij = kihT T_ij = 0. I.e., no fluctuation along the direction of propagation is allowed.

Assuming a GW that propagates along the z axis and imposing the constraints of the TT gauge, one can verify that its general form is:

hT T_ij (t, z) =    h+(t, z) h×(t, z) 0 h×(t, z) −h+(t, z) 0 0 0 0    , (2.10)

where we left implicit the dependency on the frequency ω. Thus, the TT gauge removes 4 more redundant degrees of freedom, leaving the perturbation with only 2 independent components, denoted as h+and h×in the expression above.

A typical convention, then, is to recast Eq. (2.10) in terms of polarisation tensors, that describe the independent oscillation modes of the wave. Indeed, hT T_ij can be rewrit-ten as: hT T_ij (t, z) = +_ij( ˆn)h+(t, z) + ×ij( ˆn)h×(t, z) , (2.11) where +_ij( ˆn) =    1 0 0 0 −1 0 0 0 0    × ij( ˆn) =    0 1 0 1 0 0 0 0 0    (2.12)

are the polarisation tensors. Since the wave oscillations must be transverse with respect to the direction of propagation, the polarisation tensors depend on ˆn.

Within GR, +_ij( ˆn) and ×_ij( ˆn) are the only two independent oscillation modes al-lowed for a GW propagating along the z axis in the TT gauge. The corresponding amplitudes, h+ and h×, are called plus and cross polarisations respectively, with

ref-erence to the deformations that they induce on test masses. We will describe such deformations in the next section.

The advantage of the TT gauge is therefore to simplify the description of the gravi-tational radiation, which depends only on two, spatial, independent degrees of freedom in this coordinate system: h+and h×.

2.3 Effects on test masses

The effects of the plus and cross polarisations can be visualised by inspecting their interactions with test masses. For this purpose, we consider a circular ring of point

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particles lying in the x − y plane and work out the deformations induced on it by a GW propagating along the z axis. The ring configuration is illustrated in Fig. 2.1.

x y

Figure 2.1: Circular ring of test masses in the x − y plane. In absence of gravitational waves, the ring preserves its circular shape.

Within GR, it is always possible to choose the coordinates so that the metric is Minkowskian up to its first derivatives at one particular point [111]. We choose the ring centre as such fiducial point. For simplicity, we also assume it to be the origin of the spatial coordinates, so that gµν = ηµν and ∂αgµν = 0 at the origin. Each test mass is

then characterised by its displacement with respect to the origin ~ξ = (x0, y0, 0), where

x0and y0 are the mass coordinates in the x − y plane.

The natural reference frame to study the effects of GWs on this extremely simplified version of a detector is indeed the reference frame of the instrument. The detector frame is defined as the coordinate system where, at sufficiently small mass displacements with respect to the typical variation scale of the metric, the space-time geometry is flat. In our case, this amounts to the wavelength of the GW.

The evolution of the mass displacements is determined by the geodesic deviation equation. We assume each mass to be initially at rest and non-relativistic, which is the case for ground-based detectors of GWs. Then, the geodesic deviation equation in the detector frame reads [111]:

¨

ξi _{= −c}2_Ri

0j0ξ

j_,

(2.13) where the Riemann tensor Ri_0j0is calculated at the origin.

When a GW perturbs the space-time, the latter can be expressed in terms of the gravitational perturbations. In particular, it is convenient to calculate it in the TT gauge. Indeed, in the linearised theory the Riemann tensor is invariant under the gauge transfor-mations that lead to this reference frame and the geodesic deviation equation becomes:

¨

ξi ₌ 1

2 ¨

hT T_ij ξj. (2.14)

Therefore, the masses experience tidal forces that deform their relative distances, deter-mined by the GW. As a matter of fact, even though it is always possible to cancel the

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2.4. Gravitational waves from generic matter distributions

gravitational field at one particular point through an appropriate choice of coordinates (gµν = ηµν and ∂αgµν = 0), it is not possible to eliminate the effects of a curved

geom-etry such as the tidal forces that deform the relative distances of test bodies. This offers a confirmation that GWs induce non-trivial effects.

To visualise the deformations induced on the circular ring, we consider a GW propagating along the z axis, so that the GW has the form of Eq. (2.10). The dis-placement of each test mass in presence of the GW can then be written as ~ξ =

[x0+δx(t), y0+δy(t), δz(t)], where δx(t), δy(t) and δz(t) are the deformations induced

by the GW. Since the wave is transverse, ¨ξz _{= 0. In turn, this implies that δz(t) = 0}

and the ring keeps lying in the x − y plane.

Integrating Eq. (2.14) for the plus polarisation, one finds that, at the leading order:

δx(t) = h+(t, z = 0)

2 x0 δy(t) = −

h+(t, z = 0)

2 y0. (2.15)

Recalling that each GW component is expressed in terms of plane waves and choosing the origin of times so that h(t = 0, z)T T_ij = 0, one has that h+(t, z = 0) ∼ sin(ωt).

Therefore, the displacement induced by the plus polarisations is:

δx(t) ∝ x0sin(ωt) δy(t) =∝ −y0sin(ωt) . (2.16)

We illustrate the ring deformations induced by such displacement as a function of time in Fig. 2.2. The shape of the circular ring evolves recalling a plus sign, hence the name of the GW polarisation.

Similarly, for the cross polarisation one has that:

δx(t) ∝ y0sin(ωt) δy(t) =∝ x0sin(ωt) . (2.17)

In this case, the deformations induced by the cross polarisation as a function of time recall the shape of a cross, as illustrated in Fig. 2.3.

2.4 Gravitational waves from generic matter

distribu-tions

We dedicate this section to the study of the generation of GWs. In particular, we work out the expression for GWs generated by generic matter distributions, by solving the linearised Einstein’s equations for an arbitrary source Tµν.

The astrophysical objects that produce the GWs that we observe on Earth are distant sources. Hence, their gravitational field is weak at the observation point and one can assume the related metric perturbations to be small with respect to the flat background. Therefore, GWs from such sources are described by the linearised formula of Eqs. (2.5). Moreover, we consider the low-velocity limit v c, where v is the speed of the source. Indeed, just like in the generation of EM waves, this assumption simplifies the equations and allows the multipole expansion of the gravitational radiation.

More specifically, if ω is the typical frequency of the source motion (for instance, the orbital frequency of a binary system of stars) and d is its typical length scale, one

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time (a)

x y

(b)

Figure 2.2: Deformations induced by the plus polarisation on the circular ring of figure 2.1. (a): ring distortion as a function of time. (b): ring distortion at the two instants corresponding to the maximum stretch along the x (solid line) and y (dashed line) directions. The plus polarisation deforms the ring recalling a plus sign.

has that v ∼ ωd. Assuming that the GW frequency is of the same order of ω, it follows that ω ∼ ωGW = 2πc/λ, where λ is the wavelength of the GW. Thus, the low-velocity

assumption implies that λ d: we are considering sources whose typical length scale is much smaller that the wavelength of the gravitational radiation.

The general solution of Eqs. (2.5) can be found through the well-known Green function of the d’Alembertian operator, which gives the integral of the source term at the retarded time:

¯ hµν(t, x) = 4G c4 Z d3x0 1 |x − x0_|Tµν(tret, x 0_{) .} (2.18) In the above equation, the primed coordinates extend along the spatial extension of the source, x is the observation point outside of it and tret= t − |x − x0|/c is the retarded

time.

For distant sources, r = |x| |x0| and 1/|x − x0_{| ∼ 1/r to the leading order.}

The expansion of the retarded time in the energy-momentum tensor, instead, gives

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2.4. Gravitational waves from generic matter distributions time (a) x y (b)

Figure 2.3: Deformations induced by the cross polarisation on the circular ring of figure 2.1. (a): ring distortion as a function of time. (b): ring distortion at the two instants corresponding to the maximum stretch along the bisectors of the x − y plane (solid and dashed lines). The cross polarisation deforms the ring in a cross shape.

us to consider the leading order of this expression, so that Eq. (2.18) becomes: ¯ hµν(t, x) = 1 r 4G c4 Z d3x0Tµν(t − r/c, x0) . (2.19)

The previous equation can be evaluated in the TT gauge, where the plus and cross polarisations are defined. The coordinate transformations that lead from the Lorentz gauge to the latter have been discussed in section 2.2 and can be imposed through the projector Λijkl:

Λijkl( ˆn) = PikPjl−

1

2PijPkl, Pij( ˆn) = δij − ninj, (2.20) where ˆn is the direction of propagation of the GW and δij is the Kronecker delta. Indeed,

one has that hT T

ij = Λijkl¯hkl[111] and any non-spatial component of ¯hµν is removed

by the projection.

By applying Λijklto both sides of Eq. (2.19), we get the GWs in the TT gauge:

hT T_ij (t, x) = 1 r 4G c4 Λijkl( ˆn) Z d3x0Tkl(t − r/c, x0) . (2.21)

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We are interested in expressing the above equation in terms of ρ = T00_/c2_{, that for}

non-relativistic systems is the source mass density [111]. For this purpose, we recall here that a standard result of GR is that the energy-momentum tensor in the linearised regime obeys the conservation law ∂µTµν = 0. By means of the latter, one can verify

that the following relations hold: Z d3xTkl(t, x) = 1 2 ¨ Mkl, Mkl = Z d3xT 00 c2 (t, x)x k_xl_, _(2.22)

where Mkl_{is called the second moment of T}00_/c2_.

Therefore, one can rewrite Eq. (2.21) as:

hT T_ij (t, x) = 1

r

2G

c4 Λijkl( ˆn) ¨M

kl_{(t − r/c) .} _(2.23)

Since the projector Λijklis orthogonal to δkl, then, one has that:

Λijkl( ˆn) ¨Mkl ≡ Λijkl( ˆn) ∂2 ∂t2 Mkl− 1 3δ kl_M ii , (2.24)

where the expression in brackets is the quadrupole moment of the source mass density. Higher order terms in the low-velocity expansion of the energy-momentum tensor would lead to the contributions of higher multipoles to the gravitational radiation.

Outside of the source and in the TT gauge, GWs are described by the plus and cross polarisations discussed in section 2.2. The results of Eqs. (2.23) and (2.24), however, imply that their characteristics (i.e., their explicit expressions) are determined by the source properties through its mass quadrupole moment, to the leading order.

2.5 Gravitational waves from compact binary

coales-cences

The derivation presented so far predicts the gravitational radiation emitted by the motion of generic matter distributions in the weak-field, low-velocity limit, assuming a flat background metric. Here, we consider binary systems of compact objects.

Such systems can be BBHs, BNSs, neutron star-black hole systems, white dwarf binaries etc. and are characterised by the high mass density of the component objects, hence the compactness. More exotic compact objects like boson stars and wormholes have been theorised over the years, e.g. [113]. However, so far, LIGO-Virgo confirmed the detection of GWs from BBHs and BNSs. Thus, we consider BBHs and BNSs in what follows.

Black holes are regions of the space-time where gravity is so strong that not even light can escape its attraction. Thus, no information is available on their internal struc-ture. They can be though of as objects made of pure space-time that interact only though the gravitational force.

GR predicts that black holes are completely characterised by three properties: their mass, intrinsic angular momentum, or spin, and electric charge. However, significantly

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2.5. Gravitational waves from compact binary coalescences

charged black holes are not expected in nature, as the EM repulsion in holding together a charged mass would be orders of magnitude stronger than the gravitational force. Therefore, here we consider uncharged black holes.

Black holes span a wide mass range and can be roughly classified in: stellar-mass black holes (5 − 100M), intermediate-mass black holes (100 − 105M) and

supermas-sive black holes (105_{− 10}9_M

). Moreover, GR predicts that their dimensionless spin

χ = c| ~S|/Gm2cannot exceed unity, where m and ~S are the black hole mass and spin,

respectively.

Just like stars, their motion is determined by the Einstein’s equations and in turn it induces deformations on the surrounding space-time. According to GR, they can form binary systems which merge to give rise to more massive black holes, radiating energy through GW emission during the process. Such GWs carry the imprint of the source properties. Thus, they offer a unique channel to study the physics of BBHs. Indeed, BBH mergers were observed for the first time through their GW emission [12].

Neutron stars (NSs), instead, are lower-mass objects with respect to black holes. Their masses have been accurately measured over the years and have been found to be as large as 2.01 ± 0.04M [114]. In such stars, gravity has compressed the matter to

the point that part of the electrons combined with the protons to form neutrons, hence the name.

Another key difference with respect to black holes is that, due to the presence of the star matter, one can describe the NS internal structure through an EOS. The latter relates the pressure and the density of the matter inside the object: by requiring the hydrostatic equilibrium, it allows to predict the maximum mass allowed for the NS, as well as its radius and tidal deformability as a function of its mass [105].

Indeed, unlike black holes, NSs in binary systems undergo tidal deformations due to the gravitational field of the companion when the objects are sufficiently close to each other. The tidal deformability is the parameter that expresses the capacity of the star to undergo such tidal deformations: the higher, the more the star will be deformed.

Despite the actual EOS of NSs is still unknown, various EOS models have been developed over the years [115, 116]. Each EOS predicts a specific mass range allowed for the star and the related star radius and tidal deformability. In analogy to black holes, then, NSs can spin, with maximum dimensionless spin χ = 1. However, the fastest spinning NS observed to date has χ_{. 0.4 [117].}

The evolution of CBs under the influence of gravity is fully predicted by GR and leads to the merger of the component objects. The whole process, from the two-body evolution to the formation of a single stable remnant, is known as compact binary coalescence (CBC).

Indeed, GR predicts that, if bounded in binary systems, BBHs and BNSs radiate energy away through GW emission as they orbit the common centre of mass. As a consequence, the orbit shrinks and the object’s relative velocity v increases due to the conservation of the total angular momentum.

The initial phase of the coalescence, when v/c 1, is called inspiral and is analyt-ically known in the post-Newtonian (PN) approximation. The PN formalism predicts the evolution of the CB by solving the Einstein’s equations through a perturbative

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ex-pansion in v/c of both the matter distribution and the background geometry. Thus, as opposed to the simplified framework presented in section 2.4, it also takes into account the corrections to the flat background metric due to the motion of the compact objects. Eventually, the orbit becomes unstable and the compact objects plunge into each other, finally merging into a single remnant. Because these mergers are amongst the most violent phenomena of the universe, the merger phase is a strong-field, high-curvature regime, where the non-linearities of GR become important. Thus, the merger is difficult to predict analytically. However, it can be predicted numerically by solving the Einstein’s equations of the system. Therefore, it is modelled as a superposition of pseudo-quasinormal modes: damped sinusoids whose parameters are calibrated on numerical solutions of the Einstein’s equations [118].

The final object which forms through this violent process is initially distorted, but radiates its anisotropies through GW emission, until it reaches a stable state. This last part of the coalescence is called ringdown.

For BBHs, the ringdown is theoretically known from perturbation theory, and is modelled as a superposition of quasinormal modes, whose parameters are known func-tions of the remnant black hole mass and spin. For BNSs, instead, the understanding of this last phase is complicated by the presence of the star matter and by its internal structure, which is highly unknown.

Indeed, the ringdown of BNSs is extremely difficult to model analytically, as it requires to address both the perturbations of the metric and of the NS matter simulta-neously. Moreover, depending on the initial total mass of the binary and on the stars’ EOS, the remnant could be a black hole or an NS itself. In the latter case, the NS can either reach a stable configuration through GW emission, or subsequently collapse into a black hole. For these reasons, currently, the modelling of the ringdown of BNSs is mainly based on numerical solutions of the Einstein’s equation, that assume a given EOS of the star.

Thus, CBs emit GWs throughout the coalescence and the morphology of such sig-nals depends on the properties of the compact objects. The simplified approach of the previous section, although inaccurate for BBHs and BNSs, which induce non-negligible deformations to the flat background metric, already highlights the dependence from the source mass quadrupole moment.

Indeed, direct detections of GWs from CBCs allow us to infer multiple source prop-erties simultaneously, such as the binary masses, spins and luminosity distance [13, 119, 120, 121, 122, 108, 36, 37]. Moreover, when combined together, multiple observations

enable us to study the population of compact objects of the universe [19, 35, 32, 41]. Furthermore, GWs give access to all the phases of the coalescence, including the strong-field regime of the merger. Thus, GWs from CBCs are used to perform pioneer-ing tests of GR, both on a spioneer-ingle-event and on a statistical basis [14, 123, 33, 40]. Hence, GWs from CBCs are an exciting channel to study the physics of such systems. In the next subsections, we discuss the GW signals produced by these sources.

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2.5. Gravitational waves from compact binary coalescences

2.5.1 Compact binaries in circular motion

As opposed to the merger and ringdown, GWs from CBC inspirals can be worked out in a simplified framework with minor calculations. In this subsection, we present such explicit derivation.

At the lowest order in v/c, the inspiral GWs can be modelled through the quadrupole formula of Eq. (2.23), with the appropriate energy momentum tensor. In particular, we consider CBs in circular motion.

We note that, instead of considering a flat background metric, a comprehensive derivation would include the deformations of the flat background induced by such objects. However, the simplified framework of the quadrupole formula can already highlight the main characteristics of the inspiral waveform. Therefore, we derive the inspiral signal of CBCs within this approach here and refer the reader to Ref. [111] for details on the full, post-Newtonian treatment. We will then compare the result of the computation with waveform models that consider the full derivation.

We model the CB as two point masses m1, m2orbiting around their centre of mass

(CM). We assume that the relative coordinate ~R = ~r1− ~r2 is in circular motion with

orbital frequency ωorb and that the system has an inclination ι with respect to the line

of sight of the observer. We consider the CM reference frame and choose its orientation so that the orbit lies in the x − y plane. The CB configuration is illustrated in Fig. 2.4.

y z x m1 m2 r ι φ (r, ι, φ)

Figure 2.4: Compact binary configuration. The two masses m1, m2 (black dots) orbit

the common centre of mass (the origin) in the x − y plane. An observer is placed at a distance r and zenith and azimuth angles (ι, φ) with respect to the binary reference frame.

In fact, the system loses energy through GW emission during the coalescence and the motion is non-circular. However, such emission is negligible during the first orbits of the inspiral. Therefore, here we consider the GWs produced for circular motions. In

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the next subsection we will use such GWs to estimate the effects of the energy loss on the orbit, and the resulting corrections to the circular-orbit GW emission.

Since we are assuming a flat, Newtonian background metric, the orbital frequency

ωorb of the binary motion is related to the orbital separation R through Kepler’s third

law,

ω2_orb = G(m1+ m2)

R3 (2.25)

and the trajectories of the two bodies are:

x1(t) = m2 MR cos(ωorbt + π/2) x2(t) = − m1 MR cos(ωorbt + π/2) , (2.26) y1(t) = m2 MR sin(ωorbt + π/2) y2(t) = − m1 MR sin(ωorbt + π/2) , (2.27) z1(t) = 0 z1(t) = 0 , (2.28)

where M = m1 + m2 is the total mass of the binary and the constant phase of π/2

amounts to choosing the mass initial positions along the y axis.

Through the object’s trajectories, one can compute the mass moment Mijof Eq. (2.22), that for a discrete system is:

Mij = m1xi1x

j

1+ m2xi2x

j

2. (2.29)

Recalling that hT T₁₁ = h+and hT T12 = h×, the plus and cross polarisations are obtained

by substituting the mass moment in the quadrupole formula, Eq. (2.23), with ˆn = (sin ι sin φ, sin ι cos φ, cos ι):

h+(t) = 4 r _GM c c2 5/3_ω orb c 2/3_{1 + cos}2_ι 2 cos [2ωorb(t − r/c)] , (2.30) h×(t) = 4 r _GM c c2 5/3_ω orb c 2/3

cos ι sin [2ωorb(t − r/c)] . (2.31)

In the above, we introduced the parameter Mc,

Mc=

(m1m2)3/5

(m1+ m2)1/5

, (2.32)

which has the dimensions of mass and is commonly called chirp mass. When accounting for the energy loss of the binary, this parameter determines the characteristic chirping behaviour of the GW frequency, hence the name. We will discuss this particular aspect in more detail in what follows.

Equations (2.30) and (2.31) show that GWs oscillate at double the orbital frequency of the source: ωGW = 2ωorb. Moreover, the inclination of the line of sight, ι,

deter-mines how loud the wave amplitudes are for different observers, with ι = 0, π rad corresponding to the maximum amplitudes and ι = π/2, 3π/2 rad to the maximum suppression.

Typically, the former systems are referred to as face-on (ι = 0 rad) and face-off (ι = π rad) binaries, whereas the latter are called edge-on systems, with reference to the portion of the orbital plane that is visible to the observer (either the “face” or the “edge”).

Softwareandtheoreticalmethodsforgravitational-wavedataanalysis PhDthesis

Dipartimento di Fisica E. Fermi

Corso di Dottorato in Fisica

PhD thesis

Software and theoretical methods for

gravitational-wave data analysis

Candidate:

Supervisor:

Giulia Pagano

Prof. Walter Del Pozzo

Abstract

Contents

Chapter 1

Introduction

1.1

Conventions and notation

Chapter 2

Gravitational waves: theory and

observations

2.1

Gravitational waves as solutions of Einstein’s field

equations

2.2

Propagation of gravitational waves

2.3

Effects on test masses

2.4

Gravitational waves from generic matter

distribu-tions

2.5

Gravitational waves from compact binary

coales-cences

2.5.1

Compact binaries in circular motion