Dipartimento di Fisica
Unsupervised classification of
short transient noise to improve
gravitational wave detection
Author
S. Bini
Supervisor
Prof. F. Fidecaro
Co-supervisor
Prof. M. Razzano
September 2020
Gravitational waves interferometers are complex and sensitive detectors, whose data are non stationary and non Gaussian. Short duration distur-bances, called ’glitches’, are caused by the instrument itself or by its interac-tions with the environment. These transient noises are particularly concern-ing as they can mimic gravitational wave signals, and have both high rate and high signal-to-noise ratio. Great effort has been made in the latest years to understand their causes and mitigate them: in particular as glitches differ significantly in terms of duration and frequency, their classification is crucial to trace back their origin. Due to glitches complexity, huge number and time evolving nature, machine learning techniques find great application in this field.
This thesis proposes an unsupervised clustering algorithm able to group transient noise into different classes according to their morphology in a time-frequency map, using a neural network called ’autoencoder’ and a density-based clustering algorithm, without any prior knowledge on the data it is applied to. This method is successfully tested on LIGO Hanford detector glitches, and applied to latest Virgo data, clustering one week, randomly selected, of Virgo transient noise and contributing to a candidate event val-idation. The information acquired could enhance detector characterization and transient noise mitigation, improving gravitational-wave searches.
Introduction 3
1 Gravitational waves 5
1.1 Gravitational wave theory . . . 6
1.1.1 Linearization of Einstein’s equations . . . 6
1.1.2 Energy carried by gravitational waves . . . 8
1.1.3 Sources of gravitational waves . . . 9
1.2 Advanced Virgo . . . 15
1.2.1 Fabry - Perot cavity . . . 17
1.2.2 Angular response . . . 20
1.2.3 Dark fringe detection . . . 21
1.2.4 Control and Locking . . . 22
1.2.5 Optical layout . . . 23
1.3 Data analysis techniques . . . 27
1.3.1 Matched filtering . . . 28
1.3.2 Parameter estimation . . . 28
1.3.3 Significance of an event . . . 30
1.3.4 Data analysis techniques for different GWs signals . . . 32
1.4 Advanced Virgo noise sources . . . 38
1.4.1 Quantum noise . . . 39
1.4.2 Seismic noise . . . 41
1.4.3 Thermal noise . . . 42
1.4.4 Other noise sources . . . 42
2 Short transient noise in gravitational wave interferometers 44 2.1 Impact of transient noise on GW searches . . . 45
2.1.1 Detector characterization . . . 46
2.1.2 Advanced Virgo Data Quality . . . 48
2.1.3 Significance of candidate event case of GW150914 . . . 51
2.1.4 Coincidence of transient noise and GW
case of GW170817 . . . 54
2.2 Transient noise detection . . . 56
2.2.1 Wavelet transform . . . 56
2.2.2 Q transform . . . 57
2.2.3 Omicron algorithm . . . 59
2.3 Glitches characterization with machine learning techniques . . 63
2.3.1 Introduction to Machine Learning . . . 63
2.3.2 Machine learning and GW physics . . . 68
2.3.3 Supervised VS unsupervised transient noise classification 69 2.3.4 Supervised classification of transient noise . . . 71
2.3.5 Virgo glitch families . . . 79
3 Unsupervised clustering algorithm 83 3.1 Clustering . . . 84 3.1.1 Clustering algorithms . . . 84 3.1.2 Clustering metrics . . . 87 3.2 Autoencoder . . . 90 3.2.1 Introduction . . . 90 3.2.2 Autoencoder in GW physics . . . 92 3.3 t-SNE algorithm . . . 94 3.3.1 Introduction . . . 95
3.3.2 t-SNE inevitability and weakness . . . 96
4 Application to Virgo data 98 4.1 Training & Test . . . 99
4.1.1 Data preparation . . . 99
4.1.2 Training . . . 102
4.1.3 Test . . . 107
4.2 Results on O3 Virgo data . . . 111
4.2.1 One week glitches . . . 111
4.2.2 Candidate event : S191225aq . . . 113
4.3 Future perspectives . . . 116
Conclusions 117
Appendix 118
Bibliography 124
This thesis proposes a machine learning clustering algorithm to characterize short duration noise in gravitational wave (GW) interferometers.
The first chapter provides an overview of GW physics, deriving the wave equation from Einstein’s equations and describing the main sources of GWs [1]. Particular relevance has been given to the description of Advanced Virgo interferometer [2] and to the data analysis techniques used to extract the astrophysical signals from the detector noise [1].
Among different noise typologies, gravitational wave interferometers are af-fected by short duration disturbances, called glitches, that are particularly concerning because they mimic transient astrophysical GW signals [3]. The second chapter aims to convince the reader of the importance of glitch char-acterization, describing the impact of this noise on the evaluation of the significance of GW candidate events [4] and, when transient noise and GW signals are coincident, on GW parameters estimation [5].
Due to transient noise complexity, huge number and time evolving nature, machine learning techniques find great application in this field [6]. After a brief introduction to these techniques, we overview the main studies on tran-sient noise classification, performed with machine learning algorithms. The last two chapters contain the original contribution of this thesis. The third chapter proposes a clustering algorithm able to group transient noise into different classes according to the glitch morphology in a time-frequency map. This method is based on unsupervised learning, i.e. it does not require any previous knowledge on the data it is applied on, and consists of two steps: first an autoencoder neural network compresses the input data, extracting meaningful features to distinguish glitch classes [7]. Then a clustering algo-rithm groups the compressed representation.
In the fourth chapter, we train the autoencoder with the Gravity Spy glitch dataset [8], we tune the network parameters and select the optimal
config-uration between two possible architectures and different autoencoder com-pression factors.
We successfully test the proposed method, achieving an higher perfor-mance than a related work [9].
Then the clustering algorithm is applied to the latest Virgo data, from O3 run, focusing on one week, randomly selected, and close to a candidate event, to contribute to its validation.
Gravitational waves
This chapter provides an introduction to gravitational wave physics. Since the first detection of gravitational waves (GW) in 2015 from the inspiral and the succeeding merger of two black holes [10], there have been several detections that lead useful insight in fundamental physics and astrophysics [11]. Today this field of research is extremely active, involving a worldwide community and the construction of various new facilities.
Gravitational waves are obtained from Einstein’s equations and are ex-pected to be emitted by various sources as compact binary systems, rotating pulsars and by a primordial stochastic background (Section 1.1). Up to date, only the first type has been detected by ground-based interferometers. Three detectors have already observed GWs: Advanced LIGO Livingston, Advanced LIGO Hanford [12] and Advanced Virgo [2]. The working princi-ple of ground-based GW interferometers is presented, focusing on Advanced Virgo (Section 1.2). The detector output contains large noise and specific data analysis techniques are used to extract the astrophysical signals and to estimate their main features (Section 1.3). Several noise sources affect the in-terferometer and various strategies to mitigate them have been implemented (section 1.4).
The third section, regarding data analysis techniques, will be particularly important for the further development of this work. The guiding line for this chapter is [1] but up-to-date information on the status of the detector and on the achieved results are reported.
1.1
Gravitational wave theory
Gravitational waves are obtained from the linearization of Einstein’s equa-tions for small fluctuaequa-tions of the Minkowsky metric (Section 1.1.1), and carry energy and momentum (Section 1.1.2). Once emitted, GWs travel into space without absorption or scattering event and could be detected. The main sources of GW are compact binary systems, asymmetric rotating bodies, ex-plosive events as core-collapse supernova and a stochastic GW background (Section 1.1.3).
1.1.1
Linearization of Einstein’s equations
Einstein’s equations are ten non-linear partial differential equations that de-scribe the relationship between the metric gµν and the mass and energy of
the system, contained in the stress-energy tensor Tµν,
Rµν −
1
2gµνR = 8πG
c4 Tµν (1.1)
Rµν and R being the Ricci tensor and the scalar curvature, that depends on
the first and second derivative of the metric gµν. Typically Eq.1.1 is written
with the Einstein’s tensor Gµν defined by:
Gµν = Rµν−
1
2gµνR (1.2)
For weak gravitational fields, the metric can be expanded around the flat-space metric ηµν
gµν = ηµν+ hµν (1.3)
|hµν| 1 (1.4)
Einstein’s equations can be linearized to the first order with respect to the small fluctuations hµν. Using ¯hµν = hµν − 1/2ηµνh, where h = ηµνhµν,
and the Gauge invariance ∂µ¯h
µν = 0 1 to simplify the equations as in the
electromagnetic case, the wave equation is obtained, and shows that Tµν acts
as source for the GWs:
2¯hµν = −
16πG
c4 Tµν (1.5)
1Lorentz gauge: The metric g
µν transform as gµν0 (x0) = ∂x ρ ∂x0µ ∂xσ ∂x0νgρσ(x) so for a transformation of coordinates as x0µ= xµ+ξµ(x), ∂ν¯h µνbecomes ∂νh¯0µν = ∂νhµν−∂ν∂νξµ. Therefore, if ∂ν¯hµν = fµ(x), to obtain ∂ν¯h0
µν = 0 we must chose ∂ν∂νξµ(x) = fµ(x). The
In vacuum Tµν = 0 and the solution to Eq.1.5 is a superposition of plane
waves of the form ¯hµν = Hµνeikµk
µ
with kµ = (ω/c, k) and ω/c = |k|.
Ex-ploiting the Gauge invariance, if the wave vector has components kµ = (ω/c, k, 0, 0), then H
µν can be reduced to a null trace tensor with no
time dependent components and only two degree of freedom corresponding to the amplitude of the ’plus’ and ’cross’ polarization of the GW wave. In this case the tensor hµν satisfies
h00= 0 , h0i= 0 , ∂ihij = 0 , hii = 0 (1.6)
This defines the transverse-traceless gauge (T T ) denoted by hT Tij , that de-scribes the fluctuations of the metric outside the source with two degrees of freedom and non-zero components only in the plane transverse to the direc-tion of propagadirec-tion. The field in the T T gauge is obtained from hkl of a
GW propagating in a generic direction n using a combination of projector operators Pij(n) = δij − ninj,
hT Tij = Λij,klhkl, Λij,kl = PikPjl−
1
2PijPkl (1.7) Unlike the electromagnetic case, a rotation of π/4 around the propagation axis transforms one polarization into the other and for this reason there are referred as plus (h+) and cross (h×) polarizations.
The gravitational waves generated from a source can be studied looking at the solution of the wave equation ∂µ∂µφ = f where f represents the source.
For a non relativistic source the dimension of the radiating source d is much smaller than the wavelength of the emitted waves λGW and the solution of
the wave equation hT T
ij can be expanded around v/c. The leading term is
hT Tij (t, x) = 1 r 2G c4 Λij,kl(n) d2 dt2Qkl(t − r/c), if λGW d (1.8) with Qkl = Z d3x(xkxl− 1/3r2δkl) T00 c2 (1.9)
Qkl, called the quadrupole moment, is evaluated at the retarded time t − r/c.
The fluctuations of the metric induced by GWs, according to Eq.1.8 decrease linearly with the distance r, and depend on how the time component of the stress-energy tensor varies with time [1, 15].
1.1.2
Energy carried by gravitational waves
To understand the energy carried by GWs it is necessary to go beyond the linear approximation: indeed in the first order in hµν, the Gauge used
(∂µ¯h
µν = 0) and Eq.1.5 gives ∂µTµν = 0 that means that the energy and the
mass distribution of the system are constant and GWs do not carry energy. To include higher orders a stress-energy tensor associated to the gravitational field is defined as τµν = c 4 8πG G(1)µν − Gµν (1.10) where the index on the Einstein’s tensor Gµν (Eq.1.2) represents the higher
order on hµν. G(1)µν − Gµν contains all the terms that go as O(h2) or higher
terms. For weak fields, it is enough to consider quadratic order on the metric fluctuations and far from the sources, where GWs are plane waves, the energy-momentum tensor is:
τµν '
c4
32πGh∂µh
ργ
∂νhργi (1.11)
Using the T T gauge τ00= c2 32πGh ˙h T T ij ˙h T T ij i = c2 16πGh ˙h 2 ++ ˙h 2 ×i (1.12)
where hi brackets indicate a temporal average over a few wave periods. The sum of the stress-energy tensor of the system and of the gravitational field is a conserved quantity, ∂µ(Tµν+ τµν) = 0, so that there is an exchange
of energy and momentum between the sources and the GWs. Besides, far from the sources where the metric is flat ∂µτµν = 0.
The latter consideration implies that RV d3x(∂0τ00+ ∂iτi0) = 0 and it is
useful to compute the energy flux of GWs. Indeed, the GWs energy inside a volume V is
EV =
Z
V
d3xτ00 (1.13)
so the variation of the energy over time is 1 c dEV dt = − Z V d3x∂iτ0i (1.14)
GWs propagate radially from the source and for large distances τ0r = τ00.
Therefore the outward propagating waves carry an energy flux through a surface element dA equal to:
dE dAdt = c3 32πGh ˙h T T ij ˙hT Tij i = c3 16πGh ˙h 2 ++ ˙h2×i (1.15)
In the same way the flux of the moment is derived: the momentum of the GWs inside a spherical shell V far from the source is
PVk = 1 c
Z
V
d3xτ0k (1.16)
and the momentum flux carried away by the propagating waves is dPk dt = − c3 32πGr 2 Z dΩh ˙hT Tij ∂khT Tij i (1.17) where dA = dΩr2.
Once GWs are emitted, they travel into the space without absorption or scattering event because of the smallness of the gravitational cross-section. For comparison, the mean free path for photons in the Sun is about O(1) cm, while for gravitons is higher by a factor O(1080) so the Sun is completely
transparent for GWs radiation. Instead, a significant absorption take place if the GWs impinge on a black hole or a neutron star but the probability that such compact objects lie on the path from the source to us is very small [1, 15].
1.1.3
Sources of gravitational waves
This section presents an overview of the main sources of gravitational waves. The coalescence of compact binary systems and core-collapse supernova duce short duration GWs, while asymmetric rotating bodies lead to the pro-duction of continue GW signals. Lastly, a stochastic gravitational wave back-ground given by the superposition of numerous sources or generated in the early Universe is described.
• Coalescence of compact binary systems
The inspiral and the following merger of a compact binary emit GWs. These systems can be composed of two black holes, two binary neutron stars or one black hole and a neutron star, and GWs have been already detected from these three types (see Section 1.3.4).
In a first approximation the back reaction on the motion of the system due to GWs emission can be neglected. The orbit of two masses m1, m2
can be described by their relative coordinate which performs a circular motion. Using a flat metric, the quadrupole moment Qij is computed.
From Eq.1.8 the plus and cross polarizations are: h+(t) = 1 r 4Gµω2 sR2 c4 1 + cos2(i) 2 cos (2ωst) (1.18)
h×(t) =
1 r
4Gµωs2R2
c4 cos (i) sin (2ωst) (1.19)
where i is the orbit inclination with respect to the observer and µ the reduced mass of the system. Eq.1.18 - 1.19 show that the quadrupole radiation ωGW is at twice the frequency ωs at which the masses rotate
and both the polarizations are maxima for i = 0, i.e. the plane of the orbit is perpendicular to the line of sight of the observer. Sobstituing ωs with the Kepler’s law:
ωs2 = Gm
R3 (1.20)
it turns out that the amplitudes h+, h× depend on the masses only
through a combination called the chirp mass Mc:
Mc =
(m1m2)3/5
(m1+ m2)1/5
(1.21)
The total radiated power (Eq.1.15) is P = 32 5 c5 G GMcωGW 2c3 10/3 (1.22)
Also the total radiated power depends on the masses only through the chirp mass, that for this reason is one of the parameters estimated with the highest precision.
Until now the motion of the masses was assumed to be a circular Keple-rian orbit, however the emission of GWs dissipates energy. This energy comes from the orbit of the sources, which is the sum of the kinetic and potential energy
Eorbit= Ekin+ Epot = −
Gm1m2
2R (1.23)
To compensate the loss of energy caused by GWs emission the radius R of the orbit decreases. The more R decreases the more ωs increases,
according to the Kepler’s law (Eq.1.20). Consequently also the radiated power grows (Eq.1.22). This runaway process lasts until the masses are not too close and the hypothesis of weak gravitational field is no more acceptable.
During the coalescence, having P = −dEorbit/dt, and substituting
Eq.1.22 and Eq.1.23, the frequency of the emitted GWs increases as ˙ ωGW = 12 5 2 1/3 GMc c3 5/3 ω11/3GW (1.24)
Integrating the latter equation up to a tcoal where ωGW diverges, the
time to coalescence can be inferred from the GW frequency detected. Given Mc, the higher is the GW frequency detected, the sooner the
masses will merge: for example, for a typical value of Mc = 1.2 M if
the signal is detected when its frequency is ∼ 100 Hz the interferometer catches the last two seconds of inspiral radiation, while at ∼ 1 kHz the signal will be observed only for the last few milliseconds.
A ground-based interferometer can follow the evolution of this type of signals for thousand of cycles NGW, defined as NGW =
Rtf in
tin ω(t)/2πdt,
and this is crucial for the detection as it will be explained in the next section.
From Eq. 1.18 -1.24 during the coalescence both the amplitude and the frequency increase. For stellar mass objects, the total energy released during this phase is huge, ∆Erad ∼ 8 × 10−2µc2, making these GWs
observable.
The inspiral phase proceeds until the separation between the sources is too small and the flat space appproximation is no more valid. In a Schwarzschild geometry, there is a minimum value for the radial dis-tance beyond which circular orbits are not possible, called Innermost Stable Circular Orbit (ISCO), that is
rISCO =
6G(m1+ m2)
c2 (1.25)
When the masses distance is close to this value the stars merge. This phase is dominated by non-linear behaviour and modelled only through numerical simulation of the Einstein’s equations. The remnant of the binary system can be a prompt collapse on a black hole or a neutron star which later collapses into a black hole. This phase, called ring-down phase, is expected to emit GWs but the waveforms cannot be computed analytically [1, 15].
• Asymmetric rotating bodies
This production mechanism is important for application to isolated pul-sars: their extremely stable period of pulsation means that the energy lost through gravitational and electromagnetic waves is small but, the periodic nature of their signals enhances the chance to detected their GWs emission.
To characterize a rotating body two frames are defined: a body frame attached to the body with coordinates (X10, X20, X30) and a second fixed
frame, (X1, X2, X3), such that X30 = X3. The inertia tensor of a rigid
body is Iij = R d3xρ(x)(R2δij − xixj), ρ being the mass density and
R the radius of the object. The inertia tensor is diagonal in the body frame, Iij0 = diag(I1, I2, I3), and the eigenvalues are called principal
moments of inertia. The GW signal is obtained from Eq.1.8 computing the quadrupole moment. The polarizations of the GW signal for a rotating body are:
h+= 1 r 4Gω2s c4 I1− I2 (1 + cos θ) 2 cos (2ωst) (1.26) h× = 1 r 4Gω2 s c4 I1− I2 cos θ sin (2ωst) (1.27)
GWs are emitted twice the rotating body frequency ωs only if I1 6= I2,
otherwise there isn’t a time-varying quadrupole moment. From Eq. 1.15 the total power radiated is
P = 32 5 G c5 3I2 3ω 6 s (1.28)
where is the ellipticity defined by (I1 − I2)/I3. Since the rotating
body is emitting GWs, the rotation energy of the pulsar decreases as
dErot
dt = −P and the rotation frequency ωs decreases. However only a
small fraction of the energy is lost through GW emission (Section 1.3.4) and ωS is stable [15].
• Core-Collapse Supernova
Massive stars experience violent core-collapse when nuclear produc-tion cannot sustain their own gravity any more. Core collapse is an extremely fast process, typically it lasts less than a second and some-times of the order of milliseconds, that releases a huge amount of en-ergy through mostly neutrinos, but also electromagnetic radiation and GWs. The maximum radiated energy in GWs, that can be expected in those events involving solar mass objects, is ∆Erad ∼ 10−2M c2
[1]. Neutrinos and GWs can reveal crucial information on this process since they are emitted in a short time by the central engine of core-collapse, while the electromagnetic radiation is emitted further by the turbulence in the ejected materials and their interactions with the in-terstellar medium. Since the core-collapse is a violent and rapid process the GWs radiated are very complicated to be modelled [15].
• Stochastic gravitational wave background
The stochastic GW background consists of two different contributions: an astrophysical background and a cosmological one [13, 15].
The first originated by a large number of unresolved sources. The major contributions arise from the coalescence of galactic or extra-galactic binary systems, core-collapse supernova and rotating pulsars. The observation of this astrophysical background could provide useful information among which the star formation rate, supernova rate, and mass and angular distribution of black holes. However, this background also constitutes a noise source for the detection of a cosmological back-ground.
This latter is expected to arise from the early Universe, when gravi-tons decoupled, similarly to the neutrino background or the cosmic microwave background originated by the neutrinos and the photons decoupling respectively. These primordial GWs today still retain infor-mation on the conditions in which they have been produced and their detection could provide exceptional implications.
The cosmological background is expected to be isotropic, stationary and unpolarized. Its spectrum usually is characterized by ΩGW(f )
ΩGW(f ) =
1 ρC
dρGW
d log f (1.29)
where ρC is the critical density (ρC = 3H02/8πG). Actually, the
stochas-tic GW background is expressed with h20ΩGW(f ), where h0 contains the
uncertainty in the Hubble parameter 2.
Possible mechanisms that originated the stochastic GW background are the amplification of vacuum fluctuations in inflationary models, phase transitions, bubble collisions or the relativistic oscillations of cosmic strings [13].
Some experimental bounds put constraints on ΩGW:
– the nucleosynthesis bound: nucleosynthesis successfully predicts the primordial abundances of light elements (3He,4He,7Li). These abundances depend, among others, on the freeze-out temperature, i.e. the temperature at which the rate of interaction Γ is equal to the rate of the expansion of the Universe H. H depends an
2H
all the possible form of energy present at that time; hence if the cosmological GW background was present it contributed to in-crease H and consequently the freeze-out temperature resulting in more neutrons available. Being more neutrons the predicted abundances are different, therefore experimental values of primor-dial elements abundances set a bound on the GW background. This of course concerns only GWs produced before the nuclesyn-thesis took place. This bound does not depend on the frequency and sets
h20ΩGW < 5 × 10−6 (1.30)
– cosmic microwave background fluctuations: the Sachs-Wolfe ef-fect predicts that a GW background at low frequencies should have produced a stochastic redshift on the cosmic microwave back-ground. COBE and Planck satellites measured accurately the cos-mic cos-microwave background temperature fluctuations. From the fluctuations measured by COBE satellite, a constraint on ΩGW(f )
can be set and has the form [23]
h20ΩGW(f ) < 7 × 10−11 H0 f 2 , 3 × 10−18Hz < f < 10−16Hz (1.31) while at higher frequencies, Planck satellite constrains the density to [24]
h20ΩGW < 1.6 × 10−6, f > 10−15Hz (1.32)
– Pulsar bound: millisecond pulsars provide very accurate timing and if a GW passes between the pulsar and the observer the time of arrival of the pulse is shifted. The timing fluctuations are pro-portional to the GW strain h(t). The experimental measures of pulsars periods put a constraint on ΩGW as
h20ΩGW < 4.8 × 10−9 f f∗ 2 f > f∗ = 4.4 × 10−9 Hz (1.33) [13, 15].
1.2
Advanced Virgo
Gravitational waves have been detected in 2015 for the first time by LIGO3
ground-based interferometers [10]. In this section the main features of these detectors are presented, with particularly focus on the Advanced Virgo de-tector, that is based in Italy and joined LIGO during the second observing run.
To understand how GW interferometers work it is useful to start from a simple Michelson interferometer: a monochromatic infrared light source is separated by a beam-splitter, i.e. a semi-transparent mirror, in two com-ponents with equal amplitudes, that travel into orthogonal arms. At the end of each arm, two totally reflecting mirrors reflect them back. The two beams recombine at the beam-splitter and the resulting beam goes into a photo-detector that measures its intensity.
It is convenient to write the electromagnetic field with a complex notation: E0e−iωLt+ikL·x where ωL is the frequency of the laser. The power measured
at the photo-detector, |Eout|2, is proportional to E02sin2[kL(Ly− Lx)] so it
depends on any variation on the length of the arms.
Thus the output of GW interferometers is a time-series that encloses the information of any variation of detector arms Lx, Ly. To understand the
interaction of the GWs with the detector two different reference systems could be used: the T T gauge or the proper detector frame.
• T T gauge
The coordinates are marked by the positions of free falling objects and when a GW passes their coordinates by definition do not change. The mirrors of the interferometer are suspended and can be considered in free falling in the horizontal plane as the forces that act on them can be considered static compared to the frequencies of GWs that we are looking for (O(10 Hz −1 kHz)). Hence the mirrors follow the geodesics of the gravitational field and a GW affects the propagation of light between them. The analysis of the light propagation in this reference system shows that if L λL = ωcL meanwhile photons are travelling
through the arms, h(t) changes so many times that the overall effect is cancelled out. In this frame the effect of the GWs is contained into a phase difference ∆φx(t), in the electric field at the output, of the form
∆φx(t) ' h(t − L/c)kLL, if λL L (1.34)
3Laser Interferometer Gravitational-Wave Observatory, it consists of two GW
that is formally equivalent to a variation of the arms length ∆(Lx− Ly)
L ' h(t − L/c) (1.35)
Thus the interferometer arms experience a length variation propor-tional to the amplitude of the GW.
Maximizing the total power P ∼ |Eout|2 the optimal length of each arm
should be
L ' 750 km 100 Hz fGW
(1.36) If the arms are longer the GW amplitude inverts its sign during the round trip in the arms and the effect starts to be canceled out, while if arm are shorter the light had the same sign but the effect is smaller. Typically h ∼ 10−21 and Virgo arms are 3 km long, so ∆L ∼ 10−18 m. In addition, the passage of GWs into the interferometer generates side-bands, in the light propagating in the arms, with frequencies ωL± ωGW
and amplitude proportional to O(h). • proper detector frame
In this frame the coordinates are measured by a ’rigid ruler’, with re-spect to a coordinate system with the origin fixed on the beam-splitter. The effect of a GW is to displace mirrors from their original positions. This movement is determined by the equation of the geodesic deviation. As in small scales, with respect to the variations of the gravitational field, the metric can be considered flat, this reference frame is very intuitive: the interaction between a GW and the mirrors can be de-scribed in term of Newtonian forces, but it is valid only if λGW L.
This condition is satisfied in ground-based interferometer that have an upper limit on the bandwidth ∼ 10 kHz which corresponds to a mini-mum GW wavelength of ∼ 30 km, much greater than the arm length L ∼ 3km.
From Eq.1.36 the optimal interferometer length for frequencies of a few hun-dreds Hz is several hunhun-dreds of kilometers which is impossible both for prac-tical and economical reasons. LIGO arms are long about 4 km and Virgo 3 km but an optical configuration is used to make the laser beam bounce back and forth many times in the arms increasing the path of light. This config-uration is a Fabry - Perot cavity and it is explained in the next subsection. [1, 15].
1.2.1
Fabry - Perot cavity
A Fabry - Perot (FP) cavity consists of two parallel mirrors, that at first can assumed to be plane and of infinite transverse size. The relation between the incoming field Ein, the reflected Er and transmitted field Et is
Er = rEin, Et= tEin (1.37)
where r and t are the reflection and transmission coefficients. For simplicity, the boundaries where transmission and reflection occur can be considered sharp, consequently they do not produce any phase shift and there are no losses. Thus the coefficients r and t are assumed real and r2+ t2 = 1.
When the electromagnetic field enters in the FP cavity it is partially re-flected back and partially trasmitted. The transmitted field propagates to the other mirror and it is again partially reflected and transmitted. The reflected component travels to the first mirror and continues to be reflected and transmitted. Therefore inside the cavity there is a superposition of sev-eral beams, corresponding to multiple bounces. A schematic representation is shown in Fig.1.1. In the realistic situation, where the mirrors are not sharp boundaries, the phase shift acquired is the same for all beams and so it gives only an overall factor, independent from the length of the cavity; while the losses in the mirrors are taken into account writing r2+ t2 = 1 − p where p is few parts per million in Advanced Virgo [2].
Figure 1.1: Schematic interaction of the laser beam with the two mirrors of the Fabry - Perot cavity. Various paths have been drawn spatially separated only for clarity [1].
To understand the usefulness of a FP cavity we estimate the field inside the cavity. The left mirror is set in x = 0 and labeled with 1, the right one at x = L, and labeled by 2. The incoming field Ein= E0e−iωLt+ikLx at t = t0
one (t1E0e−iωLt0) travels to the other mirror gaining a phase factor
Ecav(L) = eikLLEcav(0). Once it return to the first mirror it is again reflected
and transmitted: the total reflected field get a contribution t21r2E0e−iωLt0
since it was reflected by the second mirror once, and transmitted by the first twice. Evaluating n round trips, the reflected and transmitted fields are:
Er = E0e−iωLt0 r1− r2e2ikLL 1 − r1r2e2ikLL (1.38) Et= E0e−iωLt0 t1t2eikLL 1 − r1r2e2ikLL (1.39) The latter differs from the field in the cavity at the second mirror only by a factor t2 (Et= t2Ecav(L)).
The power of the transmitted field, and consequently the power inside the cavity, is |Et|2 = E02 t2 1t22 1 + (r1r2)2− 2r1r2cos (2kLL) (1.40) When 2kLL = 2πn with n = 0, ±1, ±2, ..., the transmitted, reflected and
cav-ity fields are proportional to 1/(1−r1r2)2hence they are huge if the reflection
coefficients are close to 1. Therefore when 2kLL = 2πn the fields show a set
of resonances, where the beams in the cavity interfere constructively and the power of the field is very strong.
Figure 1.2: Phase φ of the reflected field as a function of 2kLL. φ is defined
as a continuous function rather than reporting only the interval [0, 2π] [1]. From Eq.1.40, the distance between the maxima is ∆ωL = πcL and the
full width of the peaks at half maximum is δωL = Lc1−r√r11rr22. The capability
of build intense fields inside FP cavity is expressed through the finesse F defined by F ≡ ∆ωL δωL = π √ r1r2 1 − r1r2 (1.41)
F is related to the sensitivity of the FP cavity to changes in the arm length L.
This is particularly interesting for the purpose of GW detection: writing the reflected field as Eref l = |Eref l|eiωLteiφ, from Eq. 1.38 the dependence of
φ at the resonant point 2kLL is computed (Fig.1.2) and shows that far from
the resonant point the phase is insensitive to the changes in L, while at the resonant point a small change in L has a strong impact on the field in the cavity, and this is exactly what is needed to search for gravitational waves.
Writing 2kLL = 2πn+, for small and r2 = 1, r1 ∼ 1 the phase variation
with respect to the length variation is: ∂φ
∂ h 2F
π (1.42)
So the phase of the reflected field, that then will recombine to the other beam and they will reach the photo-detector, is extremely sensitive to length variation in a FP cavity. Furthermore F is closely related to the average time spent by photons inside the cavity τs h LcFπ, which instead in a simple
Michelson interferometer is 2L/c. Hence the higher the finesse, the higher the storage time of light in the cavity and the more the cavity is sensitive to changes in L.
A simple estimation on the reflected field in a FP cavity, when there is a GW, can be done in the proper detector frame (so it is valid only if λL L).
Considering the FP cavity oriented along x axis and a GW with only plus polarization, propagating perpendicularly along z, the variation of the cavity length is:
∆Lx(t) =
Lh0
2 cos (ωLt) (1.43)
and from Eq.1.42 with = 2kL∆L the phase shift is:
∆φF P h 4F
π kL∆L (1.44)
Thus at equal ∆L the higher the finesse the higher the phase variation. In Advanced Virgo finesse is about 440 [2].
Considering also the sidebands induced by the GW at frequencies ωL±
ωGW the phase shift in a FP interferometer depends also on the GW frequency
and on the pole frequency defined as fp h 4F Lc through
∆φF P h 4F π kL∆L 1 p1 + (fGW/fp)2 (1.45) If fGW fp the result is equivalent to Eq.1.44, while for fGW fp the
1.2.2
Angular response
To evaluate the response of the interferometer to a GW coming from an arbitrary direction and with arbitrarily polarization, it is useful to define the detector pattern function FA(ˆn)
FA(ˆn) = DijeAij(ˆn) (1.46)
where Dij is the detector tensor which connects the scalar detector output
to the strain of the GW
h(t) = Dijhij(t) (1.47)
and eA
ij is the polarization tensor, that includes the information of the GW
polarization. A first frame (x, y, z) is defined so that the interferometer arms are along the x and y axis, and a second frame (x0, y0, z0) is fixed so that the propagation direction of the GW coincides with the z0 axis. The angles θ and φ related the two reference frames. The detector output h(t) is related to the pattern function as:
h(t) = h+(t)F+(θ, φ) + h×(t)F×(θ, φ) (1.48)
and F+(θ, φ) and F×(θ, φ) can be estimated in the proper detector frame.
Studying how the cross and plus polarizations change between the two frames the resulting pattern function is:
F+(θ, φ) =
1
2(1 + cos
2θ) cos (2θ) (1.49)
F×(θ, φ) = cos (θ) sin (2φ) (1.50)
GW interferometers have a large coverage that allows to detect signals from almost all possible directions, expect from some blind spots. The drawback is that the source localization cannot be done using just one interferometer, as in the case of telescope for example. This is one of the reasons why the con-struction of a network of GW interferometers have been undertaken. With two detectors three quantities are measured: two strains h1, h2, and their
delay times δt12, but there are four unknown quantities, the polarizations
h+(t), h×(t) and the angles θ, φ. Three interferometers instead provide three
strain amplitudes and two independent delay times that allow localizing the GW source [1].
1.2.3
Dark fringe detection
In Section 1.2.1 it was pointed out that a GW produces a phase shift ∆φGW.
To understand how to extract this phase shift from the detector, we can start writing the power at its output as P (φ) = P0sin2φ where φ = φ0+ ∆φGW(t).
At first it seems that the output is most sensitive to a phase shift when φ0 = π/4, anyway this is not the best working point because the output is
also sensitive to power fluctuations in the input beam, that are much larger than the signal expected for a GW.
To mitigate input laser power fluctuations, the working point has to be where P = 0. However there also ∂P/∂φ = 0 and, since ∆φGW = O(h)
(Eq.1.44), the power fluctuations are quadratic on the GW strain, ∆P = O(h2), and the effect of the GW is invisible. To overcome this problem the
idea is to apply a phase modulation to the input laser light. This can be done using a Pockel cell, i.e. a crystal whose index of refraction depends on the applied field, and it can be easily modulated. When passing throught a time-varying index of refraction the laser acquires a time-varying phase
Ein= E0e−i(ωLt+Γ sin (Ωmodt)) (1.51)
where Γ is the modulation index, and Ωmod the frequency modulation, that is
up to tens of MHz in Virgo. The effect of the phase modulation is to generate sidebands. Higher sidebands are suppressed by higher power of Γ, and only the first two can be considered, with frequencies:
ω± = ωL± Ωmod (1.52)
The total electric field at the output of the interferometer is then:
(Eout)tot = (Eout)carrier + (Eout)ω+ + (Eout)ω− (1.53)
It turns out that when Lx = Ly both the carrier and the sidebands are on the
dark fringe ((Eout)tot = 0), but if it is set |Lx− Ly| = nλL the carrier is still
on the dark fringe, while the sidebands no more. This asymmetry is called Schnupp asymmetry. When a GW arrives the power |(Eout)carrier|2 will be
quadratic in h while the one in the carrier will be proportional to 1 + O(h). When evaluating |(Eout)tot|2 there will be three terms: one ∼ O(1), one
∼ O(h2) and one will be proportional to O(h).
Thanks to this configuration then, the detector output is linear in the GW strain h and insensitive in the power fluctuations of the input laser beam, which is on the dark fringe [1].
1.2.4
Control and Locking
Fabry - Perot cavities are extremely sensitive to change in L only if they are in resonance, while away from this condition the phase of the light losses the information on the arm length variations. Therefore the mirrors have to be in the right position, i.e. kLL = πn for some n. In order to fulfill the
resonance condition the necessary precision on the position of the mirrors δL is:
δL = λL
4F (1.54)
With a finesse of about ∼ 400, δL ∼ 10−3λL. The exact value of L is not
important as long it is a multiple of λL. In the interferometer there are also
other FP cavities that have to be in resonance (see next Section). Further-more the interferometer has to work on the dark fringe with a precision on L of δL ∼ (10−4− 10−6)λ
L.
With λL∼ 1 µm the arm length has to be kept with precision within:
δL ∼ (10−12− 10−10) m (1.55)
To lock FP cavities in resonance a feedback control system is employed, composed generally of a sensor, that produces an error signal between the actual value of a certain quantity and the desired one, and an actuator, that manages to correct the error.
The error signal is obtained using a Pound-Drever-Hall locking scheme and a laser light modulation. Similarly to what saw in the previous section, the detector output is linear in ∆φ given both by noise source or GWs. This phase fluctuation can be interpreted as the error signal which measures how well the detector is at the dark fringe with FP cavities in resonance. Even if the interferometer is always in the dark fringe, the information of GWs is contained in the feedback that has to be applied to maintain the resonance condition.
The lock acquisition consists in bringing the detector from a free state where mirrors shake to a controlled state where all the FP are in resonance [1]. Once the interferometer is locked and is in a controlled status it is in ’science mode’ and collects useful good quality data. Fig.1.3 shows the percentage of time Virgo spent in different states during the last observation run (April 2019 - March 2020).
Figure 1.3: Duty cycle of Advanced Virgo during the last observation run (O3). The interferometer acquired good quality data most of the time (sci-ence mode). Some time was spent to control the various components of the detector (locking), to improve its performance (commissioning), calibrate the detector and for weekly maintenance periods [17].
1.2.5
Optical layout
Fig.1.4 shows a simplified optical layout of Advanced Virgo. The interferom-eter is quite complex and below only the main components are described.
• input mode cleaner
This system is a triangular cavity that acts on the laser beam before it arrives at the beam-splitter. Studying the FP cavity, plane mirrors with infinite transverse size were taken into account. Of course, this is not the real case and when considering finite size mirrors some consider-ations have to be done. Indeed, the beams will have a dependence also on the coordinates perpendicular to the propagation direction and will be subject to diffraction. With plane mirrors the widening of a beam due to diffraction becomes important after only one round trip, and after multiple bounces in the FP cavity, the input beam is dispersed. The propagation on a field with finite traverse extensions is made through the definition of a paraxial propagator and it turns out that a Gaussian beam, i.e. a beam with a Gaussian profile at x = 0 and an initial transverse size ω0 ∼ cm
E(x) = ξ(x; y, z)eikLx, with ξ(x = 0; y, z) = ξ
0e−(y
2+z2)/ω2
0 (1.56)
remains Gaussian at all x and has the minimum possible spreading, according to the Heisenberg principle.
Figure 1.4: Simplified optical layout of Advanced Virgo: the laser light is produced in the gray box on the left and enters the triangular input mode cleaner cavity. At the beam-splitter (BS) it is splitted into two compo-nents with equal amplitude which enter into the two Fabry - Perot cavities. Test masses are indicated with W (west)/N(north), I(input)/E(end). The two beams recombine at the beam-splitter and reach the photo-detector, at the suspended detection bench (SDB). Power and Signal Recycling Mirrors (PRM, SRM) allow the recycling of the injected power or the GW sidebands. [2].
To avoid that a beam widens while bouncing in the FP cavity, the mir-rors surfaces have to match exactly the surface of constant phase of the beam. As the wavefronts of a Gaussian beam are almost spherical, it is useful to use spherical mirrors, in order not to increment the transverse size of the beam at each round trip in the cavity. In this way when the beam is reflected back by a mirror it is focused back towards the waist and then it re-expands toward the other mirror. In Advanced Virgo the curvature radius of FP mirrors is about 1551 m, and the beam radii in the FP cavities between 48 − 58 mm [2].
However the Gaussian beams are just one possible solution of the parax-ial propagator, and there is a complete orthonormal set of solutions (called Hermite-Gauss modes). The intensity of the traverse size for different modes is shown in Fig.1.5. Higher modes are not in resonance in the FP cavity and produce noise. To eliminate them the laser beam is sent in a input mode cleaner system, before reaching the beam-splitter, that is a FP cavity where only the Gaussian mode is in resonance and it is efficiently transmitted.
Figure 1.5: Intensity distribution of Hermite-Gauss modes. The 00 mode on the left is the desired Gaussian mode, while the others are higher order modes, where the intensity distribution becomes wider [19].
• output mode cleaner
This system is placed between the beam-splitter and the photo-detector and it removes higher modes, as the input mode cleaner. Indeed, even
if the initial beam is prepared in the Gaussian mode, due to misalign-ments and various imperfections in the mirrors, higher modes could be regenerated producing noise in the dark fringe.
• power recycling mirror:
This mirror is placed between the input mode cleaner and the beam-splitter. Previously the working point of the interferometer was set to be the dark fringe for the carrier, i.e. no light in the carrier frequency ωL goes from the beam-splitter to the photo-detector. Therefore all
the light that is in the FP cavities goes back to the laser. In the next section it will be pointed out that a powerful laser increases the detector sensitivity, but with this configuration the light that goes back to the laser seems wasted. To increase the power circulating into the interferometer, a mirror can be placed between the laser and the beam-splitter to reflect back the light. The interferometer can be seen now as a new FP cavity, formed by this recycling mirror and the other component of the detector as an ’equivalent mirror’. If this cavity is in resonance for the input laser light, the total intensity of light in the interferometer is increased up to a factor O(100).
• signal recycling mirror
This mirror is located between the beam splitter and the photo-detector, forming the signal-recycling cavity. Varying the location and the reflec-tivity of the signal-recycling mirror, the resonance frequency and the bandwidth of the interferometer change. Near the optical resonance, hence to a certain frequency, detector sensitivity is improved. Instead, if the signal recycling cavity is not resonant with the carrier detector frequency, it is called de-tuned configuration and quantum correlations between the two arms lead to beating the standard quantum limit [34].
1.3
Data analysis techniques
This section presents the main techniques to extract GW signals embedded in a much larger detector noise. The interferometer output is a time series that describes the oscillations of the suspended mirrors as a phase shift (Eq.1.45) of the laser light. It is convenient to write the detector output s(t) as a linear combination of its noise n(t) and a GW signal h(t)
s(t) = n(t) + h(t) (1.57)
where h(t) is connected to the hij tensor through the detector tensor Dij
(Eq.1.47). In this section the noise is assumed stationary and Gaussian, but the next section will discuss the limit of these hypotheses. If n(t) is stationary, its different Fourier components are uncorrelated and their ensemble average is
h˜n∗(f )˜n(f0)i = δ(f − f0)1
2Sn(f ) (1.58)
This last equation defines Sn(f ), called the noise power spectral density
(PSD), that is a crucial quantity to describe the sensitivity of a GW in-terferometer. Indeed, without any loss of generality hn(t)i = 0 and
hn2(t)i = Z ∞ −∞ df df0h˜n∗(f )˜n(f0)i = Z ∞ 0 df Sn(f ) (1.59)
If the noise n(t) increases by a factor C then Sn(f ) increases by C2. Thus the
noise of a detector can be characterised by pSn(f ), the amplitude spectral
density (ASD). In section 1.4 the main interferometer noises will be presented and they will be described by this quantity.
In practise Sn(f ) is estimated using the Welch method: first the
time-series is divided into overlapping segments that are windowed to correct their non-periodicity so that data are forced to go to 0 at the edges of the interval. Then for each segment, the Fourier transform is performed and the square magnitude computed. The resulting ’periodograms’ are averaged to obtain the power spectral density.
Sn(f ) provides a complete characterization of the sensitivity of a detector
averaged on the time interval used (usually 512s), while the presence of short transients noise is ruled out [18].
In GW detectors usually |n(t)| |h(t)| but in the next section shows how the GW signals can be identified anyway [1].
1.3.1
Matched filtering
Even if GW signals are embedded in strong noise, they can be identified if their waveform is known. This procedure is called matched filtering technique and it is derived formally optimizing the signal-to-noise ratio (SN R). The idea is to define the quantity
ˆ s =
Z ∞
−∞
dts(t)K(t) (1.60)
being K(t) a filter function. The signal-to-noise ratio is given by S/N where S is the expected value of ˆs when a GW signal is present and N the root mean square value of ˆs when there is not any GW signal. Using hn(t)i = 0 the signal-to-noise ratio is given by:
S N = R∞ −∞df ˜h(f ) ˜K ∗(f ) R∞ −∞df 1 2Sn(f )| ˜K(f )|2 1/2 (1.61)
The latter can be seen as a scalar product whose maximum is for ˜
K(f ) = const ˜h(f ) Sn(f )
(1.62) Eq.1.62 defines the optimal matched filter or the Wiener filter: if the noise does not depend on the frequency the signal itself is the best filter, otherwise if certain frequency regions have higher noise, there we must weight less the signal. The Wiener filter is the optimal filter without any assumption on the form of the signal h(f ). Inserting the optimal filter in Eq.1.61 the highest value of the signal-to-noise ratio, with known h(t), is given by:
S N 2 = 4 Z ∞ 0 df|˜h(f )| 2 Sn(f ) (1.63)
1.3.2
Parameter estimation
The matched filter technique is powerful when the waveform of the GW signal is known. In practice h(t) depends on many parameters: for instance for the coalescence of compact binary system the waveform depends, among others, on the distance of the source, the star masses and their spins. Therefore there will not be only a possible waveform but rather a family of templates h(t, θ) where θ = θ1, .., θN are the possible parameters. The parameter space
has to be discretized and for each point of this space there is a corresponding template, with which perform the matched filtering.
When for some templates the signal-to-noise-ratio (Eq.1.63) is higher than a fixed threshold, there is a candidate event for a GW detection. Before being acclaimed as a GW observation, an event has to be validated checking the detector status and other issues, as described in Section 2.1.1. The most probable value for each parameter and its error is estimated from the posterior probability. In the Bayesian approach the posterior probability is given by 4
P (hypothesis|data) = P (data|hypothesis)P (hypothesis) (1.64) where the first term on the right is the likelihood, the usual frequentist prob-ability of the data given a hypothesis; and the second term is the prior which resumes the assumptions and knowledge that the experimenter has on the hypothesis before performing trials.
Assuming the noise to be Gaussian and stationary, the probability distri-bution of n(t) is trivial and writing n(t) = s(t) − h(t, θt) with θt the unknown
true value of the parameter, the posterior probability given the observation s is p(θt|s) = N p(0)(θt) exp (ht|s) − 1 2(ht|ht) (1.65) being p0(θt) is the prior distribution and N a normalization constant.
Hav-ing the posterior probability, there are different ways to interfere the most probable value, denoted by ˆθ(s), for the true parameter value and its error. One option is define the most probable value as the one that maximizes the distribution in Eq.1.65. If the prior function is flat, i.e. the experimenter has any information to prefer certain values for θt than others, maximize
the posterior distribution is equivalent to maximize the likelihood; and the most probable value estimated is the same that gives the highest SN R with matched filtering. The error on the parameters can be defined according to the width of distribution Eq.1.65 at the peak.
Instead, if there are important priors, the values that maximize the pos-terior can be different from the value with the highest SN R in the matched filtering. In this case the estimation of the most probable value could be more complicated and could depend on how many parameters are we looking
4Bayes’ theorem states that given a set S with subsets A, B, .., P (A|B) =
P (B|A)P (A)/P (B) but P (B) = P
iP (B|Ai)P (Ai) for any B and for Ai disjoints such
that ∪iAi= S. Therefore the denominator of Bayes’ theorem is just a normalization factor
for. Besides, another way to estimated the most probable value is to compute the average of the true value over the distribution in Eq.1.65,
ˆ θi(s) =
Z
dθθip(θ|s) (1.66)
The drawback of this latter method is that it involves a multi-dimensional integral over the parameter space and can be computationally expensive. In the limit of large SN R all these estimators are consistent and provide the same estimate on ˆθ [1].
1.3.3
Significance of an event
An event is generated when the signal-to-noise ratio in Eq.1.61 is above a certain threshold. This section explains how this threshold is chosen and which is the corresponding statistical significance. A practical example of evaluation of GW event significance is discussed in Section 2.1.3, for the case of the first GW detected GW150914.
First of all, the detector noise consists of two different contributions: one Gaussian, i.e. n(t) has a Gaussian distribution, and one non Gaussian. The latter is caused by short duration disturbances caused by the instrument itself or by its interaction with the environment as, for example, human ac-tivities near the detector site or earthquakes that shake the mirrors. Setting a sufficient large threshold on the SN R allows to eliminate the Gaussian noise; instead the treatment on non Gaussian noise is more complicate be-cause it consists of events with arbitrarily high SN R. The environmental conditions and the detector status are monitored, and noisy time interval are removed applying vetoes (Section 2.1.2). In addition, to exclude that a candidate event is not a non Gaussian noise, coincidence detection on two or more detectors is required. Next chapter will be devoted to these transients noise, while at the moment only Gaussian noise is considered.
To study the statistical fluctuations of SN R, we define ρ similarly to Eq.1.61 but with ˆs at the numerator rather than its expectation value:
ρ = sˆ
N (1.67)
Therefore S/N = hρi. Recalling the definition of ˆs, ˆ
s = Z ∞
−∞
[a] [b]
Figure 1.6: In [a] the probability distribution of P (R| ¯R) for different ¯R (20,30,50). In [b] P (R| ¯R) versus R for ¯R (solid line) and without any signal (dashed) [1].
when there is no GW signal, ρ is a random variable with probability distri-bution inherited by n(t), indeed a Gaussian distridistri-bution with zero mean:
p(ρ|h = 0)dp = √1 2πe
−ρ2/2
dρ (1.69)
Instead, when a GW with signal-to-noise ratio ¯ρ is present, Eq.1.67 given ρ = ¯ρ + ˆn/N with ˆn =R dtn(t)K(t) and its probability distribution is:
p(ρ| ¯ρ)dp = √1 2πe
−(ρ− ¯ρ)2/2
dρ (1.70)
It is useful to write this latter distribution in terms of the signal-to-noise ratio in energy R ≡ ρ2, indeed ρ is the signal-to-noise in amplitude and the energy of GWs is quadratic in the amplitude. The higher the signal-to-noise ratio of a GW, the larger the mean of the probability distribution of R, which could be discriminated from the noise n(t) (Fig.1.6). The comparison between the probability distribution of SN R when there is no signal and the probability distribution when there is a GW, suggests that a true signal can be discriminated from a fluctuation due to Gaussian noise choosing a threshold in R (Fig.1.6).
For any threshold, there will be false alarm probability, i.e. how likely is that a noise fluctuation gives an SN R higher than the threshold, given by
pF A = Z ∞ Rthreshold dRP (R| ¯R = 0) = 2 Z ∞ ρthreshold dρe−ρ2/2 (1.71)
and a the false dismissal probability, i.e. how likely it is to have a true GW signal under the threshold, is:
pF D =
Z Rthreshold
0
Of course one would like to have both false alarm rate and dismissal proba-bility the smallest possible. The threshold is fixed from the maximum value of false alarm probability acceptable which depends on the number of trials that have to be done with different templates. Usually the false alarm level is set in order to have an expected number of false alarms during a run of order of few [1].
1.3.4
Data analysis techniques for different GWs signals
This section illustrates some data analysis techniques suitable for the different GW sources presented in Section 1.1.3.
• Coalescence of compact binaries
Up to date only GW signals from the coalescence of compact binaries have been detected. This is mainly due to the fact that in the last stage of the inspiral a huge amount is released in GW radiation. In ad-dition the matched filtering is very effective as accurate waveforms have been computed and the parameters of the system are known. Given some parameters, the more cycles a GW spent in the interferometer bandwidth, the higher the SN R collected.
For a binary coalescence the waveform is determined by 15 parameters: the distance, the source position (two angles), the orientation of the inspiral orbit (two angles), the arrival time in the interferometer, the orbital phase at that moment, the two star masses and their spins (6 pa-rameters). However three parameters (the arrival time, the amplitude and the phase of the signal) are extrinsic, i.e. they can be eliminated from the parameter space. Indeed all possible shifts of the arrival time of signal in the interferometer bandwidth are obtained at once with only a Fourier transform: the arrival time is given by the time that maximize the signal-to-noise ratio.
A crucial indicator of the detector sensitivity and its status is the bi-nary neutron stars range (BNS range), the average distance at which the merger of a binary neutron star system gives a SN R = 8 with the current sensitivity. The distance is averaged over all possible sky localisation of the source and binary orientations. This quantity is computed from the expression of SN R which is inversely proportional to the distance of the source, thus [1, 33] :
dsight = 2 5 5 6 1/2 c π2/3 GMc c3 5/6 Z fmax 0 df f −7/3 Sn(f ) 1/2 1 S/N (1.73)
Fig.1.7 shows the BNS range during the last observational period (O3b, from November 2019 to March 2020) for the three operative detectors.
Figure 1.7: BNS range during the last observation run (O3b) for the three detectors. Drops in BNS range correspond to periods where the detector was not in the optimal configuration due to instrumental problems or adverse environmental conditions [33].
During the first observing run (O1), from September 2015 to January 2016, three signals from the mergers of two black holes were confidently detected by LIGOs detectors. After an upgrade period the second ob-serving run (O2) starts, from November 2016 to August 2017, and the last month Virgo joined the GW interferometer network. Dur-ing O2 seven signals from the coalescence of binary black holes sys-tems and one from binary neutron star system were detected. Searches were performed by three different algorithms: pyCBC [25] and GstLAL [26] which are based on the matched filtering and coherent WaveBurst (CWB) [27] which performs an unmodelled search for short duration transients. These detections open new fields of research: they allow, for instance, to test general relativity, they provide independent estima-tion of the Hubble constant, placed constraints on the rate of compact binary system mergers, evaluating their main features as masses and spins.
Fig.1.8 shows the waveform of the detected signals from binary black holes systems, while Fig.1.9 is referred to GW170817, originated by the neutron stars binary system. The main difference between these two
systems is the chirp mass: black hole systems have a larger chirp mass, consequently the total radiated power is higher (Eq.1.22) and their waveforms evolution is faster (Eq.1.24). In addition, binary neutron star system are expected to have an electromagnetic counterpart which was extensively detected for GW170817: the counterpart covered all the electromagnetic spectrum. A short gamma ray burst, its afterglow and a kilonova emission, signature of heavy element nucleosynthesis, were detected [11, 22].
Figure 1.8: Time-frequency maps and reconstructed waveforms of the signal for the binary black holes systems detected during O1 and O2. For each event the data are from the detector where the highest SNR were observed [11]. Time-frequency maps will be explained in Section 2.2
Figure 1.9: Time-frequency map of GW170817. The chirp signal is clearly visible in the LIGO detectors, while it had a low SN R in Virgo, which was anyway crucial to constrain the localization map. In LIGO Livingston a short transient noise occurred in coincidence with the GW and it is discussed in Section 2.1.4 [22].
• Rotating bodies
Rotating bodies emit GWs at frequency twice the body rotation fre-quency (Eq.1.26 - 1.27). These periodic sources emit almost monochro-matic signal and consequently the minimum strain detectable decreases with a longer observing time. Complications arise from the fact that the signal is modulated by the motion of the Earth with respect to the source and by the intrinsic changes in frequency of the pulsar. The rel-ative motion due to the Earth’s rotation changes the apparent source position and yields an amplitude modulation of the signal as the de-tector sensitivity depends on the source position through the antenna pattern. In addition the Earth’s rotation and revolution around the Solar system barycenter produce a time-varying Doppler shift in the frequency. The signal-to-noise ratio increase with the observational time as S/N ∼√T [1].
Up to date there is no evidence for GW emission from any pulsars, and this constraints the fraction of energy that these objects lost though GW radiation (less than 0.17 % for Crab pulsar) [28].
• Burst
A supernova explosion or the final merger of compact binary systems can liberates a great amount of GW energy in a short time and so they are referred as GW bursts. Their detection is particularly difficult mainly for two reasons: first as they arise from explosive and complex phenomena it is very difficult to predict accurately their waveform and so the matched filtering technique cannot be used. Second GW bursts are hardly distinguishable from short transient noises, non Gaussian noise, that affect the interferometer. To overcome the first problem a great effort is done to provide accurate waveform templates with numerical simulations. Otherwise GW bursts are detected as excess of power in time-frequency analysis. This analysis will be deeply discussed in the next Chapter.
Regarding the second problem, to avoid confusing a GW burst for a short disturbance, coincidence in two or more detectors is required. The idea is that most of the noise in two different detectors is uncorre-lated and if a GW burst present the same energy, waveform and arrival time (considering spatially separation and relative orientations of the detectors) on different detectors, likely it has an astrophysical origin. Accidental time coincidences are evaluated through the time shifting technique (see in Section 2.1.1) [1].
The expected event rate for GW observation from core-collapse super-novae for the third-generation detectors is still one in twenty years [31]. During the second observing run (O2) Advanced LIGO and Advanced Virgo did not detect any GW burst [29].
• Stochastic background
For a stochastic background the detector output averaged over time is null hh(t)i = 0. For the detection, the quadratic order hh2(t)i could be considered but, it turns out that in this way the detector sensitivity is well below to allow the detection of the stochastic background. The op-timal strategy instead, the one that maximizes the signal-to-noise ratio similar to the matched filtering technique, is to perform a correlation between to or more detector output. Denoting with 1, 2 two different detectors, the output correlation is
hs1(t)s2(t)i = hh1(t)h2(t)i + hh1(t)n2(t)i + hh2(t)n1(t)i + hn1(t)n2(t)i
noise in one interferometer is independent from the signal in the other so hh1(t)n2(t)i = hh1(t)ihn2(t)i and if the noise is stationary hn(t)i = 0.
Furthermore if the detector are far from each other their noises are uncorrelated so hn1(t)n2(t)i = hn1(t)ihn2(t)i = 0. Hence the output
correlation provides a term quadratic in the signal and it is the optimal detection statistic [13].
From the data of the first observing run (O1) and the second (O2) there is no evidence for a stochastic gravitational wave background, and an upper limit on ΩGW is placed [30].
1.4
Advanced Virgo noise sources
In order to be able to detect a GW arriving at the Earth (typically
h0 ∼ 10−21) a ground-based interferometer with L = 3 km has to be sensitive
to a displacement ∆L ∼ 2 × 10−18 m or a phase shift of ∆φF P ∼ 10−8
rad from Eq.1.44. This section overviews the main noise sources and the strategies developed to mitigate them.
The strain sensitivity is expressed in term of Sn1/2(f ) (Eq.1.58) with
di-mension Hz−1/2. The principal sources of noise are quantum noise, related to quantum nature of light, seismic noise, caused by the continuous motion of the ground, and thermal noise. In Fig.1.10 the modelled noise budget for Advanced Virgo is presented, while Fig.1.11 is a plot of the noise budget measured with the Welch method (Section 1.3) during the last observing run.
Figure 1.10: Reference Advanced Virgo sensitivity (black line) and expected noise contributions computed using models for quantum noise, suspension thermal noise and some modeled technical noises (colored lines) [32].
Figure 1.11: Advanced Virgo sensitivity on March 1, 2020. The best sensi-tivity of that day (blue line) is compared with the reference sensisensi-tivity for O3a run (pink line) and the O3 target scenario (green) [92].
1.4.1
Quantum noise
Quantum noise affects the interferometer through two mechanisms: shot noise and radiation pressure.
The first arises from the fluctuations in the number of photons that arrive at the photo-detector and concerns high frequencies of the detector band-width. The average power at the photo-detector during an observation time T is
P = 1
TNγ~ωL (1.75)
where Nγ is the number of photons that arrive at the photo-detector during
T . For large Nγ the fluctuations of the number of photons follow a Gaussian
distribution so ∆Nγ = pNγ. This yields to fluctuations in the observed
power as (∆P )shot= 1 T∆Nγ~ωL= r ~ωL T P (1.76)
Instead, the fluctuations due to a GW signal, that induces a phase shift given by Eq.1.45, are
(∆P )GW =
P0
2 | sin 2φ0|(∆φ)F P (1.77)
Hence the signal-to-noise ratio can be written as NS = (∆P )GW
(∆P )shot and for a
the strain sensitivity through: S N = T Sn(f ) 1/2 (1.78) Substituting Eq.1.76 - 1.77 the strain sensitivity due to shot noise is com-puted. Below the complete expression for Sn1/2(f ) is reported considering
shot noise in an interferometer with FP cavities: Sn1/2(f ) shot = 1 8F L 4π~λLc ηCP 1/2q 1 + (f /fp)2 (1.79)
fp is the cutoff of the FP cavity, C is the power gained with the power
re-cycling mirror and η is the efficiency of the photo-detector. Eq.1.79 suggests that in order to beat shot noise the laser power of the recycling factor should be increased.
The second noise due to quantum nature of light is radiation pressure, caused by the pressure exerted by the photons on the mirrors. If it were a constant force a simple spring mechanism could hold the mirrors, but the number of photons hitting the mirrors fluctuates, generating a stochastic force that shakes the mirrors. The root meas square fluctuations of this force ∆F is related to the beam power through Eq.1.76:
∆F = 2∆P
c = 2
r ~ωLP
c2T (1.80)
This stochastic force acts on the mirrors that are freely moving in the hori-zontal plane, so F = M ¨x and in the Fourier space ˜F (f ) = −M (2πf )2x. The˜ complete expression of the strain sensitivity due to radiation pressure is:
Sn1/2(f ) rad = 16 √ 2F M L(2πf2) r ~CP 2πλLc 1 p1 + (f/fp)2 (1.81) where M is the mirror mass. Radiation pressure noise affects low frequencies and is inversely proportional to the the mass of the mirrors that for this reason in Advanced Virgo have been increased to about 42 kg [2].
It is crucial to note that the shot noise power spectrum is inversely propor-tional to the laser power, while the radiation pressure is directly proporpropor-tional, hence it seems there is no way to decrease one noise source without increasing the other. This is called the standard quantum limit as the existence of a limiting value of the strain sensitivity is a manifestation of the Heisenberg principle [1].
