Download Thermal lensing in Virgo and Polynomial Search: an all-sky

Thermal lensing in Virgo and Polynomial
Search: an all-sky search for gravitational
waves from spinning neutron stars in
binary systems
Sipho van der Putten
Front and back cover designed by Chiara Farinelli and Hegoi Garitaonandia. Back cover
based on ‘Fiets 1’ (1988) by Mark Raven all rights reserved.
This work is part of the research program of the ‘Stichting voor Fundamenteel Onderzoek
der Materie (FOM)’, which is financially supported by the ‘Nederlandse organisatie voor
Wetenschappelijke Onderzoek (NWO)’.
VRIJE UNIVERSITEIT
Thermal lensing in Virgo and
Polynomial Search: an all-sky search
for gravitational waves from spinning
neutron stars in binary systems
ACADEMISCH PROEFSCHRIFT
ter verkrijging van de graad Doctor aan
de Vrije Universiteit Amsterdam,
op gezag van de rector magnificus
prof.dr. L.M. Bouter,
in het openbaar te verdedigen
ten overstaan van de promotiecommissie
van de faculteit der Exacte Wetenschappen
op dinsdag 6 december 2011 om 9.45 uur
in het auditorium van de universiteit,
De Boelelaan 1105
door
Sipho van der Putten
geboren te Amsterdam
promotoren:
prof.dr. J.F.J. van den Brand
prof.dr. F.L. Linde
Contents
Introduction
1
1 Theory
1.1 General relativity and gravitational waves . . . . . . . . .
1.2 Sources of gravitational waves . . . . . . . . . . . . . . . .
1.2.1 Compact binary coalescence . . . . . . . . . . . . .
1.2.2 Bursts . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Stochastic background . . . . . . . . . . . . . . . .
1.2.4 Continuous waves . . . . . . . . . . . . . . . . . . .
1.3 Neutron stars and pulsars . . . . . . . . . . . . . . . . . .
1.3.1 Continuous gravitational waves from neutron stars .
2 Gravitational wave detection
2.1 Laser-interferometric detectors
2.2 Virgo . . . . . . . . . . . . . .
2.2.1 Optical design . . . . .
2.2.2 Suspensions . . . . . .
2.2.3 Noise sources . . . . .
2.3 Calibration of h(t) in Virgo .
2.4 Detector beam-pattern . . . .
2.5 Status of Virgo . . . . . . . .
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3 Thermal Lensing
3.1 The finite element model . . . . . . . . . .
3.2 Simulation results . . . . . . . . . . . . . .
3.2.1 Steady state thermal analysis . . .
3.2.2 Eigenmodes in thermal equilibrium
3.2.3 Transient temperature analysis . .
3.3 Optical path length . . . . . . . . . . . . .
3.4 Thermal lensing effects in Virgo . . . . . .
3.5 Conclusion and outlook . . . . . . . . . . .
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48
Contents
4 Continuous waves analysis methods
51
4.1 The gravitational wave signal model . . . . . . . . . . . . . . . . . . . . . 52
4.2 Coherent searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3 Semi-coherent searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5 Polynomial Search
5.1 Doppler shifts of binary systems . . . . . . . . . . . .
5.2 Binary system test cases . . . . . . . . . . . . . . . .
5.3 Constructing the filters and the correlations . . . . .
5.4 Coherence length . . . . . . . . . . . . . . . . . . . .
5.5 Step size in parameter space . . . . . . . . . . . . . .
5.6 Constructing the hit maps . . . . . . . . . . . . . . .
5.6.1 False-alarm probability . . . . . . . . . . . . .
5.6.2 False-dismissal probability . . . . . . . . . . .
5.7 Combination of hits . . . . . . . . . . . . . . . . . . .
5.7.1 Number count statistics . . . . . . . . . . . .
5.7.2 Renormalized number count . . . . . . . . . .
5.7.3 The sensitivity . . . . . . . . . . . . . . . . .
5.7.4 Consistent filters . . . . . . . . . . . . . . . .
5.7.5 The sensitivity with consistent filters . . . . .
5.8 Results of the simulations . . . . . . . . . . . . . . .
5.9 Discussion and outlook . . . . . . . . . . . . . . . . .
5.9.1 Follow-up and parameter estimation strategies
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6 Software and computing
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6.1 The AnaPulsar framework . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.2 Computing costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7 Summary
127
A Fourier transform definitions
131
B The correlation statistic
135
C Stepsize plots
139
D AnaPulsar inheritance diagrams
145
References
160
Samenvatting
161
Acknowledgements
165
ii
Introduction
The publication of Einstein’s first paper on general relativity in 1916 [1] paved the
way for the prediction of the existence of gravitational waves. It was first thought that
gravitational waves were nothing more than mathematical artifacts originating from the
linearization of the Einstein field equations. In 1957 Bondi [2] (and independently Weber
and Wheeler [3]) published papers in which gravitational waves were shown to be actual
physical phenomena which could be measured. The prediction from Einstein’s theory of
general relativity is that any mass distribution with a time-dependent quadrupole (or
higher multipole) moment will emit gravitational waves.
Gravitational waves are perturbations in spacetime which travel at the speed of light.
These perturbations locally disturb the spacetime metric such that measured distances
between freely falling test-masses will vary as a function of time. General relativity states
that spacetime is an elastic but stiff medium meaning that in order to produce measurable differences from flat space in the metric, the associated quadrupole moment must
be large. Furthermore, due to this stiffness of spacetime, gravitational wave signals will
not interact strongly with matter and thus the Universe will be largely transparent to
gravitational waves. This means that gravitational waves will yield a unique insight into
various astrophysical processes. Since the amplitude of a gravitational wave is related to
the magnitude of the variations in the quadrupole moment, compact objects like binary
neutron stars and black holes are the most promising sources.
In 1975 Hulse and Taylor discovered a neutron star binary system PSR 1913+16 [4].
Over the course of decades they measured the time evolution of the orbit of this system
which revealed that the system decayed (e.g. the system loses energy). Furthermore,
this decay has been shown to be consistent with the loss of energy due to the emission
of gravitational waves. This result experimentally proved for the first time since the
formulation of general relativity that gravitational radiation exists and carries energy
from a system. For this discovery Hulse and Taylor received the Nobel prize in physics
in 1993.
While Hulse and Taylor did prove the existence of gravitational radiation, their determination of the energy loss of the system constitutes an indirect measurement of
gravitational waves. Direct detection of the amplitude, phase and polarization of gravitational waves will yield much more information on the physical processes of the source.
In order to directly detect gravitational waves, the relative distance between two freely
1
Introduction
falling test masses must be measured as accurately as possible. This can be done by
using laser interferometric gravitational wave detectors like Virgo.
The Virgo detector is a Michelson interferometer with 3 km long Fabry-Perot resonant cavities as arms and is located in Italy near the city of Pisa. The detector is capable
of measuring the relative distance of two test masses up to about 6 × 10−23 at approximately 200 Hz in 1 second of observation time. With Virgo (and other interferometric
gravitational detectors like LIGO, LCGT and GEO600) it will be possible to directly
measure gravitational wave signals.
These signals, when detected, could be used amongst other things to test general relativity in the strong field regime. They can also be used to gain (additional) insight into
various astrophysical processes like supernovae or the physics of compact objects like
neutron stars and black holes. Gravitational waves can even be used to measure various
cosmological parameters and to study the primordial gravitational wave background.
Since the Virgo detector uses resonant cavities as the detection medium, the laser
power that builds up will heat up the optics. One of these thermal effects is called
‘Thermal Lensing’ and a simulation study of the thermal lensing effect in the Virgo
input mirrors is presented in this thesis. Furthermore, a new analysis method for continuous waves from rapidly rotating neutron stars in binary systems, called ‘Polynomial
Search’, is presented together with the developed analysis framework.
Outline
Chapter 1 gives a short introduction into general relativity and shows the linearization
of the Einstein field equations giving rise to gravitational waves. Furthermore, the different sources of gravitational waves are discussed with emphasis on the mechanisms of
emission of continuous waves from rotating neutron stars or pulsars.
The principles of interferometric gravitational wave detection are discussed in chapter 2. Also, in this chapter the main components of the Virgo detector are presented
as well as the current and future status of Virgo together with the status of the LIGO
detectors.
Chapter 3 presents the simulation of the heating effects of the Virgo input mirrors
of the main arm cavities. This is done by constructing a 3D model of the Virgo input
mirrors together with their recoil masses. The heating effects are computed by solving
the heat equation in this model by finite element analysis. These effects are then related
to the various time constants for heating up the mirror as well as to the change in optical
path length and the change in eigenmode frequencies as a result of this heating.
Chapter 4 gives an overview of the current developed analysis algorithms designed
for searching for continuous waves in interferometric data. Chapter 5 presents the Polynomial Search method and the tests performed on simulated data together with the
sensitivity estimate of the search in comparison to other existing methods.
Finally, chapter 6 gives an overview of the developed framework in which Polynomial
Search is implemented and the computing requirements for performing such a search on
the data obtained with LIGO and Virgo.
2
Chapter
1
Theory
1.1
General relativity and gravitational waves
Gravity is one of the four fundamental forces of Nature. In 1916 Albert Einstein published a theory which describes the gravitational interactions as curvature of the spacetime continuum. This theory is known as General Relativity (GR) [1] and relates the
energy-momentum tensor, Tµν , to the metric tensor, gµν , in a series of coupled non-linear
differential equations called the Einstein field equations.
The theory of GR predicts the existence of gravitational waves (GWs) which are
strains in spacetime propagating at the speed of light. Since these GWs are small perturbations on the background curvature, they are difficult to observe directly. Indirect
observations have been performed by following the evolution of the orbit of binary pulsar system PSR 1913+16 [4]. R.A. Hulse and J.H. Taylor received the Nobel prize in
physics in 1993 for this discovery. Recently [5], it has been shown that the measured
rate of change of the orbital period agrees with that expected from the emission of gravitational radiation, according to general relativity, to within about 0.2 percent.
Equation (1.1) shows the Einstein field equations
8πG
1
Tµν = Rµν − gµν R,
(1.1)
4
c
2
where G is the gravitational constant, c the speed of light, Tµν the energy-momentum
tensor, Rµν the Ricci tensor and R the Ricci scalar. The Ricci tensor is related to the
Riemann tensor by Rµν ≡ Rλµλν and the Ricci scalar is defined as R ≡ g µν Rµν . The
Riemann tensor can be written in terms of the connection Γρνσ , as
Rρσµν = ∂µ Γρνσ − ∂ν Γρµσ + Γρµλ Γλνσ − Γρνλ Γλµσ .
The connection can be written as a series of derivatives of the metric as
1
Γσµν = g σρ (∂µ gνρ + ∂ν gρµ − ∂ρ gµν ).
2
(1.2)
(1.3)
Note that the metric tensor, gµν , is related to the proper distance ds2 as
ds2 = gµν dxµ dxν .
(1.4)
3
Chapter 1. Theory
GR relates the energy-momentum tensor to the metric through Eq. (1.1), (1.2) and
(1.3). Equation (1.1) can be solved analytically in certain circumstances, for instance in
the weak-field limit. For weak gravitational fields the metric can be approximated by a
time-dependent perturbation of the Minkowski metric,
gµν ≈ ηµν + hµν ,
where ηµν is the Minkowski metric written as

−1 0
0 1
ηµν = 
0 0
0 0
|hµν | 1,
0
0
1
0

0
0
.
0
1
(1.5)
(1.6)
Combining Eqs. (1.1), (1.2) and (1.3) with the weak field approximation (1.5) together
with the requirement |hµν | 1 and thus neglecting terms with O(h2 ) and higher, the
Einstein equations can be linearized. When also defining the trace-reversed perturbation
h̄µν by
1
h̄µν = hµν − ηµν hλλ
2
and with the Lorentz gauge condition
∂µ h̄µλ = 0,
the Einstein equations become
h̄µν =
−16πG
Tµν .
c4
(1.7)
Note that is the D’Alambertian which, in Cartesian coordinates and flat space is
denoted by
1 ∂2
∂2
∂2
∂2
= − 2 2 + 2 + 2 + 2 .
c ∂t
∂x
∂y
∂z
Finally, taking Tµν = 0 (far away from the source), Eq. (1.7) becomes
h̄µν = 0,
(1.8)
which, when realizing that the perturbation is small and thus the space can be considered
flat, can be recognized as the wave equation and h̄µν represents the GW. Substituting
the plane wave solution
σ
h̄µν = Cµν eikσ x ,
(1.9)
where Cµν a 4 × 4 amplitude matrix and kσ is the wave vector, yields the condition
kσ k σ = 0 meaning that GWs travel at the speed of light. Choosing the transversetraceless gauge implies h̄TµνT = hTµνT . Furthermore, when also orienting the propagation
4
1.2. Sources of gravitational waves
of the wave along the z-axis, the GW strain can be written as


0 0
0
0
0 h+ h× 0 iω(t−z)

,
h(t, z) = 
0 h× −h+ 0 e
0 0
0
0
(1.10)
where the h+ and h× are the ‘plus’ and ‘cross’ polarizations of the GW. The effect of
the two different polarizations on a ring of test masses is shown in Fig 1.1.
Figure. 1.1: The effect of a GW with only the plus polarization (top) and the cross
polarization (bottom). The direction of the GW is perpendicular to the
paper. The phase of the wave is shown over one full period of time T.
Contrary to electromagnetic radiation, which can be produced by accelerating charge
dipoles, GWs can only be produced by a varying mass-quadrupole moment. This is due
to the fact that conservation of mass/energy prevents a time-dependent monopole mass
moment and conservation of momentum forbids a time-dependent dipole mass moment
from emitting GWs. Thus the first multipole moment that allows for the emission of
GWs is the quadrupole moment.
1.2
Sources of gravitational waves
There are roughly four categories of promising astrophysical sources of gravitational
waves. Binary systems of compact objects (either neutron stars or black holes), which
due to energy loss coalesce, fall in the ‘Compact Binaries Coalescence’ (CBC) category.
These CBCs have a transient but well modeled waveform which can be divided into three
5
Chapter 1. Theory
parts called the Inspiral, Merger and Ringdown phases. Unmodeled transient waveforms
fall in the ‘Burst’ category. Residual GW background from the Big Bang and/or a
large number of weak astrophysical sources fall in the ‘Stochastic Background’ category.
Finally long-lived continuous GWs originating from spinning neutron stars are called
‘Continuous Waves’ (CW).
1.2.1
Compact binary coalescence
Compact binary coalescence is the most well-known and promising potential source for
GW detection. This is due to the fact that the strains involved are relatively high. The
optimal horizon distance for neutron star neutron star (NS-NS) inspiral, neutron star
black hole (NS-BH) and BH-BH have been computed in Ref. [6] to be 33/445 Mpc,
70/927 Mpc and 161/2187 Mpc for Initial and Advanced detectors, respectively. As an
example, the predicted rate, computed by extrapolating from observed binary pulsars,
for a NS-NS inspiral event is 100 Myr−1 per Milky Way Equivalent Galaxy (MWEG).
However, this rate can plausibly range from 1 to 1000 Myr−1 MWEG−1 . These numbers
have been converted to detection rates ranging from 2×10−4 to 0.2 yr−1 for Initial LIGO
and Virgo and from 0.4 to 400 yr−1 for the Advanced detectors.
A CBC consists of three distinct phases of GW emission. First, due to loss of angular momentum the compact objects spiral towards each other in the inspiral phase.
Next, the two objects merge in the merger phase. Finally, the distorted black hole relaxes to a stationary Kerr state through emission of GWs in the ringdown phase. All
of these phases of the waveform are shown schematically in figure 1.2. The GW signal
Figure. 1.2: Cartoon indicating the three phases of an inspiral together with a
schematic of the inspiral waveform (courtesy of [7]).
produced by a CBC will increase in frequency and amplitude during the inspiral phase.
This means that the signal will enter the detector band from about 10 Hz only in the
6
1.2. Sources of gravitational waves
last few minutes or seconds (depending on the masses involved) before the merger.
The data analysis algorithms to find CBC signals in the data are based on ‘matched
filtering’. Matched filtering is done by computing analytical or phenomenological waveforms as signal templates and comparing them to the data stream. The output of each
filter-data comparison is assigned a signal-to-noise ratio (SNR). Events whose SNR
crosses a threshold are subjected to a coincidence analysis between the available detectors. The consistency of the resulting coincidence is verified by using a χ2 -test defined
in Ref. [8].
1.2.2
Bursts
A second category of potential GW sources consists of unmodeled transient phenomena,
or ‘bursts’. A wide variety of astrophysical sources is expected to produce such signals at
some level. Sophisticated models have been made in some cases, while in other cases the
amplitude and phase of the GW signals is highly uncertain. The only thing the potential
sources have in common is the fact that the duration of the signals is less than several
seconds.
Possible sources targeted by the burst search include, but are not limited to, supernovae core (SN) collapses [9] and ringdowns of black-hole mergers [10] whose inspiral
signal does not lie within the detectable region for ground-based interferometers. Furthermore, astrophysical data on bursts like Gamma-Ray Bursts (GRBs) are used as
triggers for a targeted burst search which can be run with increased sensitivity.
The rate of SN core collapse is predicted to be 0.5 yr−1 for a horizon at 5 Mpc [9] but
the actual range of burst searches from SN depends on the amplitude and phase evolution of the GWs. This strongly depends on the preferred model and thus no accurate
prediction of the event rate of these types of bursts can be made. The detection rates
of intermediate mass (20M ≤ M ≤ 106 M ) binary black hole (IMBBH) inspirals are
about 10 events per year for the Advanced detectors [10]. The rate of detection for GRBs
depends on observation distance and the field of view of the satellite used to generate
triggers as well as the amount of energy converted to GW emission. GRBs come in two
broad classes: short (commonly associated with NS-NS or NS-BH mergers) and long
(commonly associated with core collapse supernovae). Furthermore, a local population
of under-luminous short GRBs [11] seems to exist which has an observed rate density
approximately 103 times that of the high-luminosity population. When assuming that
the energy emitted in GWs equals 0.01M c2 , a field of view of 9.5 sr of the satellite
(Fermi), and Advanced Virgo and LIGO sensitivity, the expected rates are 5.38 × 10−3
yr−1 , 5.38 yr−1 and 0.08 yr−1 for the long, local and short type of GRB [12], respectively.
Since the burst search is meant to find unmodeled burst-like signals, glitches in the
detector which mimic burst signals will produce background. The search techniques for
burst searches always include correlation between detectors to reduce this background.
There are numerous techniques developed to perform these burst searches like templatebased searches [13], cross-correlation [14] and coherent methods [15].
7
Chapter 1. Theory
1.2.3
Stochastic background
A stochastic background of GWs is expected to arise as a superposition of a large number
of unresolved sources, from different directions in the sky, and with different polarizations. It is usually described in terms of the GW spectrum ΩGW (f ) ∝ f α [16] where α
is different for different emission mechanisms.
Many possible sources of stochastic GW background have been proposed. Some of
the proposed theoretical models are cosmological in nature, for example pre-big-bang
models [17]. Others are astrophysical in nature, such as rotating neutron stars, supernovae and low-mass X-ray binaries.
The analysis method used is the cross-correlation method. This method uses the output of multiple interferometers and uses the cross-correlation to filter out any common
signals from the background.
1.2.4
Continuous waves
Continuous GWs can be emitted by rapidly rotating neutron stars. Some neutron stars
are observed as pulsars and have rotational frequencies ranging from a few millihertz
up to a kilohertz. Furthermore, in order to emit GWs the neutron star must have a
quadrupole moment. This implies that there must be a non-axisymmetric asymmetry
present. This asymmetry can either be due to a geometrical asymmetry in the crust, due
to fluidic asymmetry or induced by accretion of matter onto the neutron star. These GW
emission mechanisms are discussed in some more detail in section 1.3.1. Furthermore,
the various existing data analysis methods will be summarized in chapter 4.
1.3
Neutron stars and pulsars
A neutron star is a type of stellar remnant that can result from the gravitational collapse
of a massive star during a supernova. As the core of a massive star is compressed during
a supernova and collapses into a neutron star, it retains most of its angular momentum.
The neutron star has only a fraction of its parent’s radius (around 10 km) and is formed
with high rotation speed. It gradually slows down. Neutron stars are known to have
rotation periods between about 1.4 ms to 30 seconds. Observationally, some neutron
stars are known to have very strong magnetic fields, which as a function of their age
and composition can range from 108 Gauss (milli-second pulsars) to 1015 Gauss (young
pulsars/magnetars).
The magnetic field together with the rotation of the neutron star results in the acceleration of protons and electrons on the star surface and the creation of a beam of
radiation emanating from the poles of the magnetic field. If the neutron star is positioned
in such a way that this beam sweeps the Earth, it is observed as a pulsating source of
electromagnetic radiation. Neutron stars with such properties are commonly referred to
as pulsars.
Pulsars are only observed when their radiation beam is swept into the direction of
Earth during a revolution of the neutron star. Pulsar emission is usually too weak to
8
1.3. Neutron stars and pulsars
be detected extragalactically. Due to these effects, only a fraction of the pulsars in our
Galaxy is known and more of these pulsars are close by than far away. Taking these
observational biases into account over 105 active pulsars are expected to exist in our
Galaxy [18]. However, since active pulsars last only a few tens of millions of years and
not all neutron stars pulse, the number of neutron stars in the Galaxy is expected to be
orders of magnitude larger.
Pulsars have been known to spin up or undergo sudden increases of their rotation
speed, called ‘glitches’, which range in fractional size from 10−11 to 10−4 of the spin
frequency. Most models for pulsar glitches build on the vortex unpinning model [19].
Superfluid vortices pin to defects in the crystalline crust and are prevented from migrating outward as the crust spins down electromagnetically. At some stage, many vortices
unpin catastrophically, transferring angular momentum to the crust. It is however unknown what causes this collective unpinning, this process is also thought to generate
GWs [20].
Pulsars emit a lot of energy, i.e. the Crab pulsar emits about 5 × 1038 ergs/s [21]
in wide-band electromagnetic emission. Since most of this energy is drawn from the
rotational energy of the pulsar, the rotational frequency should slowly decrease. This
decrease in rotation is characterized by the so-called spindown age τs ,
τs =
f
,
|f˙|
(1.11)
where f is the rotation frequency and f˙ is the first order time derivative of the rotation
frequency. Incidentally, when assuming1 that f˙ = −Kf n , where K is a constant and the
‘braking index’ n > 1, the spindown age is related to the age of the pulsar by
t=
τs
.
n−1
(1.12)
Experimentally, the breaking index typically has values n ' 1 − 3 [22] depending on the
specific pulsar (i.e. Crab: n ' 2.5). From the ATNF pulsar catalog [21], the left panel
in figure 1.3 shows f˙ versus f . It can be seen from this panel that all known pulsars
have f˙ < 10−9 Hz/s. Furthermore, the right panel in figure 1.3 shows the frequency
distribution of the pulsars with rotational frequency above 20 Hz. The cut off of 20 Hz
has been chosen since it roughly corresponds to the beginning of the sensitive region
of ground-based interferometers like LIGO and Virgo. The figure also shows that most
known millisecond pulsars are located in binary systems.
1.3.1
Continuous gravitational waves from neutron stars
In the frequency band relevant to ground-based interferometers there are three predicted
mechanisms for producing continuous GWs. All of these mechanisms involve neutron
stars.
1
This assumption is reasonable since it can be computed that for electromagnetic dipole radiation
˙
fEM ∝ f 3 and for GW emission f˙GW ∝ f 5 .
9
- df (Hz/s)
dt
10-8
10-9
10-10
10-11
10-12
10-13
10-14
10-15
10-16
10-17
10-18
# pulsars
Chapter 1. Theory
Isolated pulsars
Pulsars in binary systems
16
All pulsars
14
Pulsars in binary systems
12
10
8
6
4
2
10-1
1
10
102
103
f (Hz)
0
0
100 200 300 400 500 600 700
f (Hz)
Figure. 1.3: The left panel shows the first order spin-down versus the frequency of all
known pulsars. The right panel shows the frequency distribution of known
pulsars with rotational frequency over 20 Hz for isolated pulsars (total
37) and pulsars in binary systems (total 75) The data are taken from the
ATNF pulsar catalog (January 2011).
ˆ Non-axisymmetric distortions of the solid part of the star [23].
ˆ Unstable r-modes in the fluid part of the star [24].
ˆ Free precession of the whole star [25].
Furthermore, neutron stars in binary systems can emit GWs due to accretion [26].
Non-axisymmetric distortions
Non-axisymmetric distortions cannot exist in a perfect fluid star, but in realistic neutron
stars such distortions can be supported either by elastic stresses or by magnetic fields.
These non-axisymmetric distortions are akin to mountains on the Earth’s surface and
are expressed in terms of the ellipticity
=
Ixx − Iyy
,
Izz
(1.13)
where Ixx , Iyy and Izz are the moments of inertia along the x, y and z axes, respectively.
A non-axisymmetric neutron star rotating along the z-axis, at distance d with frequency frot emits GWs with amplitude ([27])
h0 =
4π 2 G If 2
,
c4 d
(1.14)
where G is Newton’s constant, c the speed of light, I the moment of inertia along the axis
of rotation and f is the GW frequency (in the case of an optimally orientated source,
f = 2frot ).
The maximum ellipticity that can be sustained by a neutron star depends on the
10
1.3. Neutron stars and pulsars
model of the neutron star used. This makes the estimation of the maximum ellipticity
highly uncertain and predictions range from 10−4 to 10−6 [23]. For a typical neutron
star the moment of inertia is in the order of I = 1038 kg m2 . Equation (1.14) can then
be parameterized as
−27
h0 = 1.068 × 10
1038
I
kg m2
10 kpc
r
f
100 Hz
2 .
10−6
(1.15)
Note that the GW amplitude scales with f 2 . It has been normalized to 100 Hz, while
the distance has been normalized to 10 kpc corresponding to a NS in our Galaxy. When
considering pulsars in a binary system frot can be as high as 700 Hz (see Fig. 1.3).
Furthermore, the nearest known pulsar in a binary system (J1045-4509) is located at
0.3 kpc. Taking these values, the GW amplitude can be as high as h0 ≈ 10−24 .
Unstable r-modes
An alternative way of generating GWs from neutron stars is through the so-called r-mode
oscillations. An r-mode is a particular torsional mode which is due to material motion
tangentially to the neutron star’s surface. Essentially, they can be seen as horizontal
currents associated with small density variations. The r-modes exist only in rotating
neutron stars and are caused by the Coriolis force generated by the rotation. Gravitational radiation couples to these modes primarily through the current multipoles, rather
than the usual mass multipoles.
It has been shown by Chandrasekhar [28] (and later generalized by Friedman and
Schutz [29]) that for these r-modes the gravitational radiation reaction amplifies the
oscillation of the rotating star as opposed to damping it. This effect can be understood
qualitatively by considering that the r-mode period is larger than the rotational period
of the neutron star (this is an intrinsic property of the r-mode oscillation). The sign of
the angular momentum carried away by the gravitational waves is the same as that of
the angular momentum of the mode pattern as measured in the inertial frame. Here, the
gravitational wave will carry positive angular momentum away from the system and to
infinity. As a result the star will spin down.
In the co-rotating frame (i.e. rotating with the neutron star) the neutron star is
at rest and the r-mode will rotate in the opposite direction. In this frame, the angular
momentum will be negative and gravitational waves will still be emitted. These gravitational waves will carry angular momentum away from the system, making the total
angular momentum even more negative for the observer in the co-rotating frame. This
implies that what started as perturbations with small negative angular momentum in
the co-rotating frame, have been transformed (in the same frame) into large amplitude
oscillations with a progressively larger negative angular momentum that will emit increasingly large amounts of gravitational waves. Therefore, these modes are unstable
and this instability is known at the Chandrasekhar-Friedman-Schutz (CFS) instability.
Ultimately, the internal viscosity of the star will stop the instability from growing.
The r-modes have been proposed as a source of GWs from newborn and/or accreting
11
Chapter 1. Theory
neutron stars [24]. Unlike the GWs originating from a non-axisymmetric mass distribution, the GWs from r-mode instabilities will be emitted at
4
f = frot .
3
(1.16)
The production mechanism of GWs from r-modes depends on the equation of state and
on the amplitude of the oscillations. The CFS instability of the r-modes in newborn
neutron stars is not a good candidate for the detection of continuous GWs since the
emission is short lived or has low amplitude. On the other hand, accreting neutron stars
are considered a better prospect for a detection of r-mode gravitational radiation since
the emission might be long-lived due to the transfer of angular momentum [30].
Free precession
When a rotating neutron star has a symmetry axis which does not coincide with its
rotation axis, the star will precess. Since this precession has a quadrupole moment, it will
cause the neutron star to emit GWs. This GW emission would occur at f = frot + fprec
where fprec frot is the precession frequency.
The GW amplitude has been parameterized in Refs. [23, 25] as
−30
h0 ≈ 4 × 10
θw
0.1 rad
10 kpc
r
frot
100 Hz
2
,
(1.17)
where θw is the angle between the axis of symmetry and the axis of deformation. It is
clear from comparing Eq. (1.17) with Eq. (1.15) that GWs from free precessing neutron
stars will even be weaker than those from non-axisymmetric neutron stars.
Accretion
When a neutron star is located in a binary system, the situation may arise that matter
is transfered from the companion star to the neutron star. These companions can range
from supergiants to white dwarves. Accretion is the process via which old pulsars are
‘recycled’ and spun-up to become milli-second pulsars. Due to the high gravitational
pull of the neutron star, accretion leads to the formation of an accretion disk around
the neutron star. Since the matter from the companion will be accelerated in this disk,
it will emit X-rays.
In the context of GW generation, accretion is a natural way of generating and maintaining crustal deformations. Accretion is not an isotropic process thus there will be
many ‘hot spots’ on the neutron star surface at which the accreted material accumulates. These hot spots can lead to the formation of crustal asymmetries, which would
lead to the emission of GWs which will carry away angular momentum. This can lead
to a situation in which the accretion spin-up is balanced by the spin-down, leading to
the continuous emission of GWs as suggested in Ref. [26].
Rapidly accreting stars are of particular interest for this sort of emission mechanism, for example in low-mass X-ray binaries (LMXBs). Under the assumption that the
12
1.3. Neutron stars and pulsars
GW emission is in equilibrium with the accretion torque, the GW amplitude is directly
related to the X-ray luminosity. The GW amplitude is parameterized in Ref. [23] as
−27
h0 ≈ 5 × 10
300 Hz
frot
1/2 10−8
Fx
erg cm−2 s−1
1/2
,
(1.18)
where Fx is the X-ray flux. The spin frequencies, frot , of LMXBs can go up to about 620
Hz.
Furthermore, according to Ref. [31], the magnetic field of a neutron star can help to
localize the accreting matter to a small area of the surface. The magnetic field lines can be
compressed into a narrow belt at the magnetic equator, which then confines the accreted
material to the poles of the neutron star. Since the magnetic axis is not generally aligned
to the angular momentum axis, this accumulation of material takes place asymmetrically,
thus creating a mass quadrupole and consequently GWs are emitted. The dissipation
of these mountains is slow because the matter is highly conductive and thus crosses the
field lines slowly. This leads to a scenario in which the pile up is matched by steady GW
emission. Reference [32] identifies the most promising targets, and lists specific actions
that would lead to significant improvements in detection probability.
13
Chapter 1. Theory
14
Chapter
2
Gravitational wave detection
As discussed in section 1.1, gravitational waves are perturbations of the spacetime curvature. These perturbations induce a strain in spacetime resulting in a time-dependent
variation of the physical distance between test masses. This strain h is in essence a
relative distance variation defined as
∆L
h≡2
,
(2.1)
L
where L and ∆L are the distance and change in distance between two test masses respectively. The detection of gravitational waves is based on measuring distances between test
masses. Note that the gravitational wave amplitude is typically a small number (10−27 )
for GWs from neutron stars (see section 1.2.4).
2.1
Laser-interferometric detectors
A laser interferometer can be used to measure the distance between test masses to great
accuracy. The simplest example of a laser interferometer is a Michelson interferometer,
shown schematically in Fig. 2.1. The figure shows that in a Michelson interferometer laser
light is split into two separate beams by using a 50/50% beam splitter. The mirrors act
as test masses and reflect the light back. The beams form an interference pattern of
which a single fringe is transmitted towards the output photodiode.
The essential idea is to maintain the interferometer in a locked position, with the
difference of its arm length fixed by measuring the light at the output and feeding
the signal back to the mirrors. The arm lengths are kept fixed in such a way that the
light interferes destructively. The correction, or error-signal, would contain the potential
gravitational wave signal since it measures the deviation from the working point or dark
fringe.
When taking the expression for the proper distance (Eq. (1.4)) and substituting the
metric tensor by the perturbed metric as shown in Eq. (1.5) together with the assumption
that the gravitational wave is +-polarized and propagating in the z-direction, the proper
distance of a photon traveling along the x-axis is given by
ds2 = −c2 dt2 + (1 + h+ (t))dx2 = 0,
(2.2)
15
Chapter 2. Gravitational wave detection
Figure. 2.1: A schematic representation of a Michelson interferometer (left) and the
effect of a passing gravitational wave (right). In the right panel, the wave
is assumed to be of plus polarization and is propagating perpendicular to
the interferometer, into the page.
where c is the speed of light. Since h is considered small, the approximation |h+ (t)| 1
can be made and will yield
1
(1 − h+ (t))c dt = dx.
(2.3)
2
Integrating Eq. (2.3) over the time it takes the light to complete a single round rip, will
yield the phase change along the x and y direction given by
Z
2Lx,y Ωl Ωl t
φx,y = Ωl ∆tx,y =
±
h+ (t0 )dt0 ,
(2.4)
c
2 t− 2Lcx,y
where Ωl = 2πfl is the angular frequency of the laser light. The phase difference δφGW =
φy − φx caused by a gravitational wave can be computed from Eq. (2.4) to be
Z t
δφGW = Ωl
h+ (t0 )dt0 .
(2.5)
t−
2Lx,y
c
Assuming that the arm lengths are equal (L = Lx = Ly ) and changes in the incident gravitational wave are much slower than the storage time in the arms, the phase
difference becomes
4πL
δφGW =
h+ ,
(2.6)
λl
where λl = 2πc/Ωl . Equation (2.6) shows that the phase difference is proportional to
the length of the arms of the Michelson interferometer. Note that Eq. (2.6) is only valid
when the wavelength of the gravitational wave is large compared to the arm length.
16
2.1. Laser-interferometric detectors
The sensitivity of a Michelson interferometer can be computed by substituting the
Fourier transform of the gravitational wave,
Z ∞
h+ (t) =
h+ (ω)e−iωt dω,
−∞
into Eq. (2.5) which yields
Z
δφGW =
∞
−∞
HMI (ω)h+ (ω)eiωt dω,
(2.7)
where ω is the angular frequency of the GW and the sensitivity of the Michelson interferometer as a function of ω, HMI (ω), is written as
2Ωl
Lω −i Lω
HMI (ω) =
sin
e c .
(2.8)
ω
c
From Eq. (2.8) it can be seen that the gravitational wave response of the Michelson
interferometer is maximized when L satisfies
Lω
π
= ,
c
2
for a given gravitational wave angular frequency ωGW . This would imply that the optimal arm length for a 1 kHz gravitational wave signal is about 75 km. Since this is
an unrealistic arm length, other ways of improving the optical design to increase the
effective arm length have been developed.
One way of accomplishing this, is the so-called delay-line Michelson interferometer.
Figure 2.2 shows a delay-line Michelson interferometer in a schematic way. In this de-
Figure. 2.2: Schematic of a delay-line Michelson interferometer. In this case the beam
is reflected 4 times (NDL = 4).
17
Chapter 2. Gravitational wave detection
sign another mirror is placed in between the beam splitter and the end mirror of each
arm. These mirrors are designed in such a way that the laser beam is reflected multiple
times before it is transmitted onto the beam splitter. The sensitivity of this delay-line
interferometer is given by
2Ωl
NDL Lω −i NDL Lω
2c
HDLMI (ω) =
sin
e
,
(2.9)
ω
2c
where NDL is the number of bounces between the two mirrors (a simple Michelson interferometer will have NDL = 2). As Eq. (2.9) shows, the sensitivity of the interferometer
increases with the number of bounces. However, it is non-trivial to produce mirrors
which allow for multiple bounces.
A more practical way of increasing the arm length is by using Fabry-Perot resonant
cavities as arms for a Michelson interferometer. A Fabry-Perot cavity is an arrangement
of mirrors that forms a standing wave cavity resonator for laser light. The sharpness of
the resonance is determined by the reflection coefficients of both the input (rI ) and end
mirrors (rE ) of the cavity. The finesse, F, is defined as the ratio of the cavity resonance
width (FWHM) to the free spectral range ∆f = c/2L, the interval between resonances.
For a general cavity, the finesse is given by
√
π rI rE
F=
.
(2.10)
1 − rI rE
In the limit that the time scale for the metric change is long compared to the storage
time of the light in the cavity, the light stored in a Fabry-Perot cavity of finesse F will
have
2F
Nφ =
(2.11)
π
times the phase shift in response to a GW of a one-bounce interferometer arm. Figure
2.3 shows the phase of the reflected beam and the fraction of the transmitted power of
a Fabry-Perot cavity. From Fig. 2.3 it can be seen that the phase of the reflected beam
is a steep function of ∆φ for large values of the finesse. Thus a small deviation from
∆φ = 0 will induce a large phase shift in the reflected light. Furthermore, due to the
sharpness of the Fabry-Perot resonance, small deviations from ∆φ = 0 will cause a large
decrease in transmitted power (or an increase in reflected power).
Two Fabry-Perot cavities can be combined in a Michelson interferometer as shown
schematically in Fig. 2.4 This setup will produce an interference pattern which is directly related to the reflected phase and power of the light from both of the Fabry-Perot
cavities.
The frequency response of a Fabry-Perot-Michelson interferometer HFPMI (ω) is written as
sin Lω
2acav Ωl
−i Lω
c
c ,
(2.12)
HFPMI (ω) =
Lω e
−2i
ω 1 − rI rE e
c
where
acav =
18
t2I rE
.
1 − rI rE
(2.13)
3
Finesse 10
Finesse 50
Finesse 100
Finesse 500
2
1
Ptrans
Phase (rad)
2.1. Laser-interferometric detectors
1
Finesse 10
Finesse 50
Finesse 100
Finesse 500
0.8
0.6
0
0.4
-1
0.2
-2
-3
-0.1
-0.05
0
0.05
0.1
∆φ (rad)
0
-0.4
-0.2
0
0.2
0.4
∆φ (rad)
Figure. 2.3: The left panel shows the phase of the reflected beam as a function of the
accumulated phase difference ∆φ during the storage time of the light. The
right panel shows the fraction of transmitted power as a function of ∆φ.
Both panels show curves for different values of the finesse.
Figure. 2.4: Schematic of a Michelson interferometer with Fabry-Perot cavities.
Figure 2.5 shows the frequency response of a delay-line Michelson interferometer, a
Fabry-Perot Michelson interferometer and a simple Michelson interferometer. As can be
seen in Fig. 2.5, the response of the Fabry-Perot interferometer shows a more smooth
response compared to that of the delay-line interferometer. Also it can be seen that
the sensitivity of the Fabry-Perot interferometer is lower than that of the delay-line interferometer in the high frequency range fGW > 100. However, the smoothness of the
frequency response as well as the fact that delay-line interferometers are more susceptible to scattered light noise makes the Fabry-Perot interferometer the most adopted
configuration for gravitational wave detectors.
Currently, there are several ground-based interferometers operational in the world:
LIGO [33], GEO600 [34] and Virgo [35]. LIGO consists of two separate interferometers
located in Livingston (Louisiana) and Hanford (Washington) in the USA. GEO600 is
19
|H(ω)|2
Chapter 2. Gravitational wave detection
HDLMI
HFPMI (finesse 50)
HFPMI (finesse 500)
HMI
104
103
102
10
1
10-1
10-2
10-3
10-4
1
10
102
103
104
105
fGW (Hz)
Figure. 2.5: The magnitude of the frequency response for the Michelson interferometer (MI), delay-line Michelson Interferometer (DLMI) and Fabry-Perot
interferometer (FPMI) as a function of the gravitational wave frequency
fGW . All frequency responses are shown relative to the Michelson interferometer response. For all curves, the base line arm length is 3 km, the laser
wavelength has been chosen to be 1 µm the finesse of the cavities is set to
50 and 500 and the number of bounces for the delay-line interferometer is
computed with Eq. (2.11).
an interferometer with 600 m long arms located near Hannover in Germany. The Virgo
project consists of a single interferometer which is located in Italy near the city of Pisa.
2.2
Virgo
Virgo [35] is a Michelson interferometer with Fabry-Perot cavities as arms each 3 km
long, located in Italy near the city of Pisa. Its objective is to directly detect gravitational
waves in coincidence with other interferometers like LIGO [33]. The most important
components of Virgo are the optics and the suspension system.
2.2.1
Optical design
The optical layout of the Virgo interferometer is shown schematically in Fig. 2.6. The
interferometer consists of a laser system, an input mode cleaner, a power recycling
mirror, two Fabry-Perot cavities and a detection system.
20
2.2. Virgo
Figure. 2.6: A diagram of the Initial Virgo design optical layout.
The laser system
The laser is a Nd:Yag system emitting at λ = 1.064 µm. This wavelength has been
chosen because lasers emitting in the infrared region are more powerful, reducing the
shot noise. Furthermore, optical components for this wavelength are readily available.
The output power in the TEM00 mode is about 20 W. The laser system is placed on an
optical table within a clean room outside of the vacuum.
The input mode cleaner
Any positional noise (e.g. coupling of seismic ‘beam jitter’ noise to the input bench) of the
laser system induces additional higher order modes to the fundamental TEM00 mode.
The input mode cleaner is a triangular cavity, 144 m long with a finesse of about 1000
which is tuned to resonate on the fundamental TEM00 mode and thus filters out these
higher order modes. It consists of two plane mirrors placed on an optical bench known as
the ‘injection bench’ and a third concave mirror suspended by a superattenuator located
in the mode-cleaner tower. The end-mirror is connected through a dedicated vacuum
pipe to the injection bench.
21
Chapter 2. Gravitational wave detection
Power recycling
Shot noise is one of the fundamental noises which limit the sensitivity of an interferometer. Shot noise is a photon counting error, which is proportional to the square-root of the
laser power incident in the interferometer. This noise is dominant at high frequencies
(>500 Hz). Since the gravitational wave signal is proportional to the incident power,
the shot-noise level of an interferometer is improved by increasing the laser power. In
order to increase the laser power in the Fabry-Perot cavities, a technique called power
recycling has been used. This involves the placement of an extra power recycling mirror
before the beam splitter. This mirror forms a resonant cavity together with the interferometer. The recycling factor, defined as the ratio between the power stored in the cavity
and the input power, is designed to be 50. With this design value a power of about 1
kW is expected on the beam splitter.
Fabry-Perot cavities
Each arm is a Fabry-Perot cavity consisting of an input mirror and an end mirror. The
cavities are 3 km long and in Initial Virgo designed with a finesse of 50. The input
mirrors and end mirrors are 350 mm in diameter. The input mirrors are flat and have
an approximate reflectivity of 98% while the output mirrors are 100% reflective (within
a few p.p.m.). Furthermore, in order to compensate for the beam divergence over the 3
km length, both end mirrors are curved with an approximate radius of curvature of 3.5
km. Both input and output mirrors (as well as the beam splitter, power recycling mirror
and the mode cleaner end mirrors) are suspended by a series of coupled pendula called
‘superattenuators’.
Detection system
The detection system is composed of the suspended detection bench placed in vacuum
and the external detection bench which is not suspended and is located outside the
vacuum. The suspended detection bench mainly holds the output telescope and the
output mode cleaner (OMC), a monolithic optical cavity 2.5 cm long of finesse F = 50,
which filters higher order modes of the beam. The main beam leaves the suspended
bench to reach the external bench, where a set of InGaAs photodiodes are located in
order to detect the signal. The output of these diodes are used for the locking of the
interferometer as well as for the signal detection.
2.2.2
Suspensions
The detection of gravitational waves is limited at low frequencies, where the seismic
noise of the ground is transmitted through the suspensions √
to the mirrors. This seismic
noise would induce position disturbances of about 10−9 m/ Hz at 10 Hz. Specific seismic isolation systems, consisting of coupled pendula called ‘superattenuators’ [36], have
been developed for Virgo. The purpose of these superattenuators is to suspend the test
masses (i.e. the mirrors) in such a way that they are virtually freely falling (e.g. the
22
2.2. Virgo
amount of external force on the test masses is minimized).
Figure
2.7
shows
a
schematic
of
a
superattenuator.
It can be seen from this figure
that a superattenuator consists
of five filter platforms. The filters are connected by suspension wires and a support marionetta. The mirror and recoil
mass are suspended from the
last filter (the so-called ‘filter 7’).
Each filter consists of a series of
blade springs which isolate the
test mass in the vertical direction. The entire chain of filters
is suspended from the ground by
a so-called ‘inverted pendulum’
which introduces yet another step
in the seismic isolation.
The superattenuator is capable of inhibiting the transmission
of seismic vibrations to the mirror above a few Hz in all 6 degrees of freedom. The superattenuators are designed to reduce the
residual motions of the mirrors.
In order to reach √
a target sensi−18
tivity of 10
m/ Hz at 10 Hz,
a local control system for the test
masses has been installed in the
superattenuators. Six full superattenuators (tall towers) are suspending the mirrors of the FabryPerot cavities, the beam splitter
and the power recycling mirror.
Because of the lesser seismic isolation requirement, three shorter
version (short towers) are utilized
to suspend the suspended injec- Figure. 2.7: Left: A schematic view of the Virgo superattenuator taken from Ref. [36]
tion bench, the far mirror of the
mode cleaner and the bench of
the detection system. See Ref. [36] for a more detailed description and performance
of the different seismic isolation systems in use in Virgo.
23
Chapter 2. Gravitational wave detection
2.2.3
Noise sources
Sn (Hz-1/2)
Several sources of noise can introduce spurious signals capable of masking the weak
effects due to an incident gravitational wave. The detector sensitivity is limited by the
intensities of these noise sources. The total expected noise is the sum of the power of
all these noise sources and results in the sensitivity curve √
of an interferometer. This
sensitivity curve is given as the amplitude spectral density, Sn , (see appendix A and
Eq. (A.15)), with Sn the sum of the powers of all these noise sources.
Figure 2.8 shows the design sensitivity of the Virgo detector broken down into the
three most prominent noise sources. Reference [37] gives a more detailed description of
all known noise sources. The main noise sources shown in Fig. 2.8 are seismic noise,
10-16
Total noise
Seismic noise
Thermal noise
Shot noise
10-17
10-18
10-19
10-20
10-21
10-22
10-23
10-24
1
10
102
103
104
f (Hz)
Figure. 2.8: Breakdown of the design sensitivity curve of the Virgo detector in the
most relevant noise components.
thermal noise and shot noise.
Seismic noise
The seismic noise curve is computed by taking the transfer function of the superattenuator and assuming a seismic noise spectral amplitude x̃seis = 10−7 /f −2 for 1 Hz < f < 103
Hz. Figure 2.8 shows that seismic noise is expected to be dominant up to about 3 Hz.
Seismic noise is the limiting factor for the sensitivity in the low frequency regime. Any
improvements to the superattenuators (e.g. lowering the resonant frequencies) will not
yield a significant improvement in the sensitivity. This is due to the so-called Newtonian
noise or gravity gradient noise [38]. This noise is an irreducible background noise caused
by stochastic gravitational field modulations due to seismic noise.
24
2.3. Calibration of h(t) in Virgo
Thermal noise
Thermal noise is due to energy loss processes of the suspension wires and test masses.
The thermal noise curve shown in Fig. 2.8 is the sum of the noise spectra of all these
processes. As can be seen, thermal noise dominates from 3 to about 500 Hz. Up to about
30 Hz the noise is due to thermal fluctuations of the pendula formed by the mirror and
its suspension wires. Above 30 Hz the tail contribution is due to the high frequency
internal modes of the mirrors. The spike at approximately 7 Hz is due to the thermal
noise associated to the main vertical mode of the marionetta-mirror system. The various
spikes in the spectrum are due to contributions to the thermal noise of the violin modes
of the suspension wires. Note that the thermal modes of the cavity mirrors (around 5.5
kHz) play a major role in chapter 3.
Shot noise
Since the arrival times of the photons at the photodiodes are random and can be considered Poisson distributed, the detected power will fluctuate. This noise decreases with
the square root of the number Nγ of photons and thus with the power contained in the
interferometer. An accurate measurement of a gravitational wave requires an accurate
measurement of the phase of the output laser light. For high frequency gravitational
waves, a more accurate measurement of the phase is needed compared to lower frequency gravitational waves. Since the responses of the MI, DLMI and FPMI (Eqs. (2.8),
(2.9) and (2.12)) interferometers are inversely proportional to the gravitational wave
frequency, the accuracy of the phase measurement degrades accordingly. In this region,
the shot noise will dominate due to the intrinsic photon-counting error, known as shot
noise.
2.3
Calibration of h(t) in Virgo
In order to produce a measurement of the strain amplitude of a gravitational wave with
the Virgo detector, a calibration of the output signal is necessary. This calibration is
needed to measure the sensitivity of the interferometer and to reconstruct the strain
amplitude, h(t), from the data stream. The calibration method is described in Ref.
[39] and the implementation in Virgo is described in Ref. [40]. In order to obtain the
sensitivity curve and the reconstructed strain amplitude, the change in arm length must
be related to the variation of the output power of the interferometer. This is done by
first calibrating the length measurement and then measuring the interferometer response
function.
To achieve optimal sensitivity the positions of the mirrors are actively controlled
in such a way that the Fabry-Perot cavities remain on resonance. Furthermore, this
control is designed to keep the output locked on the dark fringe. The mirrors can only
be controlled up to frequencies of a few hundred Hertz. This implies that the active
control of the mirrors will also attenuate the effect of a passing gravitational wave with
frequencies below these few hundreds Hertz. Above a few hundreds Hertz, the mirrors
25
Chapter 2. Gravitational wave detection
behave as free falling masses in the longitudinal direction.
The mirror active control is calibrated by actuating the mirrors and monitoring
the effect in the detector output. The actuation will induce a time dependent length
change of the interferometer arms. This length change is related to the measured phase
change and is given by Eq. (2.5). When purposefully misaligning the input mirrors the
interferometer becomes a simple Michelson interferometer. The measured phase change
is then related to the length change by
δφGW =
4π
∆L(t),
λl
(2.14)
where λl is the laser wavelength and ∆L(t) is the induced differential arm length change
of the interferometer. Since the power on the output photodiode is a function of the
phase difference, Eq. (2.14) can be used to calibrate the length measurement ∆L.
In order to measure the sensitivity of Virgo, the interferometer response function
must be measured. This response function is in essence the transfer function from the
differential arm length variation, ∆L(t), to the output power on the photo diode. This
response function has dimension m/W and is measured by injecting white noise into
the end mirror actuators. Due to the calibration of the actuators described above, the
relation between actuation and length change is known. However, since the longitudinal
controls do not act above a few hundred Hertz, the response function can only be
measured up to these few hundred Hertz. In order to measure the response function
for higher frequencies, a model is used to extrapolate the effect to higher frequencies.
This model is based on an ideal Fabry-Perot interferometer with a finesse of 50. The
Virgo sensitivity curve is measured by taking the spectrum of the output signal (in
W) and multiplying it with the response function. This yields the sensitivity in meters
which will become the sensitivity in strain amplitude when dividing by the Virgo arm
length of 3 km. Note that the sensitivity is measured a few minutes after the response
function to avoid variations of the optical gain from the noise injection. During the first
Virgo science run (VSR1) the errors on the sensitivity below 1 kHz were 4% due to the
mirror actuation calibration. The errors above 1 kHz were higher (5 to 10%) due to the
inaccuracy of the response function model at these frequencies (see Ref. [40]).
For the data analysis the output data stream of the interferometer must be converted
to a gravitational wave strain amplitude stream, h(t). Equation (2.1) states that the
strain is the relative distance variation between two test masses, in this case the endmirrors. Therefore, in order to reconstruct the strain amplitude from the detector output,
the effective differential arm length variation (∆L(t)) is required. As stated before, ∆L(t)
can be computed. However, first the effect of the active controls on the mirrors must be
corrected for. This is done by subtracting the contribution of the injected control signals
from all mirrors from the output signal. This control free signal is then multiplied by
the response function to obtain ∆L(t). Finally this is converted to h(t) by dividing by
the 3 km Virgo arm length. The systematic errors on the h(t) data stream is found to
be in the order of 4% in amplitude and 50 mrad in phase during VSR1 (see Ref. [40]).
26
2.4. Detector beam-pattern
2.4
Detector beam-pattern
Since the detector is not equally sensitive in all directions and gravitational waves have
two polarizations (‘plus’ and ‘cross’), any gravitational wave arriving at the detector can
be written as
h(t) = F+ (t)h+ (t) + F× (t)h× (t),
(2.15)
where h(t) is the detector output, h+ is the plus polarization component of the gravitational wave, h× is the cross polarization component, and F+ and F× are the so-called
beam-pattern functions. These beam-pattern functions represent the detector’s acceptance for any point in the sky. Because of the motion of Earth the beam-pattern functions are periodic functions of time with a period of one sidereal day. The detector
beam-pattern functions are computed in Ref. [41] and can be written as
F+ (t) = sin ζ [a(t, α, δ) cos 2ψ + b(t, α, δ) sin 2ψ] and
F× (t) = sin ζ [b(t, α, δ) cos 2ψ − a(t, α, δ) sin 2ψ] ,
(2.16)
(2.17)
where ζ is the angle between the arms in the interferometer (90◦ in Virgo and LIGO)
and ψ is the polarization angle of the gravitational wave source. The functions a(t, α, δ)
and b(t, α, δ) both depend on the time and location of the source (right ascension α and
declination δ) as well as on the orbital parameters of Earth and the detector coordinates.
Figure 2.9 shows the location parameters of the source as well as the polarization
angle. The coordinate system is such that the gravitational wave propagates in the z
y
Z0
x
ψ
Source
z
δ
Y0
α
X0
Equatorial
referential
Figure. 2.9: The coordinate system (x, y, z) of the source relative to the equatorial coordinate system (X0 , Y0 , Z0 ). Here, α and δ are the equatorial coordinates
of the source and ψ is the polarization angle.
direction. The x and y axes are oriented along the principal axes of deformation of the
+ polarization. The polarization angle, ψ, is the angle between the x axis and the (α,Z0 )
plane and the plane perpendicular to the source.
The acceptance of an interferometer as a function of sky position, time and polarization is often shown as the so-called antenna pattern function. The polarization-averaged
27
Chapter 2. Gravitational wave detection
antenna pattern, < F >ψ , is defined as
1
< F >ψ ≡
2π
Z
2π
0
q
F+2 + F×2 dψ.
(2.18)
δ (rad)
Note that Eq. (2.18) will yield a number between 0 and 1 as a function of sky location.
When combining Eqs. (2.16) and (2.17) into Eq. (2.18) together with the detector coordinates of Virgo, the antenna pattern of Virgo can be computed. Figure 2.10 shows
this antenna pattern as a function of right ascension (α) and declination (δ). Note that
Fig. 2.10 also depends on the time due to Earth’s motion in the solar system. It can be
3
0.9
2
0.8
0.7
1
0.6
0
0.5
0.4
-1
0.3
0.2
-2
0.1
-3
-3
-2
-1
0
1
2
3
α (rad)
Figure. 2.10: The polarization-averaged antenna pattern for the Virgo detector.
seen clearly that there are parts of the sky where the detector is blind and this must be
taken into account when performing any kind of data analysis.
In order to give an estimate of the loss of signal-to-noise ratio for a signal at a random
position in the sky, Eq. (2.18) can be averaged over sky-location as well as over time.
This computation for Virgo yields an average value of 0.42 with a variation of 0.16 over
one sidereal day as computed in Ref. [42].
2.5
Status of Virgo
The Initial Virgo and LIGO detectors have concluded their first science runs. For Virgo
this was called Virgo Science Run 1 (VSR1) which ran from May 18, 2007 to October
1, 2007. After this run the Virgo detector was shut down for an upgrade, designated
Virgo+, and returned to operation in 2009. The second science run (VSR2) ran from July
7, 2009 to January 8, 2010. There was a short science run (VSR3) from August 11, 2010
to October 19, 2010. After the end of VSR3 various small commissioning runs as well
28
2.5. Status of Virgo
Figure. 2.11: The sensitivity curves for Virgo and LIGO during VSR2/S6.
as a minor science run together with GEO600 have been planned. Finally, in September
2011 Virgo+ is planned to be taken off line for a major upgrade called Advanced Virgo.
Advanced Virgo will have a sensitivity increase of an order of magnitude across the
entire frequency interval. In 2014 Advanced Virgo is planned to have first lock meaning
that after a certain commissioning period the science data taking of Advanced Virgo
will start in 2015.
In 2007 both Virgo and LIGO collaborations signed a memorandum of understanding
which effectively merged both collaborations. The goal of this merger was to perform
coincident analyses, increasing the confidence of a possible detection of gravitational
waves. Therefore, each Virgo science run has a LIGO counterpart. For VSR1 it was
LIGO science run 5 (S5) and for VSR2 it was S6.
Figure 2.11 shows the sensitivity of LIGO and Virgo during VSR2/S6 as a function
of frequency. As can be seen in this figure, the Virgo detector is superior to the LIGO
detectors for frequencies below 50 Hz due to the superattenuators. For the intermediate
frequencies, 50 < f < 500 Hz, LIGO has a better sensitivity than Virgo while at high
frequencies the Virgo and LIGO detectors are of comparable sensitivity.
29
Chapter 2. Gravitational wave detection
30
Chapter
3
Thermal Lensing
Stable operation of the Virgo interferometer is limited by wavefront deformations caused
by thermal effects. Even with the use of highest possible quality optics available, the
absorption of laser power within transmissive and reflective optical components together
with the large optical power stored in the various parts of the interferometer will result
in thermal instabilities of the system. The thermal transients complicate the capability
to lock the interferometer.
When injecting the full power of the laser into the interferometer, the stored power
causes the mirrors to heat up in a non-uniform way. Such local heating induces a variation in the optical path length inside the mirror which results in the distortion of the
wavefront. This thermal effect, called ‘thermal lensing’, sets a limit on the laser power
which can be injected into the interferometer. Thermal lensing is an important limiting
effect which will become even more important since Advanced Virgo requires an increase
of both the injected power and the finesse of the Fabry-Perot arm cavities.
The amount of allowable power stored in the interferometer is limited by the optical absorption in the mirrors and their coatings. The temperature dependent index
of refraction and the thermal expansion coefficient of the optical materials (e.g. fused
silica) ensures that non uniform temperature gradients induced by the absorption of the
laser power will result in non uniform optical path length distortions. These path length
distortions will induce a non uniform phase distortion of the laser light. This effect may
limit both the controllability and sensitivity of Virgo.
In general, the total phase distortion, φtrans , for a collimated probe beam transmitted
through a heated optical component and propagating along the z-axis is
Z
2π h(r)
n(x, y, z, t)dz
φtrans (x, y, t) =
λ 0
Z h0
Z h0
2π
≈
n0 h0 +
∆nT (x, y, z, t)dz +
∆nE (x, y, z, t)dz
λ
0
0
2π
+ (n0 uz (x, y, h0 , t) − (n0 − 1)uz (x, y, 0, t))
λ
= φ0 + φT (x, y, t) + φE (x, y, t) + ψ(x, y, t),
(3.1)
31
Chapter 3. Thermal Lensing
where h0 is the nominal distance through the optical component [43], uz (x, y, z, t) is
the net local axial displacement of the material under thermoelastic deformation at
the coordinates (x, y, z) at time t, ∆nT is the local refractive index change induced by
temperature change (the thermooptic effect), ∆nE is the local refractive index change
induced by local thermal strain (the elastooptic effect).
For temperature increases small compared to the ambient temperature, the thermooptic effect can be written as
∆nT =
dn
∆T,
dT
(3.2)
dn
where dT
is the refractive index derivative with respect to temperature (thermooptic
coefficient) and ∆T is the temperature deviation from the ambient temperature. When
assuming cylindrical symmetry and polarization along the x-axis, the elastooptic effect
can be approximated by
∆nE = −pxx α∆T,
(3.3)
where pxx is the component of the elastooptical tensor in the direction of the polarization
and α is the thermal expansion coefficient. Combining Eqs. (3.1), (3.2) and (3.3) yields
∆S ≡
λ
(φtrans − φ0 ) = ∆ST + ∆u + ∆SE
2π
Z h0
dn
∆T dz,
=
+ α − αpxx
dT
0
(3.4)
where ∆S is the total change in optical path length, ∆ST is the change in optical path
length induced by the thermooptic effect, ∆u is the change in optical path length induced
by the thermal expansion of the component and ∆SE is the elastooptic effect.
Finite Element Analysis (FEA) yields a general, precise, and rapid means of solving
the partial differential equations which govern heat transfer and thermal expansion for
individual optical components.
The results of a FEA to compute the temperature field T (x, y, z, t) in the Virgo input
mirrors are presented in this chapter. The temperature field has been calculated for two
cases.
ˆ The steady state case: The temperature of the mirror is stationary as absorbed
power heating of the mirror is in equilibrium with the heat transfer by radiation.
ˆ The transient case: The temperature of the mirror is not stationary.
The computation of the temperature field allows for the calculation of the induced
change in optical path length as well as the temperature dependence of the mirror resonance modes. This Finite Element Model (FEM) has been developed with the COMSOL
multiphysics package [44].
32
3.1. The finite element model
3.1
The finite element model
Thermal lensing of the mirror is analyzed by solving the heat equation,
ρC
∂T
+ ∇ · (−k∇T ) = Q,
∂t
(3.5)
for a given power density due to absorption of laser power1 where T is the temperature, ρ the density, C the heat capacity, k the thermal conductivity and Q is the heat
source. Note that heat transfer by radiation is not shown in this equation but is taken
into account as boundary conditions in the full simulation. The power emitted through
radiation by an object of surface A and with emissivity is given by Stefan-Boltzmann’s
law
Prad = Aσ T 4 − T04 ,
(3.6)
where σ = 5.67×10−8 Wm−2 K−4 is the Stefan-Boltzmann constant and T0 is a reference
temperature which is taken to be room temperature (300 K).
A dynamical thermal analysis is performed which includes the heat radiation exchange with its environment, including the recoil mass of the mirror. The resulting
temperature distribution evolution is used to predict the change in optical and mechanical properties of the mirror at different time scales. These include its thermal expansion,
optical path length and the frequencies of the vibrational modes.
Figure 3.1 shows the meshed 3D-models of the geometry used for the analysis with
and without recoil mass. A mapped mesh has been used to mesh the mirror geometry
with cubic elements. The element density has been chosen higher in the center since
that is where the thermal gradient will be the highest. Finally, the recoil mass has been
meshed in a coarse way since it is only used as a heat-reflective shield. The geometry
Y
Y
Z
X
Z
X
Figure. 3.1: 3D models used in the FEA. The left panel shows the mesh model of the
input mirror and the right panel shows the mesh model for the input mirror and recoil mass.
of the input mirror is modeled as a solid cylinder without any internal structure like
1
Reference [45] gives an analytical solution to a similar problem.
33
Chapter 3. Thermal Lensing
coating layers2 . This is justified by considering that the combined radial heatflow is
much smaller than the combined axial heatflow since the summed thickness of the coating layers is much smaller than the radius of the mirror (Table 3.1). Furthermore, the
emissivity of the mirror is treated as a measured intrinsic quantity which means that
the internal structure of the mirror becomes irrelevant in terms of radiation.
In the actual setup, the mirror is suspended from a large mass by two thin steel wires.
The heat transfer through these wires has been neglected as well as the effect they might
have on the eigenfrequencies of the mirror. The power dissipated in the substrate is given
by
dPs
Ps = Plaser
ts R ≈ 1.94 mW,
(3.7)
dz
s
where Plaser represents the laser power, dP
the fractional loss (in ppm/cm) in the subdz
strate, ts is the thickness of the mirror and R the power recycling factor of the power
recycling mirror. The beam splitter contributes a factor of 12 , which is canceled by the
fact that the beam passes twice through the substrate. The power dissipated in the
coating is given by
F
Pc = Plaser δPc R ≈ 5.73 mW,
(3.8)
π
where δPc is the fractional power loss (in ppm/bounce) in the coating and F the finesse
of the Fabry-Perot cavity.
From the calculated amount of dissipated power, the heat flow for the two losses can
be calculated. The power density in the substrate is modeled in accordance with the
Gaussian shape of the beam with waist w. The power density in the coating is similarly
modeled. The total absorbed power relates to the respective heat flows as
ZZZ
Ps =
2 +y 2
w2
−2 x
Qs e
ZZ
dV,
and
Qc e−2
Pc =
V
x2 +y 2
w2
dS.
S
Here, V and S denote a volume and surface element, respectively and Qs and Qc are the
induced power densities for the substrate and the coating, respectively. A calculation
with as input the parameters from Table 3.1 yields
Qs =
W
2Ps
≈
26.6
ts πw2
m3
and
Qc =
2Pc
W
≈
7.50
,
πw2
m2
(3.9)
Note that these estimated power densities are based purely on the nominal design values
of Virgo for the given finesse and expected losses in the substrate and coating. It has
been shown that the actual losses are significantly higher than expected. This may be
due to damage and/or pollution of the mirrors and warrants further investigation.
2
The reflective coating of the input mirror consists of alternating layers made of SiO2 and Ta2 O5
each approximately 0.2 µm thick.
34
Value
0.175
0.096
8
0.022
36
50
300
2203
73.2
0.164
739
1.36
0.89
0.7
0.54 × 10−6
9.80 × 10−6
1.5 × 10−4
7.1 × 10−5
33
0.75
1.25
2700
160
900
0.2
Symbol
r
ts
Plaser
w
R
F
Tbg
ρs
E0
ν0
cs
ks
dPs /dz
α
dn/dT
(1/E0 )dE/dT
(1/ν0 )dν/dT
kc
tc
δPc
ρr
kr
cr
r
W/(m K)
µm
ppm/bounce
kg/m3
W/(m K)
J/(kg K)
K−1
ppm/cm
J/(kg K)
W/(m K)
K
kg/m3
GPa
W
m
[46]
m
The variation of the Poisson ratio has been calculated taking the derivative of the formula for ν given in Ref. [51], which yields
1 dG
1 dG
(ν + 1) E1 dE
dT − G dT , where G represents the shear modulus. The value for G dT is also taken from Ref. [51].
a
dν
dT
=
[52]
[48]
[49]
[50]
[50, 51]a
[48]
[46]
[48]
[47]
Source
Unit
Table. 3.1: The parameters used in the FEA. The value for the emissivity of oxidized aluminum was used for the recoil mass.
Recoil mass properties (Al)
Coating properties (Ta2 O5 & SiO2 )
Substrate properties (SiO2 )
ITF properties
Geometry
Property
Mirror radius
Mirror thickness
Recoil mass parameters
Laser power
Beam width
Power recycling factor
Fabry-Perot finesse (end cavity)
Background temperature
Density
Young modulus
Poisson ratio
Heat capacity
Thermal conductivity
Emissivity
Power loss
Thermal expansion
Thermooptic coefficient
Young modulus per Kelvin
Poisson ratio per Kelvin
Thermal conductivity
Summed thickness
Relative power loss
Mass density
Thermal conductivity
Heat capacity
Emissivity
3.1. The finite element model
35
Chapter 3. Thermal Lensing
3.2
Simulation results
3.2.1
Steady state thermal analysis
Results of the 3D FEA for the equilibrium state for both the power densities Qc and Qs
separately and summed are presented in Fig. 3.2. The figure shows that the contributions
x
x
z
z
x
z
Figure. 3.2: Temperature increase half-slices at y=0. The top left panel shows the temperature increase due to losses in the coating, the top right panel shows
the temperature increase due to losses in the substrate and the bottom
panel shows the total temperature increase induced by the sum of the two.
to the total temperature distribution for the coating and for the substrate differ by
almost an order of magnitude. The maximum temperature increase is obtained when
both Qc and Qs are turned on: Tmax − T0 = ∆Tmax ≈ 75 mK for the mirror with recoil
mass.
36
3.2. Simulation results
The average mirror temperature increase < ∆T > is defined as
RRR
∆T dV
V
< ∆T >= RRR
.
dV
V
(3.10)
Table 3.2 presents the average temperatures for the different cases with or without recoil
mass. It is shown that the recoil mass has a significant effect on the global temperature
Without recoil mass Including recoil mass
< ∆T >s (mK)
< ∆T >c (mK)
< ∆T >c+s (mK)
1.47
4.08
5.54
2.26
6.28
8.52
Table. 3.2: Average mirror temperatures < ∆T >c , < ∆T >s and < ∆T >c+s calculated under Qc , Qs and Qc + Qs loads.
increase. Assuming a linear response to the small power densities we can calculate the
respective thermal resistances using < ∆T >c+s from Table 3.2 and the value for Tmax
mentioned earlier. It follows that
d < ∆T >
≡ 1.11 K/W
dQ
and
dT max
= 9.78 K/W.
dQ
In order to quantitatively show the temperature distributions along key areas on
and in the mirror, four sample curves have been chosen where the (one dimensional)
equilibrium temperature field will be drawn. Figure 3.3 schematically shows these key
positions as bold lines. Figure 3.4 shows the equilibrium temperature distributions for
Figure. 3.3: The four mirror lines where the temperature distribution is calculated for
the case of thermal equilibrium: front radial, back radial, lateral and axial.
the mirror with and without recoil mass. It can be seen from the radial profiles that the
temperature of the mirror locally increases a few mK due to the thermal shielding of
37
Chapter 3. Thermal Lensing
Figure. 3.4: One dimensional temperature profiles when the mirror is in thermal equilibrium. The top left and right panels show the radial front and radial
back profiles while the bottom left and right panels show the lateral and
axial profiles as shown schematically in Fig. 3.3. The red curves with the
box-markers are the profiles computed with the recoil mass included in the
FEA and the blue curves with the circle-markers are computed without
the recoil mass.
the recoil mass. For the mirror’s outer edge (the lateral profile shown in Fig. 3.4) this
effect is more pronounced as its temperature increase is roughly doubled by the recoil
mass shielding.
38
3.2. Simulation results
3.2.2
Eigenmodes in thermal equilibrium
The eigenmodes of the Virgo input mirrors have been calculated. Figure 3.5 outlines the
two principal eigenmodes, the ‘Butterfly’ and the ‘Drum’ mode. Note that these results
are in agreement with those of previous calculations [46]. To first order the eigenfre-
X
X
Z
Z
Figure. 3.5: Principal resonance modes for a uniform temperature distribution. The
left panel shows the butterfly resonance mode predicted at 3904.0 Hz and
the right panel shows the drum resonance mode predicted at 5578.1 Hz.
√
quency is proportional to the square root of the stiffness: f ∝ E. When defining
∆f ≡ f (T = T0 + < ∆T >) − f (T = T0 ) where f (T ) is the eigenfrequency for temperature T , < ∆T > is the average mirror temperature increase as defined in Eq. (3.10) and
T0 is room temperature, it can be derived that
∆f ≈ f (T0 ) < ∆T >
1 1 dE
.
2 E0 dT
(3.11)
In Eq. (3.11) E0 is the Young modulus at room temperature. With the parameters from
Table 3.1, Eq. (3.11) yields
∆f = f (T0 ) < ∆T > × 7.5 × 10−5 K−1 ,
(3.12)
where f (T0 ) is the eigenfrequency without thermal load. From Eq. (3.12) it is possible
∆f
, for both modes
to predict the frequency dependency on the average temperate, <∆T
>
as
∆f
∆f
= 0.42 Hz/K and
= 0.29 Hz/K.
< ∆T > drum
< ∆T > butterfly
The eigenfrequencies can also be studied in a FEA with the temperature load obtained
from the equilibrium analysis. The resulting frequency shifts together with the change
in frequency per Kelvin are listed in Table 3.3. From the results shown in Table 3.3
it can be seen that the difference in frequency due to the temperature increase of the
eigenmodes is in the order of a few millihertz for initial Virgo operation. Also there is
∆f
for the drum mode with or without recoil
a noticeable difference in the value for <∆T
>
mass. Considering the fact that the only difference between the two situations is that
the temperature distributions are different, this suggests that the drum mode is more
39
Chapter 3. Thermal Lensing
Mode
Drum
Butterfly
Drum + recoil mass
Butterfly + recoil mass
∆f (mHz) ∆f / < ∆T > (Hz/K)
3.38
1.81
4.57
2.64
0.61
0.34
0.53
0.30
Table. 3.3: Temperature dependence of the resonance frequency of the various eigenmodes.
sensitive to a non-uniform temperature distribution.
Comparing the calculated numbers originating from Eq. (3.12) with the numbers in
Table 3.3 shows that the values obtained without recoil mass differ from the theoretical
values. The difference seems to decrease when considering the addition of the recoil
mass.
Previous work on the temperature dependence of the eigenfrequencies of the two
principal eigenmodes has been done and is presented in Refs. [48, 53]. In that work the
recoil mass was omitted and uniform heating of the mirrors was assumed. The present
work improves on the previous results by including the recoil mass and computing the
temperature dependency of the eigenfrequencies of both principal modes for non-uniform
temperature distributions. The values found here are in qualitative agreement with the
previous work done.
3.2.3
Transient temperature analysis
The temperature as a function of time for different points of the mirror has been calculated using the FEM. The transient temperature behavior is sampled at three different
points on the mirror, shown in Fig. 3.6. Figure 3.7 shows this transient behavior for each
Figure. 3.6: The three mirror positions where the temperature as a function of time is
shown in Fig. 3.7.
of the mirror positions listed. From this figure it can be seen that thermal equilibrium
40
3.2. Simulation results
Figure. 3.7: Evolution of the temperature in time on various points on the mirror. The
top left, top right and bottom panels show the temperature evolution on
the circumference, the beam spot and the rear of the mirror, respectively.
The data points are obtained from the transient FEA while the curves are
fits of Eq. (3.13). The blue curves represent the mirror without recoil mass
and the red curves the mirror with recoil mass.
is reached after a certain time. This equalization process can be modeled by
f (t) = A(1 − e−t/τ ),
(3.13)
where A is the equilibrium value and τ the time constant. Table 3.4 shows the resulting
time constants when fitting Eq. (3.13) to Fig. 3.7. Due to the complex structure of the
mirror and recoil mass system various time constants are expected. These time constants
pose a challenge for the control systems used to operate the Virgo detector. The time
constant of the temperature in the beam spot is determined by the beam waist and the
mirror’s thermal conductivity. The front side of the mirror will dissipate the heat much
faster through conduction to the substrate than by thermal radiation. This implies that
the time constant in the beam spot is much smaller than on the other places places on the
mirror. The rest of the mirror’s heat dissipation is dominated by radiation and therefor
the other time constants will be larger (in the order of hours). The relative difference
between the time constants with and without recoil mass on the circumference of the
mirror is smaller than at the back of the mirror. This is because the recoil mass does not
41
Chapter 3. Thermal Lensing
location
Circumference
Beam spot
Rear
τno recoil (h) τrecoil (h)
5
0.08
3
6
0.08
4
Table. 3.4: Time constants for the temperature stabilization on various points on the
mirror. These time constants have been computed by fitting Eq. (3.13) to
the temperature evolution which have been obtained from the FEA for the
various locations on the mirror.
cover the entire back of the mirror (as can be seen in the geometry in Fig. 3.2) and thus
has a smaller effect on the thermalization. Also, the equilibrium temperature is lower
than on the front side, which is to be expected for the same reason.
The time evolution of the average temperature with and without recoil mass is shown
in Fig. 3.8. From Figs. 3.7 and 3.8 it can be concluded that the recoil mass has a non
Figure. 3.8: The average temperature of the mirror as function of time. The blue curve
represents the mirror without recoil mass and red curve the mirror with
recoil mass. The time constants are computed to be τrecoil = 6 h and
τno recoil = 4 h.
negligible effect on the long term behavior of the mirror as well as on the equilibrium
temperature.
42
3.3. Optical path length
3.3
Optical path length
The local heating of the mirror will cause a variation in optical path length for the light
interacting with the mirror. This may cause problems for the interferometer locking
system at relatively hight cavity powers. The change in optical path length (∆S) can be
written in terms of the thermooptic effect and the thermoelastic effect as described in
Eq. (3.1). For this analysis the thermooptic effect is much stronger than the elastooptic
effect and the latter can be neglected. The change in optical path length at distance r
from the axis can be calculated by numerically integrating the temperature distribution,
∆T , along z by using Eq. (3.14).
∆S(r) =
dn
+α
dT
Z
ts
∆T (z, r)dz.
(3.14)
0
A measure for the severity of the thermal lensing effect in the input mirror is the effective
radius of curvature [54]. A first order approximation for the effective radius of curvature3
for the heated mirror and assuming that the mirror is initially flat, is given by
R=
w2
,
2δS
(3.15)
where w is the waist of the beam and δS is the relative change in optical path length
between r = 0 and r = w given by
δS ≡ ∆S(0) − ∆S(w).
Figure 3.9 shows the change in optical path length versus the radial coordinate for
a beam traveling through the mirror parallel to the optical axis. By using Eq. (3.15)
in combination with the data shown in Fig. 3.9, the effective radius of curvature is
calculated to be 21 km4 .
Figure 3.10 shows the transient behavior of the change in relative optical path length
(δS) together with that of the radius of curvature. The value of the time constant
roughly corresponds to the time constant calculated on the beam spot of the mirror,
τbeamspot = 0.08 h (see Table 3.4). Figure 3.10 also shows that the influence of the recoil
mass is negligible on the surface of the mirror in the region between r = 0 and r = w.
This means when using Eq. (3.15) to calculate the effective radius of curvature, the recoil
mass can be neglected. This can be understood by comparing the steady-state change
in optical path length as a function of r for the case with and without the recoil mass
as shown in Fig. 3.9. In this figure the two curves seem to differ by a constant offset,
especially for r < w. This implies that the radius of curvature will be the same for both
cases which is to be expected since the addition of the recoil mass mainly increases the
temperature in a uniform way.
3
This approximation assumes that the change in optical path length in the region 0 < r < w can be
described by a circle with radius R.
4
For comparison: the end-mirrors have a radius of curvature of 3.5 km.
43
Chapter 3. Thermal Lensing
Figure. 3.9: Change in optical path length, ∆S as function of the radial coordinate
in thermal equilibrium for the mirror with and without the recoil mass
included in the simulation.
Figure. 3.10: Left: the time evolution of δS with τrecoil = τno
time evolution of effective radius of curvature.
3.4
recoil
= 0.1 h. Right: the
Thermal lensing effects in Virgo
In section 3.2.2 the frequency of the two principal resonant modes (drum and butterfly)
of the Virgo input mirrors have been computed. Furthermore, the change in frequency
as a function of average mirror temperature has been computed. With this result it is
possible to infer the average mirror temperature by monitoring the frequency change of
these modes.
Figure 3.11 shows the power spectra of the output data of the Virgo detector. The
drum modes of both input mirrors, shown in the left panel, can be seen as the double
44
3.4. Thermal lensing effects in Virgo
10-32
10-33
10-34
10-35
10-36
10-37
10-38
10-39
10-40
10-41
10-42
10-43
10-44
Sn (Hz-1)
Sn (Hz-1)
10-32
10-33
10-34
10-35
10-36
10-37
10-38
10-39
10-40
10-41
10-42
10-43
10-44
3860
3880
3900
3920
3940
f (Hz)
5540
5560
5580
5600
f (Hz)
Figure. 3.11: The PSDs taken from Virgo at GPS time 947284200 (VSR2) created
from the h-reconstructed data, channel ‘V1:h 16384Hz’. The left panel
shows the frequency interval corresponding to the butterfly modes and
the right panel shows the frequency interval corresponding to the drum
modes.
peaks centered around 5543.2 Hz and 5545.5 Hz, respectively. The double peaks centered around 5584.5 Hz and 5586.3 Hz, respectively represent the drum modes of the
end mirrors. The difference between the two sets of peaks are due to the fact that the
end mirrors are slightly thinner. In the right panel of Fig. 3.11 the same double peak
structures corresponding to the butterfly modes can be seen. In this case the input mirrors modes are centered around 3884.2 Hz and 3885.2 Hz while the end mirror modes are
centered around 3917.7 Hz and 3918.8 Hz. The simulated values for these modes, shown
in section 3.2.2 deviate at maximum 0.6 % and 0.5 % from the actual values for the
drum and butterfly modes, respectively. Note that the difference in absolute frequency
between the end and input mirrors of the modes are small and thus the temperature of
all mirrors can be inferred by using the results of this simulation.
From the temperature dependence of the modes together with the output of the interferometer, the average temperature of the mirror can be inferred. A method to measure
the frequency of the modes and convert them to average temperatures is presented in
Ref. [48]. In order to verify the simulated values of the temperature dependence of the
mirror modes shown in section 3.2.2, the long term temperature evolution of the West
End (WE) building is compared to the temperature of the mirror as computed from the
resonant modes. The result is shown in Fig. 3.12. It can be seen that the simulated values of the frequency evolution of the drum and the butterfly modes are consistent with
the global temperature. Furthermore, plots like Fig. 3.12 can be used to identify which
resonance corresponds to which mirror as the frequency of the modes can be linked to
temperature sensors in the various buildings.
For comparison it is noted that when measuring the power contained in the FabryPerot cavities and using this as input for the FEM, it is possible to simulate the heating
and cooling cycles of the mirrors. This has been done with a more simple FEM discussed
in Ref. [55] (Punturo’s model) which did not include a recoil mass. The result of this
45
∆ T (K)
Chapter 3. Thermal Lensing
0.4
0.35
0.3
WE temperature
Drum mode
Butterfly mode
0.25
0.2
0.15
0.1
0.05
0
-0.05
0
20
40
60
80
100
120
time (h)
Figure. 3.12: The deviation of the temperature from room temperature (300 K) of the
West End (WE) mirror computed from the frequency of the drum mode
(‘+’-sign) and the butterfly mode (the dots). The solid curve shows the
global temperature evolution in the WE building as measured by temperature sensor ‘EM TE SUWE10’. In this plot, the 0 hour point corresponds to GPS time 840240000. The data from the Drum and Butterfly
modes have been taken from Ref. [55] and have been rescaled such that
the more accurate results of the temperature dependence of the eigenfrequencies computed for non-uniform heating and with a recoil mass have
been used.
analysis compared to the temperature of the mirror which is computed from the drum
mode for the West Input (WI) mirror is shown in Fig. 3.13. From this figure it can be
seen that the simulation agrees reasonably well with the data. However, the measured
mirror temperature if roughly a factor of 10 larger than the simulated temperature.
A similar study shows that the temperatures of the North Input (NI) mirror are approximately a factor 4 higher than the nominal value. These results indicate a larger
absorption of laser light than previously assumed (Table 3.1). This discrepancy should
be investigated with a more accurate FEM which includes the recoil mass. Furthermore,
in order to estimate the actual losses in the mirror, more FEM studies are needed which
should include the recoil mass.
To actively control the length of the cavities, the input beam is phase modulated
and the reflected light is examined to obtain an error signal. This modulation is done at
a certain frequency ωm and will result in taking power from the carrier at frequency Ωl
and splitting this into two sidebands at frequencies Ωl + ωm and Ωl − ωm . This modulation frequency is chosen such that the sidebands are antiresonant in the optical cavity
when the carrier is resonant. This implies that if the cavity is slightly shorter than on
resonance, the reflected carrier suffers a small phase shift and will combine with the
46
3.4. Thermal lensing effects in Virgo
Figure. 3.13: The average temperature deviation from room temperature as computed
with Punturo’s model (solid curve) and the average mirror temperature
computed from the frequency of the drum mode. This FEM uses the
measured power stored in the Fabry-Perot cavity as input and the mirror temperature is computed with the old uniform value of 0.61 Hz/K, as
derived in Ref. [53].
fully reflected sidebands. The combined signal will oscillate at ωm out of phase with the
original modulation signal. If the cavity is slightly longer it will oscillate in phase with
the modulation signal.
It has been shown in Ref. [43] and [56] that the sidebands are especially sensitive to
thermal lensing effects. The heating of the mirrors will cause the sidebands to be lost
which will unlock the interferometer. These losses are especially strong in the recently
proposed Marginally Stable Recycling Cavities (MSRC) optical scheme for Advanced
Virgo. The loss of these sidebands seems to be governed by a series of time constants.
These could be linked to the simulated time constants shown in section 3.2.3. However,
since this loss of sideband power has to do with the actual phase of the carrier and sidebands a more advanced study including the phase of both beams should be performed.
A preliminary study is performed in Ref. [57].
47
Chapter 3. Thermal Lensing
In order to compensate these thermal lensing effects in Virgo, a ‘Thermal Compensation System’ (TCS) has been developed (see Ref. [58]). This TCS consists of a CO2
laser which is tuned to emit a ring-shaped beam which is incident on the mirror in
order to locally heat the mirror such that the wavefront distortions will be negated. A
first TCS for the input mirrors has been installed in Virgo in April 2008. Note that
more stringent constraints must be placed on Virgo’s thermal compensation system in
order to guarantee successful operation at high power with the MSRC optical scheme of
Advanced Virgo.
3.5
Conclusion and outlook
A finite element model has been described that models the Virgo input mirrors as a
cylinder and its recoil mass with the dimensions as realistic as possible. With the assumed power dissipated in the substrate and coating, Ps = 1.94 mW and Pc = 5.73 mW,
the corresponding power densities on the substrate and the coating can be calculated:
Qs = 25.6 W/m3 and Qc = 7.50 W/m2 , respectively. With these numbers, the temperature evolution in the mirror can be obtained taking into account the heat transfer via
radiation.
In thermal equilibrium the volume-average temperature increase is 8.52 mK and 5.54
mK for a model with or without recoil mass. From this difference it can be inferred that
the recoil mass acts as a heat-reflective shield and must be included in realistic calculations.
The eigenmodes of the mirror are a function of the temperature distribution, since
both the Young modulus and the Poisson ratio depend on temperature. Eigenfrequencies
with and without a thermal load have been calculated for the non-uniform temperature
distributions. The change in frequency per unit temperature has been calculated for the
drum and butterfly mode. The results are 0.53 Hz/K for the drum mode and 0.30 Hz/K
for the butterfly mode.
It is important to realize that the FEA result significantly differs from the case of
uniform heating of the mirror: 0.42 Hz/K for the drum mode and 0.29 Hz/K for the
butterfly mode. The value for the butterfly mode only deviates 3% from the uniform
heating value while the drum mode deviates 26%. The butterfly mode is less sensitive
to a Gaussian-like non-uniform temperature distribution compared to the drum mode
due to the spatial shape of the mode. These results allow measurement of the average
mirror temperature by measuring the frequency change of the drum and butterfly modes
together with the computed temperature dependence of the frequency of these modes.
The simulation has shown that the addition of the recoil mass has a non-negligible effect
on this temperature dependence.
The transient temperature analysis reveals the different time scales of the system,
varying from several minutes for the temperature in the beam spot to many hours for
the global temperature. The presence of the recoil mass increases the long term time
constant from 4 to 6 hours.
With the temperature information gained in the equilibrium and transient analyses,
48
3.5. Conclusion and outlook
the change in optical path length and the effective radius of curvature can be calculated.
The change in optical path length in equilibrium along the optical axis varies between
0 and 24.5 nm. The effective radius of curvature in equilibrium is calculated to be 21
km. The transient analysis shows that the time constant involved is 0.1 h. This result
has been obtained by using an approximation for the effective radius of curvature which
assumes circular deformation of the optical path length around the optical axis.
All simulations have been done assuming that = 0.2 for the emissivity of the recoil
mass (a typical number for the emissivity of oxidized aluminum). This number should
be verified since it impacts the simulation.
It has been shown that according to the measurement of the average temperature of
the mirrors that the optical losses are significantly higher than the expected values (up
to an order of magnitude). In order to measure the actual optical losses in the mirror
it may be possible to use the FEA together with the measured power in the cavity as a
function of time. The resulting temperature can then be compared to the measured average temperature of the mirror. However, such an analysis would require further study.
The sidebands have been shown to be extremely sensitive to the thermal lensing effect
due to the observation of various time constants in the measured optical power in the
sidebands. Further study is needed to determine the exact nature of this behavior and
the origins of these time constants.
49
Chapter 3. Thermal Lensing
50
Chapter
4
Continuous waves analysis methods
In section 1.3.1 it has been discussed that non-axial symmetric neutron stars are the
most promising continuous sources of GWs in terms of expected amplitude. From Eq.
(1.15) it is apparent that the typical strain may be in the order 10−27 to 10−24 . Figure
2.11 shows that the power spectral density of the LIGO and Virgo interferometers is
3 × 10−23 Hz−1/2 and 6 × 10−23 Hz−1/2 respectively in the most sensitive region. Thus
the GW amplitude is weak compared to the noise level.
Continuous GWs can be integrated over longer periods of time. This is optimally
done via matched filtering as explained in appendix B. The signal-to-noise (SNR) ratio
for the optimal filter h̃(f ) is
2
Z ∞
S
|h̃(f )|2
2
SNR =
=4
df,
(4.1)
N
Sn (f )
0
where Sn is the single sided power spectral density of the noise. Matched filtering is often
referred to as ‘coherent integration’. For a search based √
on this coherent integration
technique, run over a coherent time T , the SNR scales as T .
The scaling defined above can be used to find signals with amplitudes which are
weak compared to the noise level of the detector. For example, an integration
time of 1
√
year allows for the detectability of GW amplitudes hmin = (2 × 10−4 ) × Sn . However,
this sensitivity will only be achieved if an optimal filter, also called a ‘template’, can
be constructed and compared to the data. In order to construct this template, the
physical parameters of the system must be known to sufficient precision such that it will
remain coherent with the signal over the entire observation time. When the parameters
are not known, or not known well enough, a series of templates can be constructed,
compared to the data and a significance can be assigned to each of these templates.
For a small number of templates and with certain parameters constrained well enough,
this template-based search is the preferred analysis method. These types of searches are
referred to as ‘coherent searches’ and will be covered in more detail in section 4.2.
The template-based search is not ideal when performing a search for GWs from
neutron stars of which the parameters are (partially) unknown. The number of templates
scales as a power of the coherence time and renders such a search computationally
unfeasible for coherence times above a few days. In order to perform a blind all-sky
51
Chapter 4. Continuous waves analysis methods
search, shorter coherent sections
√ of the data must
√ be combined. In general, the sensitivity
of such methods will scale as 4 T instead of T in the case of coherent searches. These
methods are known as ‘semi-coherent searches’ and will be covered in section 4.3.
4.1
The gravitational wave signal model
If the GW signal from a neutron star were truly monochromatic, then a simple delta
function in the frequency domain would be the optimal filter. The data analysis would
then be trivial as a sufficiently long Fourier transform would be able to pick the signal
out of the background. However, as discussed in section 1.3, pulsars suffer from glitches
and spin-down. Furthermore, the signal originating from the neutron star will be Doppler
shifted by Earth’s motion (daily and yearly). Additionally, when the neutron star is in a
binary system, the orbit of the neutron star will introduce Doppler shifts. Moreover, the
detector does not have a uniform sensitivity over the entire sky, as discussed in section
2.4. A signal model can be constructed which includes all these previously mentioned
effects. This signal model can be used to construct templates for a coherent search.
When introducing χ as the angle between the deformation and rotation axis of a
neutron star and ι as the angle between the rotation axis and the propagation direction,
the GW signal from Eq. (2.15) can be written in terms of the GW amplitude h0 (see
Eq. (1.15)) as shown in Ref. [41] as
h(t) = F+ (t)(h1+ (t) + h2+ (t)) + F× (t)(h1× (t) + h2× (t)),
(4.2)
where h1 and h2 are used to decompose the signal into Φ and 2Φ components and are
given by
1
h0 sin 2χ sin 2ι cos Φ(t),
8
1
h2+ (t) =
h0 sin2 χ(1 + cos2 ι) cos 2Φ(t),
2
h1+ (t) =
(4.3)
(4.4)
and
1
h0 sin 2χ sin ι sin Φ(t),
4
h2× (t) = h0 sin2 χ cos ι sin 2Φ(t).
h1× (t) =
(4.5)
(4.6)
In Eq. (4.2), t is the time in the detector frame, Φ(t) is the phase of the gravitational
wave and F+ (t) and F× (t) are the beam-pattern functions of the interferometer and are
discussed in section 2.4. Note that the beam-pattern functions depend on the polarization
angle ψ as well as on the sky-coordinates of the source, α and δ. Furthermore, the
functions h1+ , h2+ , h1× and h2× depend on the phase Φ(t) of the GW.
Within good approximation, isolated neutron stars (e.g. neutron stars without a
companion) will have a constant velocity with respect to the solar system barycenter
(SSB). For these systems it is sufficient to consider only the Doppler shifts due to Earth’s
daily and yearly motion through the solar system. Figure 4.1 shows a diagram the
52
4.1. The gravitational wave signal model
Figure. 4.1: Diagram of the relevant position vectors of an isolated neutron star relative to the detector. The origin is located at the SSB, ~k is the GW vector,
~rd is the position of the detector on Earth, ~vd is the velocity vector of the
detector relative to the SSB and ~n0 is the position vector of the neutron
star N S.
relevant position vectors for an isolated neutron star system. When assuming a neutron
star emitting GWs at constant frequency f0 (e.g. no spin-down), the instantaneous
Doppler shifted frequency f is
f = f0
~vd · n̂0
1+
c
,
(4.7)
where ~vd is the velocity vector of the observer with respect to the source and c is the speed
of light. If the product ~vd ·n̂0 is constant (e.g. the neutron star has a constant velocity and
Earth would stand still), the analysis would be trivial again since the detected frequency
would simply gain a constant offset. However, the velocity of the detector with respect
to the source changes in time due to Earth’s rotation and its revolution around the Sun.
The change in detected frequency over integration time T can be computed to be
(∆f )Doppler = f0
(∆vd )T
,
c
(4.8)
where (∆v)T is the change in velocity during the integration time. In order to retain the
strength of the signal in a single frequency bin with size ∆f = 1/T one can calculate the
maximally allowed coherence time. This is done by requiring that the frequency shift is
at most half a frequency bin. Equating the Doppler shift from Eq. (4.8) to ∆f /2 and
solving for T together with taking the detector at latitude 40◦ and R⊕ = 6.38 × 106 m
53
Chapter 4. Continuous waves analysis methods
will yield
1/2
1 kHz
≈ 30 min
,
f0
1/2
1 kHz
≈ 120 min
,
f0
rot
Tmax
orb
Tmax
(4.9)
(4.10)
rot
orb
for the daily (Tmax
) and yearly (Tmax
) motion. Equation (4.9) shows that the maximum
integration time when performing a simple Fourier transform is about 30 minutes for a
1 kHz GW. Furthermore, when choosing the location of the source in such a way that
the Doppler shift is maximal, the maximum frequency variation for both the yearly and
daily motion can be computed to be
f0
rot
−3
(∆f )max ≈ 2.4 × 10 Hz
,
(4.11)
1 kHz
f0
orb
(∆f )max ≈ 0.2 Hz
.
(4.12)
1 kHz
The signal model should contain the phase modulation introduced by these Doppler
shifts. This it done by computing the phase with the relation
Z
Φ(t) = 2π f (t)dt.
(4.13)
In the case of binary systems the orbit of the neutron star around its companion
introduces additional Doppler shifts, depending on the orbital parameters of the neutron
star. The maximum coherence time such that the signal from a neutron star in a binary
system will remain in a single frequency bin is computed in Ref. [59] to be
2/3 −1/6 −1/2
(1 + q)1/3
P
mNS
f
kepler
Tmax = 131.6 sec
, (4.14)
√
q
1 day
1.4 M
1 kHz
where q = m2 /mNS represents the ratio of the masses of the two objects, P the period of
the binary, mNS the mass of the neutron star and f the GW frequency. For an example
binary system with parameters m2 = mNS = 1.4 M , P = 2 h and f0 = 1 kHz we find
kepler
= 20 s, much less than for isolated neutron stars. This coherence time allows for
Tmax
only a marginal gain of the signal-to-noise ratio compared to the PSD of the detector.
Section 5.1 gives a detailed description of the Doppler shifts induced by a neutron star
in a binary system.
As discussed in section 1.3, pulsars tend to lose angular momentum and spin-down.
These spin-down values are given in terms of the derivative of the rotation frequency, f˙.
This means that the frequency of the signal changes by an amount of ∆f = T f˙ during
observation time T . The maximum coherence time can then be parameterized as
s
10−9 Hz/s
spindown
≈ 8.78 hour
Tmax
.
(4.15)
−f˙
54
4.1. The gravitational wave signal model
An optimal filter has to take into account all Doppler shifts as well as the spin-down
in case a coherence time larger than 9 hours is required. This spin-down can be taken
into account by adding so-called spin-down parameters to the phase model. These spindown parameters are simply the direct derivatives of the rotational frequency and can
be included up to certain order.
The phase of the GW takes it simplest form when the time coordinate used is tNS ,
the proper time in the neutron star rest frame. The phase in this frame, ΦNS , is written
as
s
(n)
X
fNS
tn+1
(4.16)
ΦNS (tNS ) = Φ0 + 2π
NS ,
(n
+
1)!
n=0
(0)
where Φ0 is the initial phase, fNS is the instantaneous frequency of the GW in the
(n)
NS rest system, fNS are the spin-down parameters in the NS rest frame and s are the
number of spin-down parameters included in the phase model. In order to include all
previously discussed Doppler shifts in the phase model, the phase shown in Eq. (4.16)
must be rewritten in terms of the detected phase on Earth.
In the coordinate system of which the origin is located in the solar system barycenter
(SSB), the position of the detector is given by ~rd , the GW wave vector is given by ~k and
the neutron star is denoted by NS. This is shown in Fig. 4.2. The detected phase, Φd (t),
Figure. 4.2: Diagram of the relevant position vectors of a neutron star in a binary system relative to the detector. The origin is located at the SSB, ~k is the GW
vector, ~rd is the position of the detector on Earth, ~n0 is the direction towards the center of mass µ of the Keplerian orbit and ~rK is the position
vector of the neutron star N S relative to µ.
55
Chapter 4. Continuous waves analysis methods
can be computed by rewriting the proper time tNS in terms of the time in the detector
frame t. The phase observed at the detector at local time t was emitted by the NS at
coordinate time t0 such that
|~k(t0 )|
t = t0 +
,
(4.17)
c
where ~k is defined in Fig. 4.2 and c is the speed of light. Finally, the relation between
the time t0 and tNS is given in Ref. [41] by
p
k0
0
tNS = 1 − βNS t +
,
(4.18)
c
where βNS = vNS /c and k0 = |~k(t = −~k(0)/c)|. Note that Eq. (4.18) takes special
relativity effects into account. Also note that in order to rewrite the proper time as
a function of the detector time, Eq. (4.17) must be inverted. This inversion is highly
non-trivial since ~k in Eq. (4.17) contains all the orbital parameters (Earth’s daily and
yearly motion together with the Keplerian parameters) thus the functional dependency
of Eq. (4.17) can be extremely complicated. The detected phase in the interferometer,
Φd , will be of the form
Φd (t) = ΦNS (tNS (t)).
(4.19)
Even though Eq. (4.19) has no straightforward analytical form, it does contain all the
Doppler shifts discussed before.
In Ref. [41] an analytical approximation of the detected phase for the case of isolated
pulsars has been derived from Eq. (4.19). Using the coordinate system shown in Fig.
4.1, this phase is
Φd (t) = Φ0 + 2π
s
X
n=0
(n)
f0
s
n
X
tn+1
2π
(n) t
+
n̂0 · ~rd (t)
f0
,
(n + 1)!
c
n!
n=0
(4.20)
(n)
where f0 denotes the n-th order derivative of the frequency (spin-down parameter1 ),
c the speed of light. Note that since Eq. (4.20) is derived for the isolated neutron star
case, n̂0 = n̂0 (α, δ) is a unit vector pointing towards the neutron star originating from
the SSB. Equation (4.20) has been derived under the assumption that relativistic effects
can be neglected as well as the assumption that f0 < 1 kHz and that the spin-down
age τs > 40 yr. The phase model for isolated NS (substituting Eq. (4.20) into Eq. (4.2))
contains all the necessary parameters to describe the GW induced by a neutron star with
a fixed quadrupole moment. Note that glitches have not been included in the model.
The phase model shown in equation (2.15) depends on 8 + s parameters: h0 , Φ0 ,
f0 , α, δ, ι, χ, ψ and f (1) . . . f (s) . This implies that the parameter space for performing
a template-based coherent search is 8 + s dimensional. When adding (non-relativistic)
binary systems to this signal model, five more parameters need to be taken into account.
These additional parameters as well as a modified phase model will be discussed in
section 5.1.
1
(n)
The spin-down parameters f0 are measured in the SSB and are not equal to the instantaneous
(n)
spin-down parameters fNS from Eq. (4.16). The relation between the two is given in Ref. [41].
56
4.2. Coherent searches
4.2
Coherent searches
As explained before, the matched filtering procedure will be optimal. Such a search is
called a ‘fully coherent search’ and requires the precise signal shape to√be known in order
to create templates. The SNR of such a template will then scale as T where T is the
observation time.
In the case of targeting known isolated pulsars, radio observations can be used to constrain parameters like α, δ, f0 and the spin-down values. For these pulsars it is possible
to confine a GW search to a few and in some cases a single signal template. This allows
for the coherent integration of all available data with this matched filtering algorithm to
achieve the best possible sensitivity. For sources where a single template is sufficient, the
integration is done in the time domain and the parameters of the source are calculated
with Bayesian inference. When the parameters are not known with sufficient precision
through astrophysical observations, the search can still be performed in a coherent way.
However, the number of templates required will scale as T 5 [60]. This implies that such
coherent searches are limited by allowing only a small mismatch between the observed
parameters and the template parameters. This ‘wide parameter’ search is performed
in the frequency domain for computational reasons. The resulting detection statistic is
called the F-statistic (see Ref. [41]).
Both the narrow and wide parameter coherent search has been applied to the Crab
pulsar [61]. For the single-template search the joint (i.e. multi-detector) 95% upper limit
on the GW amplitude is measured to be h95%
= 2.7 × 10−25 . From this upper limit one
0
can derive that less than 4% of the energy loss of the Crab pulsar, observed in the
spin-down, is due to the emission of GWs. For the wide-band search the upper limit will
be worse due to the number of templates applied and in this analysis less than 73% of
the observed energy loss is due to GWs. These results have been obtained by analyzing
LIGO’s fifth science run (S5).
The template or F-statistic search has also been applied in the case of unknown
neutron stars. Such a search is called an all-sky search and assumes no prior knowledge
about the sky position of the neutron stars which will potentially emit GWs. This search
has been performed with the Einstein@Home distributed infrastructure [62]. At the time
of the analysis, the Einstein@Home project contained about 105 computers. The search
was performed with a coherence time of 30 hours and the templates were applied with
50 Hz < f0 < 1500 Hz and −f /τs < f˙ < f /τs where τs > 1000 years for signals below
400 Hz and τs > 8000 years for signals above 400 Hz. In the most sensitive region of the
LIGO detector, between 125 Hz and 225 Hz, the 90% upper limit has been measured to
be h90%
= 3 × 10−24 for all continuous GW sources. These results have been obtained
0
by analyzing S5.
The F-statistic search has also been applied to the known Low Mass X-Ray Binary
(LMXB) Scorpius X-1 (Sco-X1) [23]. Sco-X1 is a neutron star orbiting a low-mass companion with a period of 18.9 hours. The sky-location of Sco-X1 is well known due to
X-ray observations but the binary orbital parameters have large uncertainties. Also, the
rotational frequency of the neutron star is unknown as well as the associated frequency
derivatives. This search is limited to 6 hours of coherent integration due to the compu-
57
Chapter 4. Continuous waves analysis methods
tational constraints and has been performed on the frequency bands 464 − 484 Hz and
= 1.7 × 10−22 and
604 − 624 Hz 2 . The upper limits computed from this search are h95%
0
h95%
= 1.3 × 10−21 for both bands respectively. Note that this result has been produced
0
by analyzing the second LIGO science run (S2) which had limited sensitivity.
When attempting to perform an all-sky search for GWs from neutron stars in binary
systems, one must be able to create templates in a 13 + s dimensional parameter space.
It has been estimated [59] that the number of templates required for performing an
all-sky search is about 1026 . This number applies to a search for neutrons stars in binary
systems where sky-coordinates α and δ are fixed. Furthermore, there are constraints on
the maximum eccentricity emax = 0.8, semi-major axis ap = 1011 cm, orbital angular
velocity ωmax = 10−4 rad/s and maximum GW frequency fmax = 1 kHz. The assumed
5
coherence time is 107 seconds. In this case the number of templates scales with a5p , fmax
and linearly with T . When attempting to perform an actual all-sky search, the number
of templates will scale as a large power of T . This clearly shows that a coherent all-sky
search for GWs from neutron stars in binary systems is computationally unfeasible.
4.3
Semi-coherent searches
Since all-sky searches for GWs are computationally challenging or in some cases unfeasible, several computationally less demanding search analysis methods have been developed. Such methods cannot obtain the same sensitivity as the coherent searches. These
analysis methods generally utilize short stretches of coherent integration and combine
the results per coherent stretch in a incoherent way. Two different semi-coherent analysis
methods have been developed, called the ‘Hough search’ [63] and the ‘PowerFlux’ search
[64].
These methods are based on summing measures of strain power from many coherent stretches. Each method also corrects explicitly for sky-position dependent Dopplermodulations of the apparent source frequency due to Earth’s motion as well as for the
spin-down as shown schematically in Fig. 4.3. This requires a search in a four-dimensional
parameter space over the parameters f0 , f (1) , α and δ. The PowerFlux method also includes the polarization angle ψ in the search.
The PowerFlux and Hough methods mainly differ in the way the measures of strain
power are defined. PowerFlux measures strain power as the measured power per frequency bin weighted with the beam-pattern functions for circular and linearly polarized
gravitational waves. The detection statistic here is the estimated amplitude for linear
or circular polarized gravitational waves. The Hough method measures strain power by
selecting frequency bins where the power exceeds a predetermined threshold. The Hough
detection statistic is the number count of the selected frequency bins. The Hough and
PowerFlux methods of combining separate FFTs are based on the same principle and
thus the sensitivity will scale in similar ways as a function of observation time.
The Hough search is based on dividing the total observation time, Tobs , into N seg2
The interval has been chosen since Sco-X1 exhibits quasi-periodic oscillations and as discussed in
Ref. [23], these QPOs might be linked to the rotational frequency of the neutron star.
58
4.3. Semi-coherent searches
Figure. 4.3: An illustration of the discrete frequency bins of the short Fourier transforms (SFTs) of the data versus the discrete start time of the SFT. The
dark pixels represent the signal in the data. Its frequency changes with
time due to the Doppler shifts and spin-down. By sliding the different
SFTs in frequency, the signal bins can be lined up and summed after appropriate weighting to create a detection statistic.
ments of length T which are Fourier transformed. By applying a threshold on the power
spectrum certain bins will be selected for each segment. These selected bins produce a
distribution in the time-frequency plane. This distribution will contain all the Doppler
shifts and spin-downs discussed in section 4.1. The Doppler modulation of the signal
depends on the sky position of the source. Since this dependence is known a priori (Eq.
(4.8)), the sky position (α, δ) consistent with a certain measured f (t) can be computed3 .
Furthermore, the spin-down parameters introduce another shift in the time-frequency
plane. Also for this effect the functional dependency is known: a linear decrease of the
frequency for a single spin-down parameter, a quadratic decrease for two spin-down parameters, etc. The Hough search uses the Hough transform4 to sum the frequency bins
consistent with a certain sky position and spin-down in order to generate a detection
statistic.
The sensitivity of a Hough search with a false alarm rate of 1% and a false dismissal
rate of 10% is computed [63] to be
r
r
8.54 Sn
Sn
1/4
= 8.54N
.
(4.21)
h0 = 1/4
N
T
Tobs
The second equality is valid under the assumption that Tobs = N T which is not the case
if there are gaps in the data. Note that the power of the signal must be contained in a
3
Strictly speaking the set of sky positions consistent with a certain f (t) form a circle in the celestial
sphere centered around the vector ~v .
4
The Hough transform is a robust parameter estimator of multidimensional patterns in images
initially developed by Paul Hough [65] to analyze bubble chamber pictures at CERN.
59
Chapter 4. Continuous waves analysis methods
single frequency bin. Section 4.1 shows that the maximum coherence time is about 30
minutes for a 1 kHz signal. It is also possible to perform the Hough search on so-called
demodulated data which are data which have been resampled such that the Doppler
shifts have been corrected for. In that case there is no such restriction on T . However
such an approach would require much more CPU power as it involves generating signal
templates for each point in the sky.
The best Hough upper limit result for the fourth LIGO science run (S4) is published
in Ref. [64] to be h95%
= 4.25 × 10−24 . This result has been obtained with T = 30
0
minutes, Tobs = 29.5 days and N = 2966 in the band 140.00 to 140.25 Hz using the data
from H1, H2 and L1.
As stated before, a coherent all-sky search for GWs from neutron stars in binary
systems is computationally unfeasible. A semi-coherent approach based on the principle
of the signal not leaving a single frequency bin like the Hough search will yield a short
coherence time. Furthermore, the parameter space will increase to 13 + s parameters.
There are currently two analysis methods which target unknown neutron stars in binary
systems: the ‘TwoSpect’ method and the ‘Polyomial Search’ method.
TwoSpect, described in Ref. [66], uses short Fourier transforms such that the power
of the signal remains in a single frequency bin. From these spectra, a time-frequency
plot is computed which will contain the Doppler shift induced frequency variation of the
signal. The method then exploits the periodicity of the Keplerian orbit by performing
another Fourier transform on this time-frequency diagram. The signal will be concentrated around the central GW frequency and its Doppler shift induced harmonics. When
first demodulating the Doppler shifts induced by Earth’s motion, these harmonics are
purely determined by the binary orbit of the neutron star. These harmonics are then
summed with an incoherent harmonic summing technique which identifies certain regions of interest in the parameter space. Subsequently, templates are applied to these
regions of interest and a detection statistic is generated by weighted summing of the
second Fourier power. The parameter space used for these templates are the GW frequency f , the binary period P and the modulation depth (e.g. the frequency difference
of the signal between the apastron and periastron) ∆f .
The TwoSpect search is based on the same principles as the other incoherent searches:
weighted summing of a statistic where the weights are defined by the templates applied.
The sensitivity will scale as
r
Sn
,
(4.22)
h0 ∝ N 1/4
Tobs
where the proportionality constant is defined by how many templates are applied, what
the false alarm and false dismissal probability is, etc. Note that TwoSpect is less sensitive
to eccentric orbits and is limited to relatively slow binaries since the search is dependent
on containing the signal in a single frequency bin. This thesis presents ‘Polynomial
Search’ developed at Nikhef. Chapter 5 gives the details of Polynomial Search together
with the sensitivity estimates and tests on simulated gravitational wave signals in white
noise.
60
Chapter
5
Polynomial Search
In section 1.3.1 it has been discussed that non-axial symmetric neutron stars emit continuous gravitational waves. The amplitude of these waves are orders of magnitude lower
than the noise level of the LIGO and Virgo detectors. However, when the continuous
signal√is integrated over an observation time T , the signal-to-noise ratio will increase
with T which will allow us to obtaining a lower detection limit.
From the ATNF pulsar catalog shows that the majority of the known pulsars with
rotational frequencies above 10 Hz are in binary systems as can be seen in Fig. 1.3. Furthermore, the gravitational wave amplitude for a neutron star with a given quadrupole
mass moment scales with f 2 as shown in Eq. (1.15). Thus an all-sky analysis method
specifically targeted towards the detection of gravitational waves from neutron stars in
binary systems is desirable. However as stated in section 4.1, the Keplerian orbit of the
neutron star introduces a Doppler shifts to the signal in addition to the Doppler shift
from the detector motion, causing a coherent analysis method to be unfeasible.
In order to target these unknown binary systems, the ‘Polynomial Search’ method has
been developed. Polynomial Search employs a limited number of templates (or filters)
and aims to increase the possible coherence time by more than an order of magnitude
compared to existing semi-coherent methods. This is accomplished by taking the templates to be Taylor expansions of the phase of the gravitational wave which will match
the signal over a certain part of the orbit. Furthermore, Polynomial Search is designed
to include highly eccentric orbits as well as completely circular orbits.
5.1
Doppler shifts of binary systems
The phase of the gravitational wave is Doppler shifted. Equation (4.20) shows the phase
as a function of all the Doppler shifts for isolated neutron stars. The magnitude of these
shifts (the Earth’s daily and yearly motion as well as the spin-down) have been discussed
in section 4.1. The fourth contribution to the Doppler shift is given by the Keplerian
orbit of the neutron star and the magnitude of the modulation will depend on the orbital
parameters.
Polynomial Search is based on the fact that the detected phase of the gravitational
61
Chapter 5. Polynomial Search
wave signal can be written as a Taylor expansion up to a certain order N during an
observation time T ,
N
X
φn n
Φ(t) ≈ 2π
t ,
(5.1)
n!
n=0
where φn are the Taylor components. The required number of orders to be taken into
account N , strongly depends on the coherence time T as well as on the binary parameters
which are to be covered by the search. To make statements about the limits and errors on
the Taylor parameters in the expansion shown in Eq. (5.1), a description of the relative
motion of the neutron star with respect to the center of mass of the binary system and
the relative motion of the detector with respect to the Solar System Barycenter (SSB)
must be made.
When assuming non-relativistic motion, the orbit of a neutron star can be computed
with Kepler’s laws. Figure 5.1 schematically shows the orbit of a pulsar in an eccentric
orbit. In this coordinate system, the position of the neutron star in polar coordinates r
and θ can be computed with the following equations
t−τ
2π
= E − e sin(E)
(5.2)
P
r
θ
E
1+e
tan
=
tan
(5.3)
2
1−e
2
(1 − e2 )a
r(t) =
.
(5.4)
1 + e cos(θ)
The polar coordinate θ is computed by by solving Kepler’s equation (Eq. (5.2)) for the
angle E (the eccentric anomaly) and substituting it into Eqs. (5.3). Note that in Eq.
(5.2) τ is the time of periastron passage (i.e. the time the NS passes point P in Fig.
5.1). The r coordinate is computed by solving Eq. (5.3) for θ and substituting it into
Eq. (5.4). Finally, the length of the semi-major axis, a can be computed from Kepler’s
third law:
r
2
3 G(M + mns )Pb
a=
,
(5.5)
4π 2
where G is Newton’s constant, M the mass of the companion object, mns the mass of the
neutron star and Pb is the period of the orbit. It can be seen that the (non-relativistic)
description of the orbit of a neutron star depends on four orbital parameters (P , a, e
and τ ). Combining these orbital parameters with an additional orbital inclination angle
i, makes the parameter space for binary orbits five dimensional.
In order to set upper limits on the possible observable phase changes in the signal
due to the Doppler shifts, the extreme values of the orbit and their time derivatives
must be computed. These extreme values occur in the periastron and can be computed
by combining Eqs. (5.2), (5.3) and (5.4) together with the fact that the Keplerian-orbit
induced phase modulation, ΦK , is given by
ΦK (t) = 2πf0 t +
62
2πf0
n̂ · ~rK (t).
c
(5.6)
5.2. Binary system test cases
Auxillary circle
y
NS orbit
NS
r
E
θ
µ
O
P
x
ae
a
Figure. 5.1: Schematic representation of the motion of a neutron star in an eccentric
orbit in the orbital xy plane. O denotes the center of the elliptical orbit
with the semi-major axis length a and eccentricity e as well as the center
of the circle with radius a. P is the periastron (the point of closest approach). NS is the neutron star which has polar coordinates coordinates
(r(t), θ(t)) as a function time. The origin of this polar and corresponding
Cartesian coordinate system is located at the center of mass, µ.
Note that the n̂ and ~rK are defined in Fig 4.2 and that |~rK | = r(t) as computed by
solving the Kepler equations.
5.2
Binary system test cases
In order to define a region of parameter space where Polynomial Search can be applied,
the maximum values of the Doppler-shifts will be computed for two extreme cases.
Figure 5.2 shows the orbital periods and eccentricities of known pulsars in binary systems
with spin frequencies above 20 Hz. To illustrate the computation of the limits of the
63
Chapter 5. Polynomial Search
P (days)
histo
M > 0.5 Msol
M < 0.5 Msol
103
102
10
1
10-1
10-7
10-6
10-5
10-4
10-3
10-2
10-1
1
Eccentricity
Figure. 5.2: Orbital period versus eccentricity for the known pulsars with frequencies
above 20 Hz in binary systems (total 78) from the ATNF pulsar catalog
[21]. The triangles indicate neutron stars with a companion with a median
mass M < 0.5M . The squares indicate neutron stars with a companion
with M > 0.5M .
parameters used in Polynomial Search, two extreme systems are considered with fgw = 1
kHz for both cases.
1. A 1.4 M neutron star orbiting an object with the same mass in a circular orbit
with P = 2 h.
2. A 1.4 M neutron star orbiting an object of 14 M in an eccentric orbit with
e = 0.6 and P = 6 h.
These two cases roughly correspond to the binaries located in lower right corner (high
eccentricity, small orbital period and high companion mass) and lower left corner (low
eccentricity and small orbital periods) of Fig. 5.2.
The parameters of the Keplerian orbit are displayed in Table 5.1. Figure 5.3 shows
parameter binary 1 binary 2
h0
f0
P
ap
e
10−27
1000 Hz
2h
1.65 ls
0
10−27
1000 Hz
6h
9.68 ls
0.6
Table. 5.1: The parameters used for the different test binaries.
64
5.2. Binary system test cases
y (ls)
x (ls)
the position of the neutron star in these binary orbits as a function of time. The x and
y coordinates of the neutron star are defined in Fig. 5.1. The origin of this system is
set on the center-of-mass of the binary system (see Fig. 5.1). Figure 5.4 shows the first
4
2
0
-2
-4
-6
-8
-10
-12
-14
0
6
4
2
0
-2
-4
-6
2
4
6
8
10
12
t (h)
0
2
4
6
8
10
12
t (h)
Figure. 5.3: The left panel shows the x coordinate of the neutron star in the circular
orbit (solid curve) and in the eccentric orbit (dashed curve). The right
panel shows the situation for the y coordinate. Note that both positions
are computed in light seconds (ls).
four derivatives of these orbits. The phase modulation of the gravitational wave signal
for these orbits is dominated by the Doppler shift of the binary motion. By using Eq.
(5.6) the maximum change in the n-th order derivative of the phase can be written as

2πf0


2πf0 t +
n̂ · ~rK |max for n = 0,


c


2πf0
d~rK dn ΦK for n = 1,
= 2πf0 + c n̂ · dt (5.7)
dtn max 
max

n


2πf0
d ~rK 

n̂ ·
for n > 1.
c
dtn max
n
In Eq. (5.7) d dtΦnK max is the maximum value for the nth derivative of the detected phase
n
on Earth and ddt~rnK max is the nth derivative of the position of the neutron star in the
direction of maximum variation and at the time of maximum variation (i.e. in the
periastron). Equation (5.7) allows for the interpolation of limits on Polynomial Search
parameters.
The maximum values for the first, second and third-order derivative of the phase
can be computed with the information from Fig. 5.4 together with Eq. (5.7). With
these values, the limits on Polynomial Search parameters, φn |max up to fourth order are
calculated and are shown in Table 5.2. From Fig 5.4 it can be seen that for the eccentric
binary the absolute values of the derivatives are smaller than for the circular system for
about 80 percent of the time. The maxima occur close to the periastron, resulting in
higher values for the extremal Taylor components.
The maximum values of the consecutive derivatives decrease as powers of the orbital
65
1000
×103
d2x(t)
(m/s2)
dt2
dx(t)
(m/s)
dt
Chapter 5. Polynomial Search
500
400
200
0
-200
-400
-600
-800
-1000
-1200
-1400
0
0
-500
2
4
6
8
10
12
t (h)
d4x(t)
(m/s4)
dt4
d3x(t)
(m/s3)
dt3
-1000
0
1.5
1
0.5
2
4
6
8
10
12
t (h)
2
4
6
8
10
12
t (h)
0.005
0.004
0.003
0.002
0
0.001
-0.5
0
-1
-0.001
2
4
6
8
10
12
t (h)
0
×103
d2y(t)
(m/s2)
dt2
dy(t)
(m/s)
dt
-1.5
0
1500
1000
500
0
2
4
6
8
10
12
t (h)
0.5
d4y(t)
(m/s4)
dt4
d3y(t)
(m/s3)
dt3
-500
0
0
-0.5
2
4
6
8
10
12
t (h)
0.004
0.003
0.002
0.001
0
-1
-0.001
-1.5
-0.002
-0.003
-2
0
800
600
400
200
0
-200
-400
-600
-800
0
2
4
6
8
10
12
t (h)
-0.004
0
2
4
6
8
10
12
t (h)
Figure. 5.4: The first four derivatives of the x and y coordinates as a function of time
for the neutron star in the circular orbit (solid curve) and the eccentric
orbit (dashed curve).
frequency ωorbit |max . The latter is evaluated at a certain point in the orbit, torbit , where
66
5.3. Constructing the filters and the correlations
Parameter
Circular orbit
φ1 |max (Hz)
1000 ± 1.45
φ2 |max (Hz/s) ±1.26 × 10−3
φ3 |max (Hz/s2 ) ±1.10 × 10−6
φ4 |max (Hz/s3 ) ±9.62 × 10−10
Eccentric orbit
1000 ± 5.27
±2.77 × 10−3
±5.07 × 10−6
±1.74 × 10−8
Table. 5.2: The extremal values of the polynomial search parameters for the circular
and eccentric orbit (see text) for gravitational wave frequencies up to 1
kHz.
the derivatives are maximal. This maximal orbital velocity is given by
~r(torbit ) × ~v (torbit ) ,
ωorbit |max = |~ω |max = r(torbit )2
(5.8)
where ~r and ~v are the velocity vector and position vector of the neutron star, respectively.
In the circular case ωorbit = 2π
= ωorbit |max . When the coherence time Tcoh is sufficiently
P
1
smaller than ωorbit | , the higher-order contributions can be ignored1 .
max
5.3
Constructing the filters and the correlations
Instead of attempting to create templates in the 13 + s dimensional parameter space,
Polynomial Search employs an empirical phase model given by the Taylor expansion of
the phase and creates templates in a parameter space given by the components φn . As
stated in section 5.1, the Doppler shift induced phase modulation is proportional to the
powers of the maximum orbital frequency
n d φ
∝ (ωorbit |max )n ,
(5.9)
dtn max
where ωorbit |max represents the angular frequency at a certain point, torbit , in the orbit
where the velocity is maximal. For circular orbits the maximum contributions, occurring
in either the peri- or apastron, of the nth and the (n + 2)th derivative coincide with the
roots of the (n + 1)th derivative2 . Therefore, the maximum of the first and third-order
derivatives occur at the same value of torbit , where the third-order contribution is reduced
with a factor (ω)3 . This implies that the next contributing order will be the fifth order
which will be reduced by (ω)5 . This, together with the fact that in Ref. [67] is has been
computed that the number of templates to consider scales3 as T n(n+1)/2 , leads to the
P
.
In the case of the circular orbit this statement simply says Tcoh 2π
For example: in the periastron, the orbital velocity (the first derivative) is maximal, but the derivative of the orbital velocity is zero.
3
Strictly speaking this scaling is valid for a full bandwidth search for all allowed values for the filter
parameters which obey the Shannon-Nyquist theorem.
1
2
67
Chapter 5. Polynomial Search
choice to truncate the expansion at the third-order in the current implementation of
Polynomial Search. The phase, up to third-order, is given by
α 2 β 3
~
Φ(t; ξ) = 2π φ0 + f0 t + t + t ,
(5.10)
2
6
where the vector ξ~ = (φ0 , f0 , α, β) lives in the four dimensional template (or filter)
~ is given by
parameter space. Note that the frequency f (t, ξ)
~
~ = 1 dΦ(t; ξ) = f0 + αt + β t2 .
f (t; ξ)
2π dt
2
(5.11)
The filters (based on Eq. (4.2))
~ = sin Φ(t; ξ)
~
F (t; ξ)
(5.12)
are created in this parameter space and are to be applied to the data in stretches of
time T . These filters are valid representations of the signal shown in Eq. (4.2) as long
as T is short enough such that the beam-pattern functions are constant. The maximum
values for α and β can be calculated from the Kepler orbits assuming that the dominant contribution in the frequency dependence on time arises from the Doppler shifts
as discussed in section 5.1. The values were calculated for the two specific binaries and
are shown in Table 5.2.
Polynomial Search is performed by stepping in parameter space and creating a collection of filters. The filters are then applied to the data by shifting the data over the
~ could be computed in the
filter and computing the correlation. The correlation C(τ ; ξ)
time domain as shown in Eq. (A.3) but this requires the integral to be evaluated for each
value of the lag. This computation will be much more computationally efficient when
done in the frequency domain. When combining the computation of the correlation in
the frequency domain with the expression of the signal-to-noise for matched filters, an
expression for the correlation is derived in appendix B. The result of this derivation is
Z ∞
~
D̃(f )F̂ ∗ (f ; ξ)
~
q
C(τ ; ξ) =
e−2πif τ df,
(5.13)
1
−∞
S
2 n
where τ is the lag, D̃(f ) is the Fourier-transformed data and F̂ ∗ denotes the complex
conjugate of the Fourier-transformed filter normalized to its total power. Furthermore,
the power spectral density of the noise has been taken to be independent of frequency
(white). In the case of discretely sampled data with sample frequency fs,data during time
T
Nf
1 X D̃(fk )F̂ ∗ (fk ) −2πifk τj
q
C(τj ) =
e
,
(5.14)
T k=0
1
S
2
n
where τj is the j th time sample, k denotes the frequency bin number, Nf = fs,data T /2 is
the number of frequency bins and fk is the k th frequency bin.
68
5.3. Constructing the filters and the correlations
Equation (5.13) shows that the correlation between a filter and the data can be
computed for all lags by a single inverse Fourier transform. Note that by using the
correlation theorem (Eqs. (A.3) and (A.4)) integration time T , Eq. (5.13) is equivalent
to the expression
r Z T
2
~ =
~
C(τ ; ξ)
D(t)F̂ (t − τ ; ξ)dt
(5.15)
Sn 0
as explained in appendix B. It can be seen from Eq. (5.15) that the correlation statistic
can be considered Gaussian distributed with unit variance and can also be interpreted
as a signal to noise ratio. Furthermore, the filter is a function of t − τ , implying that the
filter is effectively shifted in time with lag τ . With the substitution t0 = t − τ , the phase
of each shifted filter can be written as

!
!


α
β
β
2
3
2
Φ(t0 ) = 2π 
 φ0 − f0 τ + 2 τ − 6 τ + f0 − ατ + 2 τ t

|
{z
} |
{z
}
φ0 (τ )
f0 (τ )

1
+
2
!
α − βτ
|
{z

1 3
t + βt 
,
6 

2
}
(5.16)
α(τ )
From Eq. (5.16) it follows that the lag parameter creates an interdependency of the filter parameters. This property can be exploited to avoid generating filters for each point
in parameter space and will drastically reduce the amount of CPU time and memory
needed.
In generating and applying the filters it is important to minimize the number of computational steps. This is accomplished in two ways. Firstly, the dependence of the lag
parameter τ is calculated by time shifting (employing a phase shift in frequency space;
see appendix B). Secondly, each filter is frequency shifted to obtain the dependence of
f0 . The computational steps will be elaborated upon next.
In order to use the lag parameter as a time shift, it must be possible to shift the filter
within the block of data between 0 and T . Thus the filters are zero-padded for t > T /2
and only the lags 0 < τ < T /2 are considered, implying that Eq. (5.16) is valid for
τ < t0 < τ + T /2 and that values for τ > T /2 should be ignored. For each value of the
lag a new shifted filter will be generated. The dotted curve in figure 5.5 schematically
shows the frequency of a typical filter as a function of time by using Eq. (5.11). The
dashed curve in Fig. 5.5 represents the frequency of a potential gravitational wave signal
as a function of time. As can be seen, the filter does not match the frequency evolution
of the signal. However, the filter matches the signal when it is shifted in time by τ = 300
s (solid curve) as the horizontal arrow indicates. Furthermore, while the parameters of
the signal (shown in Fig. 5.5) differ from those of the filter, the time shift causes the
filter to match due to the functional dependencies of the filter parameters on the lag
69
f (Hz)
Chapter 5. Polynomial Search
data: f0 = 4.42, α = -0.0029, β = 1e-05
8
time-shifted filter: f0 = 4.0, α = 0.0001, β = 1e-05, τ = 300.0
filter: f0 = 4.0, α = 0.0001, β = 1e-05, τ = 0
7.5
frequency-shifted filter: f0 = 5.0, α = 0.0001, β = 1e-05, τ = 0
7
6.5
6
5.5
5
frequency shift
time shift
4.5
4
0
200
400
600
800
1000
1200
t (s)
Figure. 5.5: An example of the frequency and lag evolution of a typical filter. The dotted curve represents the unshifted filter. The dashed curve is an example
of what a gravitational wave signal may look like during observation time
T = 1200 s. The horizontal arrow indicates the time-shift (τ = 300 s)
needed for the filter to match the signal, resulting in the solid curve overlapping with the signal. The vertical arrow indicates a potential frequencyshift (a shift in f0 ) such that the same filter can be used to match a signal
with higher frequency. The legend shows the actual filter parameters used
in this figure.
shown in Eq. (5.16).
Since the gravitational wave signal shown in Fig. 5.5 can match arbitrary filter parameters, Polynomial Search must be able to search over the parameter space (e.g. the
stepping in parameter space is discussed in more detail in section 5.5). This is done
by creating a series of filters with different parameters each representing a point in parameter space. Naively speaking, each point in parameter space requires the numerical
re-computation of the filter function F (t). While this is true for each new value for α and
β, the f0 parameter can be varied by simply shifting the power spectrum of the filter
in the frequency direction. This is shown in Fig. 5.5 by the vertical arrow, depicting
the same filter as before but shifted in the frequency direction by 1 Hz. In practice, the
filters are sampled at a sample frequency fs,filter fs,data . In this way, the filters can be
applied to different frequency intervals of the data by comparing a different part of the
power spectrum of the data to the filter. This process is shown schematically in Fig. 5.6.
In order to make sure the frequency of the filter does not fall in the 0 Hz or fN bins, a
buffer zone has been chosen.
~ for
When discretizing the filter, nlags = T fs,filter /2 values of the correlation C(τ, ξ)
the lags between 0 and T /2 are obtained, where each value of τ represents a new filter
with parameters φ(τ ), f0 (τ ) and α(τ ) as shown in Eq. (5.16). In other words: in this
70
5.4. Coherence length
Figure. 5.6: A schematic representation of the power spectrum of a typical filter (left)
and the power spectrum of a block of data (right). In the left panel, the
hatched region represents the ‘buffer’ zone, fN,filter = fs,filter /2 the Nyquist
frequency and ∆ffilter is the active frequency range of the filter. In the
right panel the gray area represents the active frequency range of the filter
and fN,data = fs,data /2 the Nyquist frequency of the data.
~ yields a collection of nlags correlations which are obtained by
way a single filter F (t; ξ)
a single computation of the inverse Fourier transform.
In order to reduce the number of correlations that must be processed, only the maximum value of C from all the lags from Eq. (5.13) and the corresponding values for
τ, f0 , α and β are stored. Note that the value for φ0 has been chosen to be 0. The
freedom to choose an arbitrary value for φ0 arises from the fact that all values of φ0
are sufficiently covered when all lags τ are considered. Since C can also be interpreted
as a signal-to-noise ratio, a threshold in terms of the SNR can be set. When applying
such a threshold to the collection of maxima the resulting numbers are called hits. Hits
are displayed by plotting the frequency of the filter as a function of time f (t; ξ~hit ) by
using Eq. (5.11), where ξ~hit represents the parameters of the filters for which the correlation exceeds the threshold. Such plots are called hit maps and examples of hit maps
of simulated gravitational wave data are shown in Figs. 5.32 and 5.34.
5.4
Coherence length
In order to determine the optimal coherence length of the filters used in Polynomial
Search, a criterion for matching a filter to a known signal must defined. As the objective
of this study is to determine the optimal coherence length of the filters it is advantageous
to consider the correlation in the time domain. The correlation between a normalized4
~ over a coherent integration
signal (without noise) ŝ(t) and a normalized filter F̂ (t; ξ)
4
The definition of a normalized signal is given in appendix A.
71
Chapter 5. Polynomial Search
time of Tcoh is given by
Z
cŝF̂ =
Tcoh
~
ŝ(t)F̂ (t; ξ)dt.
(5.17)
0
Note that since the lag is not a filter parameter (e.g. the lag just allows for an efficient
computation of the correlation) it is set to 0 in Eq. (5.17). The functions ŝ and F̂ are
periodic functions of time. This implies that −1 < cŝF̂ < 1; the correlation will be +1
if F̂ matches ŝ during Tcoh and −1 if F̂ matches ŝ but is π out of phase during Tcoh .
Furthermore, if ξ~ is such that the filter does not match the signal, the correlation will lie
in between these extremal
values. As discussed in appendix B, the SNR of an optimal
√
filter will scale as Tcoh . Therefore, the quality factor is defined as
Z Tcoh
p
~
~
Q(Tcoh ; ξ) ≡ Tcoh
ŝ(t)F̂ (t; ξ)dt.
(5.18)
0
~ is the normalized
In Eq. (5.18) ŝ(t) is the normalized signal without noise and F̂ (t; ξ)
√
filter. Note that if a filter correlates fully during the integration time, then Q = Tcoh .
This corresponds to the SNR scaling for a fully coherent search.
As a demonstration, the optimal length of the coherence time Topt of the filters was
calculated for the binary systems mentioned in section 5.1, where Topt is defined as the
time length of the filter for which the quality factor is maximal. The orientation of
the binary system with respect to the detector is in all cases chosen for the worst-case
scenario: the contribution of the neglected orders in the phase derivatives is maximized.
For example, for a first order filter (f0 6= 0 and α = β = 0) the second order derivative
of the phase (n = 2 in Eq. (5.7)) is chosen to be maximal. This is done by maximizing
2
n̂ · ddt~r2K which would result in taking n̂ in the direction of the acceleration. Furthermore,
the filter parameters ξ~ are computed to match the derivatives of the phase at the time
in the orbit torbit , where these derivatives are maximal. The integration over the filter
and data multiplication shown in Eq. (5.18) is then taken from torbit − Tcoh
to torbit + Tcoh
.
2
2
Note that this represents the worst case scenario in terms of the phase modulation; the
actual phase modulation will always be less.
For the aforementioned circular orbit, Fig. 5.7 shows the quality factor versus the
coherence time for filters up to fourth order (γ/24t4 ). From this figure the values for
Topt have been inferred and are displayed in Table 5.3. These numbers give the lower
limit for the optimal coherence time for a Polynomial Search for neutron stars in binary
systems with circular orbits with periods larger than 2 hours for different orders of the
filter.
For a real search, the filters can be applied with a longer coherence length and
still remain coherent. From Table 5.3 it can be seen that the gain in quality factor
when adding a third-order term to the phase model is about a factor of 1.6. Adding an
additional fourth-order term will increase the quality factor with a factor 1.3, while the
10
number of extra filters that must be computed will scale as Tcoh
[67].
For the second, eccentric binary, Q depends on torbit since the values of the derivatives
depend on the position in the orbit. Figure 5.8 shows the three-dimensional plots of the
time in the orbit versus the coherence time with the Q-factor as the greyscale for the
72
Q
5.4. Coherence length
30
4th order
3rd order
2nd order
1st order
25
20
15
10
5
0
200
400
600
800 1000 1200 1400 1600 1800 2000
Tcoh (s)
Figure. 5.7: The quality factor versus the coherence time of the filter for different orders of the filter. The quality factor has been computed with the worstcase orientation of the system.
Maximum order considered Topt (s) Qmax
1
2
3
4
32.7
195
511
955
5.07
12.9
21.2
29.4
Table. 5.3: The optimal coherence length of the filters for neutron stars in binary systems with a circular orbit (first binary).
eccentric orbit. It can be seen from this figure that the value of Q depends on where
in the orbit the filter is applied. This is to be expected since Fig. 5.4 shows that the
derivatives of the phase are strongly dependent on the orbital phase.
Figure 5.9 shows that the optimal coherence length is approximately 250 seconds in
the periastron for a third-order filter. With this coherence time, a third-order polynomial
filter will be coherent at any point in the orbit for 250 seconds. In other words, when
taking Tcoh = 250 seconds all gravitational waves with Taylor components up to 3rd order
given in Table 5.2 are guaranteed to be covered. However, when relaxing this constraint
the coherence time can be taken longer (1900 seconds for a 3rd order filter). Another
point is that Figs. 5.8 and 5.9 have been created assuming the worst case orientation
for each point in the orbit and each applied order of parameter space which means that
in reality the actual Topt should be higher.
The coherence length chosen in the analysis is Tcoh = 600 seconds. Therefore, the
length of the data stretches is T = 2Tcoh = 1200 seconds, since the filter is zero padded
73
Chapter 5. Polynomial Search
50
103
60
Tcoh (s)
Tcoh (s)
60
50
103
40
40
30
2
10
20
30
102
20
10
10000
15000
20000
torbit (s)
0
60
50
103
10
0
5000
10000
15000
20000
torbit (s)
50
103
40
40
30
30
102
10
0
0
60
Tcoh (s)
5000
Tcoh (s)
10
0
10
102
5000
10000
15000
20000
torbit (s)
20
20
10
10
0
10
0
5000
10000
15000
20000
torbit (s)
0
Figure. 5.8: The quality factor Q, as a function of the time in the orbit torbit and the
coherence length Tcoh computed for the eccentric binary. The Q-factor for
a first, second, third and fourth order filter are shown in the top left, top
right, bottom left and bottom right panel, respectively. The quality factor
has been computed for the worst-case orientation of the system.
for half the time.
5.5
Step size in parameter space
Polynomial Search uses filters constructed from a four-dimensional parameter space.
The aim of the search is to reconstruct the phase evolution of a gravitational wave by
approximating it with the phase evolution of the filters that give a hit. These filters
are applied to the data by computing the correlation statistic defined in Eq. (5.13).
This correlation statistic can be interpreted as an inner product between two vectors
in parameter space. In order to determine a step size in the parameter space, the inner
product between two neighboring filters, represented by ξ~i and ξ~j , is computed. This
inner product gives rise to a metric for the parameter space. The separation between
74
Topt (s)
5.5. Step size in parameter space
104
1st order
2nd order
3rd order
4th order
103
102
10
0
5000
10000
15000
20000
torbit (s)
Figure. 5.9: The optimal coherence length Topt versus the time in the orbit for maximum Q at that time in the orbit.
two filters in parameter space is defined as
∆ξ~ij = ξ~i − ξ~j = (∆f0 , ∆α, ∆β),
(5.19)
where ∆f0 , ∆α and ∆β represent the difference in their respective parameter values.
Since Polynomial Search is based on the assumption that a potential gravitational wave
signal will behave like a polynomial during the coherent time |∆ξ~ij | can be interpreted
as the mismatch between the potential signal and the nearest filter.
In order to perform Polynomial Search, a filter bank must be created which implies
that the parameter space must be discretized. To find the optimal step size one can
attempt to compute the metric corresponding to the inner product. This metric has
been computed in the literature [67] for a binary search and restricted to low eccentric
orbits with a different way of parameterizing the phase. Their phase expansion was
written as
(m)
∞
j
X
uj (m)
(m)
t − tmid ,
(5.20)
φ (t) =
j!
j=0
(m)
where m is the data segment index, the uj parameters are the instantaneous phase
(m)
derivatives and tmid is the midpoint of the mth segment. It can be seen from Eq. (5.20)
(m)
that this Taylor expansion is centered around a certain point tmid . The difference with
Polynomial Search is that the τ parameter to match the filter with the data at the
optimal time is obtained by taking the maximum correlation of each filter-data compar(m)
ison. This essentially implies that tmid would be an additional parameter in Polynomial
Search. Furthermore, the step size in this parameter is implicitly chosen by the sample
frequency and the coherence time of the filter. These reasons make the metric computed
75
Chapter 5. Polynomial Search
in Ref. [67] unsuited for usage in Polynomial Search. The lag makes analytic computation of the parameter space metric non-trivial. Consequently, a different approach has
been adopted.
In order to demonstrate the correlation of the filters as a function of the step sizes,
slices from the parameter space metric are computed by considering the correlations of
a filter bank with a data set with known filter parameters. This filter bank has been
generated with filters with small step sizes in α, β and f0 .
Table 5.4 shows the different data sets to which the filter banks were applied and
Table 5.5 shows the different filter banks which have been applied to the data sets.
Data set f0 (Hz) α (Hz/s) β (Hz/s2 )
1
2
3
4
600
600
600
600
4 × 10−6
4 × 10−5
1 × 10−3
1 × 10−3
1 × 10−7
2 × 10−6
2 × 10−6
1 × 10−7
Table. 5.4: Data sets used to determine the step sizes in parameter space by applying a
filter bank to them. The data are sampled at 4000 Hz for a stretch of 1200
seconds.
Filter bank frange , δf0 (Hz) αrange , δα (Hz/s)
1
2
3
4
0.5, 10−3
2, 10−3
1, 10−3
1, 10−3
10−4 ,
10−3 ,
10−4 ,
10−3 ,
10−6
10−5
10−6
10−5
βrange , δβ (Hz/s2 )
4 × 10−7 , 8 × 10−9
10−6 , 5 × 10−8
10−6 , 5 × 10−8
3 × 10−7 , 5 × 10−8
Table. 5.5: The parameters of the filter banks applied to the different data sets shown
in Table 5.4. The range and step size are given for each parameter per filter
bank which was applied to a particular data set. The central value of each
parameter is chosen to be the true value shown in Table 5.4. The data are
sampled at 4000 Hz for a stretch of 1200 seconds.
In order to visualize the parameter space, the filters from the filters banks from Table
5.5 are applied to the data sets from Table 5.4. The results are drawn as slices of parameter space, for example the correlation statistic as a function of the value for f0 versus α
for different slices of β as is shown for set 1 in Fig. 5.10. The remaining permutations of
the parameters are shown in Figs. 5.11 and 5.12. In these plots the gray scale shows the
correlation normalized such that a correlation of 1 means that the filter and the data
are equal. The plots for the remaining data sets can be found in appendix C.
When considering Figs. 5.10, 5.11 and 5.12 two observations can be made. Firstly,
around the true values for data set 1 (f0 = 600 Hz, α = 4 × 10−6 Hz/s and β = 1 × 10−7
Hz/s2 ) a band of values appears where the correlation is close to one. This is due to
76
5.5. Step size in parameter space
-3
×10 β (Hz/s^2) = -3e-07
0.1
0.5
0
599.9
600
×10
0.1
-3
600.1
β (Hz/s^2) = -6e-08
599.9
600
600.1
×10-3 β (Hz/s^2) = 1.8e-07
0.1
0.5
0
599.9
600
600.1
599.9
600
×10
0.1
600.1
β (Hz/s^2) = 2e-08
599.9
600
×10
0.1
600.1
β (Hz/s^2) = 2.6e-07
600
1
0.5
0
599.9
600
×10
0.1
600.1
β (Hz/s^2) = 1e-07
1
0.5
0
599.9
600
600.1
×10 β (Hz/s^2) = 3.4e-07
0.1
-3
1
0.5
0
-3
×10 β (Hz/s^2) = -1.4e-07
0.1
-3
1
0.5
0
599.9
1
0.5
-3
1
β (Hz/s^2) = -2.2e-07
0
-3
1
0.5
0
×10
0.1
-3
1
600.1
0.5
0
599.9
1
600
600.1
Figure. 5.10: The step size plots for data set 1 (f0 = 600 Hz, α = 4 × 10−6 Hz/s
and β = 1 × 10−7 Hz/s2 ). The parameters of the filter bank used in
this figure are shown in Table 5.5. The plots show α (vertical axis) versus
f0 (horizontal axis) for different values of β. The color codes shows the
correlation, normalized to unity.
the fact that the interdependency shown in Eq. (5.16) together with the lag as a free
parameter can be used to compensate for a certain amount of mismatch. This effect can
be clearly seen in Fig. 5.11 when comparing the panel where α = 4 × 10−6 (the middle
one) to the following 4 panels: the mismatch in α can be (partially) compensated by
taking a lower value for β and a higher value of f0 . This compensation only works if τ is
left to be a free parameter and the maximum value of the correlation is taken for each
computation of the correlation of a filter with the data.
Another observation from Figs. 5.10, 5.11 and 5.12 is that the parameter space slices
are not symmetrical around the true injected value. This is due to the fact that the
injected signal has a positive α and β and thus filters with negative parameters will not
fit as well as filters with positive parameters regardless of the value of the lag as no
amount of shifting in time will increase the correlation.
The requirements on the step sizes are such that for only signal in the data, at least
77
Chapter 5. Polynomial Search
×10
-6
0.5
α (Hz/s) = -7.6e-05
×10
-6
1
0.5
α (Hz/s) = -5.6e-05
0.5
0
600
×10
-6
0.5
600.1
α (Hz/s) = -1.6e-05
599.9
600
0.5
×10
600.1
α (Hz/s) = 4e-06
600
×10
-6
0.5
600.1
α (Hz/s) = 4.4e-05
599.9
600
0.5
×10
599.9
600.1
α (Hz/s) = 6.4e-05
0.5
0
600.1
0.5
×10
600.1
α (Hz/s) = 2.4e-05
1
0.5
599.9
600
×10
-6
1
0.5
599.9
600.1
α (Hz/s) = 8.4e-05
0.5
0
600
600
0
-6
1
599.9
0.5
0
599.9
1
0.5
-6
1
0.5
0
α (Hz/s) = -3.6e-05
0
-6
1
0.5
0.5
0
599.9
×10
-6
1
1
0.5
0
600
600.1
599.9
600
600.1
Figure. 5.11: The step size plots for data set 1 (f0 = 600 Hz, α = 4 × 10−6 Hz/s
and β = 1 × 10−7 Hz/s2 ). The parameters of the filter bank used in
this figure are shown in Table 5.5. The plots show β (vertical axis) versus
f0 (horizontal axis) for different values of α. The color codes shows the
correlation, normalized to unity.
one filter should yield a correlation larger than a certain cutoff value. This optimal step
should be independent of the shape of the potential signal. When comparing the step
size plots made with data from data set 1 (Figs. 5.10, 5.11 and 5.12) to the step size
plots of the other data sets (appendix C), it can be seen that the shape of the signal
also changes the regions in parameter space where the correlation is non-zero. Therefore,
there is no single step size where the correlation is optimal and independent of the signal.
Choosing a fixed sample frequency for the filter effectively sets a step size in the lag. The
correlation is obtained for all lags in one operation (i.e. the inverse Fourier transform
in Eq. (5.13)) and due to taking the maximum of C, the filter parameters f0 , α and β
are computed for a certain value of τmax . Thus the step sizes have been computed in a
more empirical way.
By recording the maximum correlation, Polynomial Search automatically finds the
value of the lag where the filter matches best with a potential signal. When |∆ξ~ij | is
78
5.5. Step size in parameter space
×10
-6
0.5
f 0 (Hz) = 599.96
×10
-6
1
0.5
f 0 (Hz) = 599.97
0.5
0
0.5
×10
f 0 (Hz) = 599.99
×10
0.1
×10
-6
0.5
f 0 (Hz) = 600.0
0.5
0
0
×10
f 0 (Hz) = 600.02
0.5
0
0.5
×10
f 0 (Hz) = 600.03
×10
0.1
0
×10
-6
1
0.5
0
×10
0.1
×10
0.1
-3
f 0 (Hz) = 600.04
0.5
0.5
1
0.5
0
-3
0
1
0.5
-3
-6
0
f 0 (Hz) = 600.01
0
×10
0.1
1
×10
-6
1
×10
0.1
-3
0
0.5
-3
0.5
×10
0.1
0
-6
0.5
-3
0
1
1
0
-3
-6
f 0 (Hz) = 599.98
0.5
0
0
×10
-6
0.5
1
×10
0.1
-3
0
×10
0.1
-3
0
Figure. 5.12: The step size plots for data set 1 (f0 = 600 Hz, α = 4 × 10−6 Hz/s and
β = 1 × 10−7 Hz/s2 ). The parameters of the filter bank used in this figure
are shown in Table 5.5. The plots show α versus β for different values of
f0 and the color codes show the correlation, normalized to unity.
small, the assumption is made that at this lag the filter and the adjacent filter (or the
data) are matched in phase, frequency and (to a lesser extend) in α. Using these assumptions, the step in β is defined as the step that leads to a phase difference of π/2
radians at the end of the filters. This yields a step size of
1
π
= 2π δβ
2
6
3
T
96
→ δβ = 3 ,
4
T
(5.21)
where δβ is the step size in β and T is the duration of one FFT of the data in seconds.
The step size shown in Eq. (5.21) means that the phase of the filter can maximally
deviate π/4 from the signal at the end of the filter. Note that T = 2Tcoh where Tcoh is
the coherence time of the filter as described in section 5.4.
The step in α can be derived from Eq. (5.16) by noting that α(τ ) = α − βτ , meaning
that the lag will step through all α values between α and α − βτ . The step in α thus
79
Chapter 5. Polynomial Search


 |β|T for |β| > δβ
2
δα =
(5.22)
8


for
|β|
<
δβ
T2
where δα is the step size in α and the absolute value of β is taken since all step sizes are
defined to be positive. When |β| < δβ, i.e. when the β term does not contribute to the
phase difference anymore, the step in α becomes the minimal step. This minimal step is
defined, like the step in β, to lead to a phase difference of π/2 at the end of the filters.
When following Eq. (5.16) it would seem that f0 (τ ) gives the step size in f0 to be
2
T
α 2 + β2 T2 . However, this step size would allow for the possibility that a signal will
be missed which can be verified by considering Figs. 5.10, 5.11 and 5.12 and the ones
in appendix C. The figures show that it is not possible to have the maximum step in β
(δβ), the maximum step in α (δα) and the maximum step in f0 at the same time.
The way the step in f0 is computed is to assume that at a certain value for the lag,
τi , the filter will match φ0 and f0 in the middle. The phase difference at the end of the
filter will be
"
2
3 #
∆α T
∆β T
∆Φend = 2π
+
,
(5.23)
2
4
6
4
becomes
where ∆α and ∆β are the differences in α and β, respectively. By construction
∆β
∆Φend − 2π
6
3
T
π
<
4
2
(5.24)
due to the definition of the step in β. Note that α will traverse all values between α and
α + βτi , thus ∆α = βτi . In order to prevent the step in α becoming too large such that
a potential signal is missed, τi is limited to such a value that the phase difference at the
end of the filter does not exceed π/2. Substituting Eq. (5.23) in (5.24) and solving for
τi yields
8
τi = 2 ,
(5.25)
T |β|
which represents the maximum value the lag can take such that the phase difference is
less than π/2. The step in f0 is then defined as the step which allows for at most a phase
difference of π/2. Assuming symmetry around the point where f0 and φ0 are matched
(e.g. in the middle of the filter at t = T /4), the step in f0 can then be computed with
Eq. (5.11) to be
δf0 = f (T /4 + τi ) − f (T /4 − τi ).
(5.26)
Equation (5.25) yields two cases which lead to two different step sizes in f0 . The first one
is τi < T /4 meaning that the value of β is too large to take the maximum step in f0 . In
this case the step in f0 is given by the substitution of Eq. (5.25) in Eq. (5.26). When τi >
T /4 it means that the maximum step size can be taken so the lag-dependence of f0 (τ )
from Eq. (5.16) will give the step in f0 . Finally, when either of the previously discussed
cases yields a step in f0 which is smaller than the intrinsic resolution of the FFT, the
80
5.6. Constructing the hit maps
step in f0 automatically becomes this minimum step. These arguments combined gives
the step in f0 as

T
T


for τi <
2|α|τi + |β|τi


2 4

 2
T
δf0 = |α| T + |β| T
(5.27)
for τi >


2
2
2
4



1
1
for δf0 < .
T
T
When considering Eq. (5.25) together with Eq. (5.27) it can be seen that the maximum
step in frequency, δf0 |max , occurs when τi = T /4. When taking T = 1200 s, the β value
where the maximum step in frequency occurs can be computed by inverting Eq. (5.25)
and yields β = 1.85 × 10−8 Hz/s2 . When taking αmax = 4 × 10−3 Hz/s (see Table 5.6)
then δf0 |max = 2.4 Hz. The minimum step in frequency δf0 |min = 1/T becomes 8.4×10−4
Hz.
5.6
Constructing the hit maps
In order to discriminate between noise and signal, a threshold will be set on the collection
of maximum correlations. Each filter of which the correlation exceeds this threshold is
called a ‘hit’. The threshold depends on the amount of trials done and the desired number
of noise hits. The false-alarm probability is used to quantify this threshold.
Assuming that a gravitational wave signal is present in the data, the threshold set
will give rise to a probability that the signal will be missed. This probability is known as
the false-dismissal probability and can be used, together with the false-alarm probability,
to estimate the sensitivity of Polynomial Search per stretch of data.
A filter bank (Table 5.6) is constructed by using the limits on the polynomial search
parameters given in Table 5.2 together with the circular and eccentric binary (parameters
given in Table 5.1). A Polynomial Search with these parameters covers the parameter
parameter
Value
f0
α
β
Nfil
T
1000 ± 8 Hz
±4 × 10−3 Hz/s
±2 × 10−5 Hz/s2
4830694
1200 s
32 Hz
fs, filter
Table. 5.6: The filter bank used to perform Polynomial Search. Here, f0 ,
the parameters of Polynomial Search and the values represent
value and the range of each parameter, Nfil are the number of
ated with the supplied ranges in the parameters, T = 2Tcoh is
the data and fs,filter is the sample frequency of the filter.
α and β are
the central
filters crethe length of
81
Chapter 5. Polynomial Search
space such that the signals with frequency derivatives smaller than the values in the
table will be detected. This filter bank is used to verify the false-alarm probability and
compute the false-dismissal probability of Polynomial Search applied to a single stretch
of data. The search bandwidth, ∆fPS is defined as the frequency interval of the data
to which the filters are applied. In the case of the parameters shown in Table 5.6, the
search bandwidth is
∆fPS = (f0 )max − (f0 )min = 16 Hz.
(5.28)
The hit maps themselves are constructed by taking the filters which passed the
threshold and superimposing them. These hit maps will describe the frequency evolution
of the gravitational wave signal.
5.6.1
False-alarm probability
The probability that, given a threshold and in absence of a signal, at least one filter
will give a hit is called the false-alarm probability. Note that as described in section 5.3,
the maximum correlation for all the lags is taken. The correlation C for each lag is a
random number drawn from a probability density function (pdf) f (C). Reference [68]
shows that the pdf, ρ(C), of the order statistic for the maximum of N random numbers
drawn from f (C) can be written as
ρ(c) = N F (C)N −1 f (C),
(5.29)
where F (C) is the cumulative distribution function of f (C).
In the case of Polynomial Search the number of lags N = nlags is given by
nlags =
fs,filter T
,
2
(5.30)
where the factor 2 arises because of the fact that half the filter is zero-padded. This
number of lags is in essence the number of sample points in the filter as each sample
point represents a filter which is compared to data. Furthermore, the pdf f (C) is assumed
to be Gaussian,
2
(C−µ)
− √
1
2σ 2
f (C) = √
e
,
(5.31)
2πσ
where σ 2 is the variance of the noise and µ is a measure for the amplitude of a potential
signal hidden in the data.
With the Gaussian assumption together with Eqs. (5.29) and (5.30), the pdf for
finding a maximum correlation of C in nlags lags becomes
"
#nlags −1
2
C−µ
(C−µ)
− √
nlags 1 + erf( √2σ2 )
2σ 2
ρ(C|µ, σ, nlags ) = √
×e
.
(5.32)
2
2πσ
In the case of pure noise (no signal), µ = 0 and σ = 1. Figure 5.13 shows ρ(C) from Eq.
(5.32) for Polynomial Search applied to white Gaussian noise with parameters shown in
82
ρ
5.6. Constructing the hit maps
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
6
C
Figure. 5.13: The probability density function, ρ, for finding a correlation C in a series
of nlags = 19, 200 lags in the case of pure noise (µ = 0 and σ = 1).
Table 5.6 (nlags = 19, 200). From this figure it can be clearly seen that the pdf is nonGaussian. This type of pdf is known as a ‘Gumbel’ distribution [69] and has a positive
mean with nonzero skewness.
The probability that in absence of a signal none of the maximum correlations out of
nlags lags passes a threshold Cthr can be computed from Eq. (5.32) to be
!nlags
Z Cthr
)
1 + erf( C√thr
2
pnone =
ρ(C, µ = 0, σ = 1) dC =
.
(5.33)
2
−∞
The step sizes defined in section 5.5 will yield a total number of filters, Nfil . Assuming
that the filters are independent it is advantageous to consider the number of filters per
Hz of searched bandwidth Nfil0 = Nfil /∆fPS where ∆fPS is defined in Eq. (5.28). The
false-alarm probability per Hz searched bandwidth pfa then becomes
0
Nfil
pfa (Cthr ) = 1 − (pnone )
=1−
where
Ntrials =
1 + erf( C√thr
)
2
Nfil nlags
.
∆fPS
2
!Ntrials
,
(5.34)
(5.35)
In order to verify Eq. (5.34), Polynomial Search is applied to Gaussian white noise.
The parameters employed in the polynomial search are given in Table 5.6. The correlation of the filters with the data are computed and the false-alarm probability has then
been computed for each value of the threshold. In order to do this the filters are counted
83
Chapter 5. Polynomial Search
false alarm probability
by binning the frequency of the middle of the filter, in this case, in 1 Hz bins. The
false-alarm probability is then computed by registering as a function of the threshold 1
if at least one filter out of Nfil filters gave a hit or 0 if no hit occurred. These 0s and 1s
have then been averaged over 1 Hz slices of the search bandwidth (e.g. ∆fPS = 16 Hz
for this particular filter bank) for each FFT and the results are then averaged over 24
FFTs. The false-alarm and the error were given by the mean and the standard deviation,
respectively.
In practice the filters are correlated, meaning that the false-alarm probability calculated with Eq. (5.34) is overestimated. This is due to the fact that the internal correlations between the filters reduce the amount of independent trials. In order to calculate
the false-alarm probability with the correlations between the filters taken into account,
one needs to compute all the correlations between the filters individually. This computation would be impractical when dealing with a large number of filters due to the
combinatorics. Therefore, these correlations have been taken into account by using an
effective number of trials, Neff , fitting Eq. (5.34) to the data points. Figure 5.14 shows
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
5.5
Uncorrelated: Ntrials = 5.797 × 109
Simulation: Neff = 2.998 × 109
6
6.5
7
7.5
Cthr
Figure. 5.14: The false-alarm probability for the uncorrelated case where Ntrials =
5.797 × 109 (dashed curve) which is computed with Eqs. (5.34) and
(5.14). The data points represent the false-alarm probability calculated
by applying the filters to Gaussian noise and the solid curve is a fit to
these points yielding Neff = 2.998 × 109 .
the false-alarm probability as a function of Cthr for two cases. The ‘uncorrelated’ case
is the false-alarm probability assuming that the filter bank is completely uncorrelated
and that Ntrials can be computed with Eq. (5.35). The second case, termed ‘simulation’
is the result of applying Polynomial Search to the simulated data.
From Fig. 5.14 it can be seen that the expected, theoretical, false-alarm probability
(Eq. (5.34)) overestimates the actual false-alarm probability. The difference between the
84
5.6. Constructing the hit maps
theoretical curve and the simulated curve in terms of the effective number of trials is a
measure of the degree of internal correlations of the filters within the filter bank. The fit
value of the number of trials can be interpreted as the effective number of independent
trials, Neff = 2.998 × 109 .
In order to be able to separate noise hits from signal hits, a threshold on the falsealarm probability must be set. For any desired false-alarm probability the threshold is
given by inverting Eq. (5.34) which yields
i
h
√
1
−1
Neff
Cthr = 2 erf
−1 .
(5.36)
2(1 − pfa )
When taking a false-alarm probability (per Hz) of 1%, 10% or 50%, the corresponding
values for Cthr for the parameters in Table 5.6 are 6.81, 6.51 and 6.24, respectively for
Gaussian noise.
5.6.2
False-dismissal probability
The false-dismissal probability is defined as the probability that no correlation between
filter and data passes a given threshold in the presence of a signal. The false-dismissal
probability can be calculated when the correlations of all filters with the signal with
amplitude Asig are known. Also the assumption is made that out of the nlags (Eq. (5.30))
lags, the maximum correlation will give the correlation with the signal while the others
can be neglected.
The probability that none of the Nfil filters will cross the threshold Cthr , while a
signal is present in the data can be computed. The false-dismissal probability becomes
Neff −Nfil

Z Cthr
1 + erf C√thr
2
fil

pfd (SNR, Cthr ) = ΠN
ρ(C, µi , σ = 1)dC × 
, (5.37)
i=1
2
−∞
where the first factor represents the Nfil filters which yield a mean correlation of µi .
The second factor represents all the other lags which were not considered since only
the maximum correlation out of nlags lags was taken. Also, according to the assumption
stated above, these correlations were assumed to be too small to cross the threshold.
The fact that the filters in the filter bank are correlated is also taken into account by
using the effective number of trials, Neff , instead of Ntrials .
After computing the integral in Eq. (5.37), the false-dismissal probability becomes
 
Neff −Nfil

1 + erf Cthr −µ√i2(SN R)
1 + erf C√thr
2
fil
×

pfd (SNR, Cthr ) = ΠN
. (5.38)
i=1
2
2
From Eq. (5.38) it can be seen that in the limit of SNR → 0, the false-dismissal probability reduces to 1 − pfa . This is to be expected as when SNR → 0 only the noise
contributes. This implies that the probability to miss a (weak) signal is equivalent to
not having a false alarm (1 − pfa ).
85
Chapter 5. Polynomial Search
The pdf from which the correlations are drawn is shown in Eq. (5.32) and is a function of the expectation value µ of the correlation statistic. In the presence of a signal
s(t), the expectation value for the i-th filter µi is given by Eqs. (B.16) and (B.18)
h
i
(5.39)
µi = SNR × ci = SNR × F −1 ŝ(f )F̂i∗ (f ) ,
r
where
Asig √
T
√ N
=
(5.40)
Sn
Anoise 2
as derived in appendix B. Note that 0 < i < Nfil .
In order to compute the false-dismissal probability with Eq. (5.38), the actual signal
shape is needed. Even though this can be accomplished by means of simulations, the
false-dismissal probability obtained this way will be only relevant for the specific signal
shape and filter bank used in the simulation. A simplistic estimate can be given when
assuming that the filter bank is orthonormal meaning that all filters are independent
from each other. This will imply that only a single of these orthonormal, filters will give
a correlation of c = 1 and all the other filters will have negligible correlations with the
signal. This assumption will decouple the choice of filter bank and signal shape from the
false-dismissal probability. Equation (5.38) will then reduce to
SNR ≡ Asig

pfd (SN R, Cthr ) = 
1 + erf
Cthr√
−SN R
2
2
 

1 + erf
2
C√thr
2
Neff −1

.
(5.41)
Since Eq. (5.41) is based on the assumption that the filters are uncorrelated, it will be
defined as the ‘uncorrelated’ case, while Eq. (5.38) is based on the actual filter bank.
Figure 5.15 shows both the uncorrelated case and the actual case for the filter bank
given in Table 5.6 for both the circular and eccentric binary. The correlations resulting
from the application of the filter bank to the signal directly give ci described in Eq. (5.39).
In order to compute the data points shown in the figure, the false-dismissal probability
given by Eq. (5.38) has been computed for each FFT and subsequently averaged over a
single period of the signal (11 FFTs in the circular case and 35 FFTs in the eccentric
case). The error bars represent the standard deviation to the mean value over this single
period.
When comparing the two panels in Fig. 5.15 it can be seen that the false-dismissal
probabilities for both signals are similar. However, in the eccentric case the variation
of the false-dismissal probability is somewhat larger than in the circular case. This can
be explained by the fact that the coherence time has been set such that the periastron
will not be covered optimally with this particular filter bank as explained in section 5.4.
Despite this minor difference, it is deduced that the filter bank covers both the eccentric
and the circular binary sufficiently so that the false-dismissal probabilities for both cases
will be identical.
From Fig. 5.15 it can be seen that the uncorrelated false-dismissal probability does
not match the actual false-dismissal probability. The filter bank was chosen to be dense
86
1
1% FAP
10% FAP
50% FAP
Uncorrelated
0.8
0.6
1
1% FAP
10% FAP
50% FAP
Uncorrelated
0.8
0.6
0.4
0.4
0.2
0
0
false dismissal probability
false dismissal probability
5.7. Combination of hits
0.2
1
2
3
4
5
6
7
8
9 10
SNR
0
0
1
2
3
4
5
6
7
8
9 10
SNR
Figure. 5.15: The false-dismissal probability for the circular (left panel) and eccentric
binary (right panel) averaged over a single period of the binary. The data
points represent the average and the error bars the standard deviation
of the false-dismissal probability. The dashed curve represents the falsedismissal probability assuming perfect correlation for a single filter. The
data points represent the actual situation where the simulated signals
and the filter bank are used. The false-alarm probability has been set to
1%, 10% and 50%, respectively.
enough so that several filters (at least 20) will produce a correlation larger than 0.7.
This means that the false-dismissal probability, calculated according to Eq. (5.38), is
reduced with respect to the false-dismissal probability for an orthonormal filter bank.
5.7
Combination of hits
In sections 5.6.1 and 5.6.2 the statistics of the hits for a single FFT of data have been
explained. However, the gain in sensitivity for detecting gravitational waves from periodic sources lies in the integration of the signal over long stretches of time. Therefore,
the hits from each SFT must be combined. Polynomial Search employs incoherent combination by constructing a number count statistic from the number of filters which cross
a certain threshold. The statistical treatment of this statistic is analogous to that of the
Hough search described in Ref. [63].
The number count is constructed per frequency band. Then, there are two ways of
constructing the number count, either counting the number of hits for each FFT and
adding them up, or counting the number of FFTs which have one or more hits. The former will be less robust since non-stationarities and instrumental lines will have a larger
effect on the number count. The latter will suffer from a lack of statistics when dealing
with fewer FFTs to process.
87
Chapter 5. Polynomial Search
5.7.1
Number count statistics
The probability that khits out of Nfil filters cross the threshold per Hz of search bandwidth
is defined as P (khits ). The probability that kffts out of Nfft FFTs have at least one
hit in them per Hz is defined as P (kfft ). Both probabilities are given by the binomial
distribution as
Nfil !
pkhits (1 − phit )Nfil −khits and
khits !(Nfil − khits )! hit
Nfft !
P (kffts ) =
pkffts (1 − pfft )Nfft −kffts .
kffts!(Nfft −kffts )! fft
P (khits ) =
(5.42)
(5.43)
Here, phit is the probability that a single filter will cross the threshold on the correlation
and pfft is the probability that at least one hit will be found in a single FFT. When no
signal is present the false-alarm probability as defined in section 5.6.1 (e.g the probability
that at least one filter or FFT gives a hit while no signal is present) is related to phit and
pfft . In the case of counting hits, Eq. (5.42) gives the probability of finding khits which
can be related to the false-alarm probability by considering that
pfa = 1 − P (khits = 0) = 1 − (1 − phit )Nfil .
→ phit = 1 − (1 − pfa )
(5.44)
1
Nfil
Inverting Eq. (5.44) will give phit as a function of pfa . When counting FFTs with hits in
them the relation between the pfa and pfft is simply
pfa = pfft .
(5.45)
The relation given in Eq. (5.44) implies that when the false-alarm probability for a single
FFT is computed, the probability for finding an arbitrary number of hits in an arbitrary
number of FFTs is fully determined by Eq. (5.42). The same holds true for Eq. (5.45)
when counting FFTs with hits in them. However, Eq. (5.44) implies that the probability
for each filter to cross the threshold is independent. It has been shown in section 5.6.1
that this situation can be approximated by using the effective number of independent
filters. On the other hand when deriving Eq. (5.45) no such assumption has been made.
The probability of finding khits hits in a single FFT is given by substituting Eq. (5.44)
into Eq. (5.42). When applying Polynomial Search to Nffts independent FFTs and search
bandwidth ∆fPS , the expectation value µn and the variance σn2 of the number of noise
hits are
µn = Nfil phit Nfft ∆fPS ,
σn2 = Nfil phit (1 − phit )Nfft ∆fPS .
(5.46)
(5.47)
It can be seen in Eq. (5.46) that the expectation value consists of the term Nfil phit which
is the mean of the binomial distribution (5.42). The term Nfft ∆fPS arises from the fact
that the FFTs and frequency bands are independent. The same argument holds for the
variance (Eq. (5.47)). Note that phit depends on the false alarm probability as shown in
88
5.7. Combination of hits
Eq. (5.44).
When counting FFTs with hits in them, Eq. (5.45) together with Eq. (5.43) gives
the mean and variance for this counting strategy as
µn = Nfft pfft ∆fPS ,
σn2 = Nfft pfft (1 − pfft )∆fPS .
(5.48)
(5.49)
Here, the false alarm probability is identical to pfft .
Because Polynomial Search employs a scheme with FFTs having a 50% overlap, the
total observation time as a function of the number of FFTs can be written as
Tobs =
Nfft + 1
T.
2
(5.50)
For the filter bank with parameters shown in Table 5.6, the expected number of noise
hits is computed for the different total observation times shown Table 5.7. The result is
Tobs
Nfft
1200 seconds
1
1 days
143
30 days
4,319
1 year
52,595
Table. 5.7: The total observation times for which the number count statistics are computed together with the associated number of overlapping FFTs according
to Eq. (5.50).
shown in Fig. 5.16. This figure shows the linear scaling of the expected number of noise
counts as a function of the number of FFTs or the observation time. Furthermore, it
shows that the number of expected noise hits falls exponentially when the threshold is
raised. When performing Polynomial Search and counting the hits, these number counts
can be analyzed for statistical excess in a particular frequency band, depending on which
counting strategy is used.
In the case that a signal is also present in the data, the probability of finding k hits
is still the same as given in Eq. (5.42) and (5.43). However, phit and pfft are now related
to the false-dismissal probability since this is the probability that no filters (or FFTs)
cross the threshold. In this case pfd is related to phit and pfft as
pfd = P (khits = 0) = (1 − phit )Nfil
pfd = 1 − pfft ,
(5.51)
(5.52)
for counting hits or FFTs with hits in them, respectively. The observed number count
is related to the false-dismissal probability by inverting Eq. (5.51) or Eq. (5.52) and
substituting the result in Eq. (5.48). This results in the mean number of hits when a
signal is present µs :
1
µs = Nfil Nfft (1 − pfd ) Nfil ∆fPS
(5.53)
89
106
1200 seconds of data
1 day of data
30 days of data
1 year of data
105
104
µn
µn
Chapter 5. Polynomial Search
106
30 days of data
104
103
103
2
102
10
10
10
1
1
-1
-1
1 year of data
10
10
-2
10
1200 seconds of data
1 day of data
105
6
6.2
6.4
6.6
6.8
7
Cthr
10-26
6.2
6.4
6.6
6.8
7
Cthr
Figure. 5.16: The expected number of noise hits (µn ) for 1200 seconds (1 FFT), one
day, one month and one year observation time as a function of Cthr . The
left panel shows the expected number of noise hits in the case of counting the number of filters and the right panel shows the number of noise
hits when counting the number of FFTs which contain at least one noise
hit.
for counting filters and
µs = Nfil Nfft (1 − pfd )∆fPS
(5.54)
107
µs
µs
for counting FFTs. Note that pfd depends on SNR through the relation given in Eq.
(5.38). The mean number of hits in the presence of signal as a function of SNR and
observation time is shown in Fig 5.17.
1200 seconds of data
1 day of data
30 days of data
1 year of data
6
10
105
107
105
104
104
103
103
102
102
10
10
1
1
-1
-1
10
0
1
2
3
4
5
1200 seconds of data
1 day of data
30 days of data
1 year of data
106
6
7
8
9 10
SNR
10
0
1
2
3
4
5
6
7
8
9 10
SNR
Figure. 5.17: Expected number of hits as a function of the SNR of a potential signal
for a false-alarm threshold of 10%. The left panel shows the result when
counting individual filters and the right panel shows the result when
counting FFTs with hits in them. The results are computed for 1200 seconds, 1 day, 30 days and 1 year of data, respectively.
The threshold for both panels has been set to a false-alarm probability of 10% (Cthr =
6.51). The expected number of hits is related to (1−pfd )1/Nfil and (1−pfd ) for the left and
90
5.7. Combination of hits
right panel, respectively. With these two remarks, it can be confirmed by considering
Fig. 5.15 that signals with SNR above about 6 will have a non-zero value for µs for a
single FFT (1200 seconds of data). Furthermore, the scaling as a function of observation
time is again linear as in Fig. 5.16. In the low SNR limit, the number of expected hits
will reduce to the false-alarm probability. Finally, the number of expected hits in the
high SNR limit will reduce to the total number of filters or the total number of FFTs
for the left and right panel, respectively.
5.7.2
Renormalized number count
In order to acquire possible gravitational wave candidates, a threshold on the number
count must be set. This threshold will be of the form
nthr = µn + λσn ,
(5.55)
where nthr denotes the number count threshold, µn the expectation value of the noise
hits, λ the significance level and σn the standard deviation of the noise hits. When
taking η to be the observed number count in a particular frequency interval, it is more
convenient to define the renormalized number count as
Ψ(SNR) ≡
η − µn
.
σn
(5.56)
In this way, the significance level λ is a direct threshold on the renormalized number
count. Given the the fact that the noise is Gaussian and white, the renormalized number count is normally distributed. Substituting Eqs. (5.53) and (5.54) for the observed
number of hits (η) in (5.56) will give the expectation value of the renormalized number
count as a function of SNR. This renormalized number count is shown in Fig. 5.18 for
different observation times. Note that if NFFT and/or Nfil are large enough, Eq. (5.56)
is drawn from a normal distribution with an expectation value is related to the SNR of
a potential signal (Fig. 5.18) and variance 1.
The number count false-alarm probability, p?fa , is defined as the probability that at
least one renormalized number filter or FFT crosses the significance level λ defined in
Eq. (5.55) in the case that SNR= 0. As the renormalized number count for this case is
a standard normal distributed (mean 0 and variance 1), the number count false-alarm
probability can be written as a function of the threshold as
Z ∞
1 − 1 x2
1
λ
?
2
√ e
pfa =
.
(5.57)
dx = 1 −
1 + erf √
2
2π
2
λ
Which means that a significance level of λ = 3 corresponds to p?fa = 0.13% and λ = 6
corresponds to p?fa = 9.9 × 10−8 %
When setting this threshold λ, and with the false-alarm and false-dismissal probabilities given in the previous section, the sensitivity of Polynomial Search can be computed
for the filter bank from Table 5.6.
91
Chapter 5. Polynomial Search
105
Ψ
Ψ
105
1200 seconds of data
1 day of data
30 days of data
1 year of data
104
104
103
103
102
102
10
10
1
0
1
2
3
4
5
1200 seconds of data
1 day of data
30 days of data
1 year of data
6
7
8
9 10
SNR
1
0
1
2
3
4
5
6
7
8
9 10
SNR
Figure. 5.18: The expectation value of the renormalized number count as a function
of the SNR of a potential signal for a false-alarm threshold of 10%. The
left panel shows the result when counting individual filters and the right
panel shows the result when counting FFTs with hits in them. The results are computed for 1200 seconds, 1 day, 30 days and 1 year of data,
respectively.
5.7.3
The sensitivity
The sensitivity of Polynomial Search is computed by setting the threshold λ on the
renormalized number count and by solving Eq. (5.56) for SNR. Since the false-dismissal
probability is computed with simulations, the sensitivity must also be computed with
these simulations. When making the assumption that the SNR of any search for continuous waves will go as a certain power of the observation time, the sensitivity of the
search can be parameterized as
−p2
SNR = p1 Nfft
,
(5.58)
where p1 and p2 are parameters to be determined from the simulated binaries. Note that
SNR here means the minimum SNR which can be detected given a value for λ and Cthr .
Figure 5.19 shows the sensitivity of Polynomial Search as a function of the number
of FFTs for different false-alarm probabilities at λ = 6 and λ = 3. It can be seen from
the difference between the top and bottom panels in Fig. 5.19 that the sensitivity for
counting FFTs with hits in them is slightly worse at fixed number of FFTs. This follows from the fact that the simulation shows that a higher SNR is needed for the same
significance, number for filters and false alarm probability. However, the scaling as a
function of FFT number is slightly better. Furthermore, as stated before, the top panel
shows that due to a lack of statistics a certain number of FFTs are required in order to
surpass the significance level. This can be seen by comparing the curves made for λ = 6
to the ones made for λ = 3. The former will require more FFTs for a signal to trigger
enough hits (FFTs in this case) to become significant. Also for the λ = 6 curves it can
also be seen that for higher false-alarm probability more FFTs are needed to produce a
significant number count. On the other hand it might be more advantageous to count
FFTs with hits instead of individual hits when the noise is non stationary. This count-
92
SNR
5.7. Combination of hits
7
FAP = 1%, λ = 6
FAP = 10%, λ = 6
FAP = 50%, λ = 6
FAP = 1%, λ = 3
FAP = 10%, λ = 3
FAP = 50%, λ = 3
6.5
6
5.5
5
4.5
4
SNR
3.5
1
10
102
103
6.5
104
105
Nfft
FAP = 1%, λ = 6
FAP = 10%, λ = 6
FAP = 50%, λ = 6
FAP = 1%, λ = 3
FAP = 10%, λ = 3
FAP = 50%, λ = 3
6
5.5
5
4.5
4
3.5
3
1
10
102
103
104
105
Nfft
Figure. 5.19: The SNR as a function of the number of FFTs for different values of the
false-alarm probability and for significance level of λ = 6 and λ = 3. The
solid curves are fits of the sensitivity parameterization. The top panel
shows the SNR curve when counting FFTs with hits in them and the
bottom panel shows the SNR curve for counting individual filters.
ing strategy will be more robust since a single noisy FFT will count only once in the
number count statistic while when counting filters a noisy FFT will produce many more
hits. This effect is due to the excess power with respect to the estimated PSD will cause
the C-statistic to be artificially increased and thus many filters will cross the threshold.
Also, when counting individual hits, the internal correlations of the filter bank are not
properly described which will result in a non-Gaussian distributed renormalized number
93
Chapter 5. Polynomial Search
count. This is due to the fact that Eq. (5.51) is derived under the assumption that the
internal correlations in the filter bank can be neglected.
Table 5.8 shows the values for the parameters p1 and p2 for this filter bank for different values of the false-alarm probability and of the significance level. From this table it
FAP (%) λ
1
10
50
1
10
50
6
6
6
3
3
3
{p1 , p2 (×10−2 )}filcount
5.35,
5.58,
5.81,
5.04,
5.23,
5.40,
4.15
4.59
5.10
4.15
4.56
5.06
{p1 , p2 (×10−2 )}fftcount
6.20,
6.70,
7.39,
5.73,
6.21,
6.74,
4.58
5.21
5.97
4.31
5.10
5.77
Table. 5.8: The parameters of the sensitivity model as a function of the false-alarm
probability and the significance
are given for the case
level. The parameters
−2
of counting individual filters ( p1 , p2 ×10
) and for counting
filcount
FFTs with hits in them ( p1 , p2 ×10−2 fftcount ).
can be seen that when lowering the significance level from λ = 6 to λ = 3, the minimal
SNR of a detectable signal goes down by about 7% at fixed number of FFTs for both
the counting strategies. Furthermore, lowering the threshold on the correlation (e.g. by
increasing the false-alarm probability) will increase the scaling of the sensitivity as a
function of the number of FFTs. However, it will also reduce sensitivity at fixed number of FFTs with respect to setting a higher correlation threshold. The cross-over is at
about 2, 000 FFTs for counting individual filters while the crossover for counting FFTs
is > 105 due to the lower statistics.
Another observation from Fig. 5.19 is that there is some gain in increasing the observation time by adding the hits over a set of FFTs. The sensitivity of other semi-coherent
−1/4
searches as mentioned in section 4.3, scales as Nfft . The scaling of the sensitivity of
Polynomial Search as a function of the number of FFTs is given by the p2 parameter in
Eq. (5.58). As can be seen in Table 5.8 Polynomial Search scales with a lower power of
the number of FFTs with respect to the traditional semi-coherent searches. This apparent disadvantage of Polynomial Search can be explained by the following two effects.
Firstly, the threshold on the false-alarm probability per filter given in Eq. (5.44)
(phit ≈ 1.9 × 10−7 for pfa = 10%) is much lower than in the Hough search (typically
phit = 0.1). This implies that the threshold on the correlation is much higher than for
a Hough search. This effect can be mitigated by lowering the threshold and indeed the
slope of the curves in Fig. 5.19 will increase. Since Polynomial Search employs a bank
of filters, the probability that a hit will be a false-alarm will depend on the number of
filters applied. This imposes a lower limit on the threshold which will be higher than in
the Hough case. Also, maximizing the correlation introduces extremal value statistics.
In that case setting the threshold to for example Cthr = 3, will cause virtually all the
filters to be a hit (see also Fig. 5.13), making such thresholds unusable. This implies
that the Hough-like scaling of the signal-to-noise ratio will not be achieved by simply
94
5.7. Combination of hits
lowering the threshold.
Secondly, when considering a Hough search, the frequency bins which have been
selected by setting the threshold on the power. These bins must be combined in such
a way that the signal will add incoherently. In the case of isolated neutron stars this
combination is done by binning in the sky positions (α, δ) and spindown parameters
(f 1 , f 2 , . . . , f s ). If the same incoherent combination for Polynomial Search is to be
achieved, then the filters have to be combined in a way which is consistent with a
binary system with the Keplerian orbital parameters (P, ap , e, ω0 and τ ). This would
imply that the hits have to be binned not only in the sky position and spindown parameters, but also in the 5 additional
Keplerian parameters which would be computationally
√
4
unfeasible. Note that the T sensitivity gain in the incoherent summation of the Hough
search is obtained for each of the (α, δ, f 1 , . . . , f s ) bins. The number of these bins can
be seen as a number of trials and will increase as a function of T . Furthermore, if one
would want to apply a Hough search to binary systems, the coherent integration time
would be limited to about 20 seconds due to computational arguments as explained in
section 4.1.
When combining the definition of SNR given in Eq. (5.40) and Eq. (5.58), an expression for the minimal detectable gravitational wave signal can be constructed. This
expression is of the form
r
Sn
−p2
h=
p1 Nfft
.
(5.59)
T
Note that the h parameter is equivalent to the Asig parameter in Eq. (5.40). This implies
that the actual gravitational wave amplitude will have to be higher since the beampattern function has been absorbed in this parameter.
When assuming a false-alarm probability of 10%, a significance level of 6, together
with the parameterization of the SNR, the sensitivity curve shown in Fig. 5.19 and
counting FFTs with hits in them, the minimal detectable gravitational wave amplitude
is
p
−5.21×10−2
h = 0.19 × Nfft
× Sn .
(5.60)
This minimal detectable gravitational wave amplitude can computed with the Advanced
Virgo design sensitivity by inserting the design PSD in Eq. (5.60). Figure 5.20 shows the
sensitivity of Polynomial Search when assuming Advanced Virgo sensitivity for various
observation times.
The sensitivities shown in Fig. 5.20 are valid for the case that the amplitude modulation can be ignored during a single FFT. This assumption is reasonable in case the
length of the FFT is small compared to the Earth’s daily motion. Furthermore, Poly
<
nomial Search with these
parameters will cover all gravitational waves with df
dt
2 4 × 10−3 Hz/s and ddt2f < 2 × 10−5 Hz/s2 as long as the higher-order terms obey
3 d f 3
dt3 < 3.7 × 10−10 Hz/s . This includes residual Doppler effects like the Earth’s motion
(yearly and daily) as well as other frequency modulation induced by glitches and spindown.
As stated before, the advantage of adding multiple FFTs to increase the signal-to-
95
h
Chapter 5. Polynomial Search
10-21
1200 seconds of data
1 day of data
30 days of data
1 year of data
10-22
10-23
10-24
10-25
10
102
103
f (Hz)
Figure. 5.20: The sensitivity of Polynomial Search as a function of the frequency for
the Advanced Virgo design sensitivity. The sensitivity estimates have
been computed with Cthr = 6.51 (F AP = 10 %), λ = 6 (a 6σ significance
level), counting FFTs with hits in them and ignoring the effects of the
beam-pattern function of the detector.
noise ratio is rather limited. This means that in order to get a statistical significant
sample, more FFTs have to be included in the analysis. Furthermore, in order not to
be limited by the antenna pattern of the interferometer, it is advantageous to have an
observation time of at least a few days. Since this beam pattern function is periodical
over one day, the effect it has on the renormalized number count will be averaged out
when enough FFTs are taken.
5.7.4
Consistent filters
The gain of adding more FFTs to the analysis can be increased when the threshold on
the correlation can be lowered. This lowering can be achieved by exploiting the fact that
a potential signal is continuous and thus the filters which correspond to this signal should
be continuous over the FFT boundaries. When imposing constraints on the difference
of the filter derivatives between hits in consecutive FFTs, inconsistent hit pairs can be
discarded. By keeping the consistent hit pairs, the threshold on the correlation can be
lowered and this improves the sensitivity of Polynomial Search.
A consistent hit pair is defined as a hit from the first FFT and a hit from the second
FFT of which the frequency, first and second order derivatives match within a certain
tolerance. These derivatives are compared by extrapolating to a time tm . The quantity
tm is defined as
tend,1 + tbegin,2
tm =
,
(5.61)
2
96
5.7. Combination of hits
where tend,1 and tbegin,2 are the end time of hit 1 and the start time of hit 2, respectively.
The tolerance T is defined in an empirical manner as the allowed frequency mismatch
between two filters at this intermediate time. With this definition and in combination
with the step sizes in the parameters defined in section 5.5, the allowed mismatch of the
higher-order derivatives will scale as T −n . Consistent filters obey the relation
δf (n) (tm ) < ξn
T
,
Tn
(5.62)
false alarm probability
where the tolerance T is some empirical value, δf (n) (tm ) represents the difference of the
n−th order derivatives of the filters evaluated at time tm and ξn is a constant for each
derivative. Ideally, ξn depends on the actual step size of the filter bank for each parameter. However, since the step size depends on the parameters of the filters themselves
and ξn might also depend on tm , optimal selection of ξn is non-trivial. Therefore, ξn has
been set to unity for each of the derivatives making Eq. (5.62) very restrictive.
Testing for consistency between the hits in two adjacent FFTs is a combinatorial
operation (e.g. each hit in the first FFT is matched to all of the hits in the second
FFT). To keep the combinatorics involved in checking the consistency of filter under
control an initial threshold on the correlation of Cthr = 4.5 has been set. Figure 5.21
shows the false-alarm probability when applying the consistency criterion to each pair of
FFTs with the filter bank described in Table 5.6. The data points have been computed
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
4.5
Tolerance 0.5
Tolerance 0.1
Tolerance 0.01
Tolerance 0.001
No tolerance
5
5.5
6
6.5
7
7.5
Cthr
Figure. 5.21: The revised false-alarm probability by using the consistency criterion
(see Eq. (5.62)) for different values of T . The dashed curve represents the
fitted value of the false-alarm probability when the consistency criterion
is not applied, see also Fig. 5.14.
by using the same procedure as described in section 5.6.1. However, in this case the
false-alarm probability is the probability that at least one valid pair of filters crosses the
97
Chapter 5. Polynomial Search
threshold. A pair of filters is considered valid when both filters have a correlation larger
than the threshold and adhere to the consistent filter criterion. This different interpretation means that Eq. (5.34) does not apply. Modeling the false-alarm probability in
this case would require a way to model the probability that two random filters match in
parameters and that both filter pass the threshold. Since this is highly non-trivial as the
filter bank has internal correlations and the filters are not spaced at equidistant points
in parameter space, the values for the threshold at a given false-alarm probability will
be computed by interpolation.
It can be seen from Table 5.9 that by applying the consistency criterion, the thresholds can be reduced while maintaining the same false-alarm probability. The lowering
T
1%
Cthr
10%
Cthr
50%
Cthr
−
0.5
0.1
0.01
0.001
6.81
6.14
5.60
5.06
4.72
6.51
5.87
5.44
4.88
−
6.24
5.66
5.26
4.69
−
1%
10%
50%
Table. 5.9: Thresholds for various consistent filter criteria. Cthr
, Cthr
and Cthr
are
the thresholds for false-alarm probability 1%, 10% and 50%, respectively.
of the threshold will result in a more favorable scaling of the sensitivity as a function of
the number of FFTs considered. However, the tolerance cannot be arbitrarily decreased
for a the reasons listed below
ˆ The lower limit on the tolerance is defined by the frequency (and higher order)
resolution given by the FFT time. In reality, the chosen step size in section 5.5
determines the lower limit for the value of T for each order of the derivatives.
ˆ The filters are an approximation of the actual signal. Therefore, the filters will have
a mismatch in their derivatives at the end and beginning which will increase when
noise is present. When the tolerance is set too stringent, this mismatch will cause
the false-dismissal probability to increase which in turn will negatively impact the
sensitivity.
ˆ The number of hits strongly increases with the lowering of the threshold as can
be seen in Fig. 5.13. Since applying the consistency criterion is a combinatorial
problem, the amount of computing power scales with the number of hits squared.
This implies that the lowering of the threshold will be also limited by the amount
of computing power available.
ˆ A single filter tests for correlations for all the lags between 0 < τ < T /2. Since the
maximum correlation for all these different τ values is taken, the correlations will
be Gumbel distributed [69] and have a positive mean and nonzero skewness (see
Fig. 5.13). Setting a low threshold (i.e. Cthr = 3) will then yield practically all the
filters, negating the noise-discriminating properties of computing the correlation.
98
5.7. Combination of hits
The false-dismissal probability can be computed by running Polynomial Search, including the consistency criterion, on the simulated binaries. The resulting correlations
between the filters and the signals are inserted in Eq. (5.38) which yields false-dismissal
probability. Figures 5.22 and 5.23 show the false-dismissal curves for different values of
T . Note that due to computational restrictions, the threshold for T = 0.001 can only be
computed for the case of a 1% false-alarm probability, as shown in Fig. 5.21. Therefore,
only this value for the false-alarm probability can be used to compute the false-dismissal
probability shown in the lower left panels of Figs. 5.22 and 5.23. The aforementioned
1
false dismissal probability
false dimissal probability
Tolerance = 0.01
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
1
Tolerance = 0.1
1
2
3
4
5
6
1
7
8
9 10
SNR
0
0
1
2
3
4
5
6
7
8
9 10
SNR
false dismissal probability
Tolerance = 0.001
0.8
1% FAP
10% FAP
50% FAP
0.6
FDP without tolerance
0.4
0.2
0
0
1
2
3
4
5
6
7
8
9 10
SNR
Figure. 5.22: The false-dismissal probability for the circular binary evaluated for different values of T . The upper left plot corresponds to T = 0.1, the upper
right plot corresponds to T = 0.01 and the lower left plot corresponds to
T = 0.001. The error bars and shaded regions represent the variation of
the false-dismissal probability over a single period of the binary system.
effect of increasing the false-dismissal probability can be seen in the difference between
the curves of the different panels in Figs. 5.22 and 5.23. This effect is most clearly seen
in the third panel in both figures as the variation of the 1% curve is larger than the
no-tolerance curve. This is due to the fact that there is a larger chance to miss the
signal in the FFTs where the signal has large derivatives and thus the mismatch of the
filters between the FFTs will be larger. In the eccentric case this effect is even more pro-
99
Chapter 5. Polynomial Search
1
Tolerance = 0.01
False dismissal probability
false dismissal probability
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0
0
1
Tolerance = 0.1
0.2
1
2
3
4
5
6
1
7
8
9 10
SNR
0
0
1
2
3
4
5
6
7
8
9 10
SNR
false dismissal probability
Tolerance = 0.001
0.8
1% FAP
10% FAP
50% FAP
0.6
FDP without tolerance
0.4
0.2
0
0
1
2
3
4
5
6
7
8
9 10
SNR
Figure. 5.23: The false-dismissal probability for the eccentric binary evaluated for different values of T . The upper left plot corresponds to T = 0.1, the upper
right plot corresponds to T = 0.01 and the lower left plot corresponds to
T = 0.001. The error bars and shaded regions represent the variation of
the false-dismissal probability over a single period of the binary system.
nounced since the variation was intrinsically higher due to the fact that the periastron is
not matched well by the chosen filter bank. Also, the average false-dismissal probability
at high SNR is larger than when not using the consistent filter criterion. This is due to
the fact that these filters will be rejected by the strict tolerance. At lower SNR there is
still some gain when employing the consistent filter criterion with T = 0.001, because
the threshold can be set at a lower value and thus the search will be more sensitive to
low SNR signals.
Both Fig. 5.22 and 5.23 show that when setting T = 0.001 the false-dismissal probability will increase compared to the larger values of T implying that a tolerance of 0.001
is too strict. However, setting T = 0.01 does not have this effect and therefore T has
been set to 0.01 for the remainder of the analysis. For this value of T a gain in sensitivity
of about 30 − 40 % is observed for false-dismissal probabilities between about 0.05 − 0.95
with respect to the sensitivity without using the consistent filter argument. Note that
taking T = 0.01 is an ad-hoc choice for the tolerance. Conceivably, one could refine this
100
5.7. Combination of hits
value of the tolerance, however the relative gain of such a fine tuning is expected to be
smaller than 5%.
5.7.5
The sensitivity with consistent filters
The expectation value of the renormalized number count for Polynomial Search with
a false-alarm probability of 10% which, for Fig. 5.21 and Table 5.9, corresponds to a
threshold of 5.40 is shown in Fig. 5.24. When comparing the results to those presented
105
Ψ
Ψ
105
1200 seconds of data
1 day of data
30 days of data
1 year of data
104
104
103
103
102
102
10
10
1
0
1
2
3
4
5
1200 seconds of data
1 day of data
30 days of data
1 year of data
6
7
8
9 10
SNR
1
0
1
2
3
4
5
6
7
8
9 10
SNR
Figure. 5.24: The expectation value of the renormalized number count as a function of
the SNR of a potential signal for a false-alarm threshold of 10% and for
T = 0.01. The left panel shows the result when counting individual filters
and the right panel shows the result when counting FFTs with hits in
them. The different curves are computed for 1200 seconds, 1 day, 30 days
and 1 year of data, respectively. All curves have been computed assuming
that the middle frequency of the filters have been counted in 1 Hz bands.
in Fig. 5.18 it can be seen that Polynomial Search will be more sensitive to lower SNR
signals when employing the consistent filters criterion. This sensitivity is shown in Fig.
5.25 and in Table 5.10.
From the difference between the values in Tables. 5.8 and 5.10 and Figs. 5.19 and
5.25 it can be seen that there is an improvement in sensitivity when testing filters
for consistency. Lowering the threshold on the correlation could possibly improve the
sensitivity as well. However, as stated before, the computational burden would become
too high. Another improvement could be to extend the consistency argument over more
than two FFTs (i.e. demanding that the filter derivatives are consistent over 2 FFT
borders instead of 1). This would reduce the false-alarm probability. This would also
increase the computational power required and would be more susceptible to potential
holes in the data stream. When comparing Figs. 5.19 to 5.25 it can also be seen, especially
in the top panels, that the SNR is not well fitted to the model shown in Eq. (5.58).
This can be explained by the fact that the model applies to uncorrelated filters. When
applying the consistency criterion, the interdependencies of the filter bank become more
pronounced (e.g. the probability of finding a random consistent filter pair with low
101
SNR
Chapter 5. Polynomial Search
6
FAP = 1%, λ = 6
FAP = 10%, λ = 6
FAP = 50%, λ = 6
FAP = 1%, λ = 3
FAP = 10%, λ = 3
FAP = 50%, λ = 3
5
4
3
2
1
SNR
0
1
10
102
6
103
104
105
Nfft
FAP = 1%, λ = 6
FAP = 10%, λ = 6
FAP = 50%, λ = 6
FAP = 1%, λ = 3
FAP = 10%, λ = 3
FAP = 50%, λ = 3
5
4
3
2
1
0
1
10
102
103
104
105
Nfft
Figure. 5.25: The SNR as a function of the number of FFTs for different values of the
false-alarm probability and for a significance level of λ = 6 and λ = 3.
The solid curves are fits of sensitivity parameterization. The top panel
shows the SNR curve when counting FFTs with hits in them and the
bottom panel shows the SNR curve for counting filters. The figure has
been produced for T = 0.01.
values of α and β is higher since more filters have a low α and β value). This becomes
less of an issue when considering FFTs with hits in them since an FFT can only count
once and thus setting the threshold from λ = 3 to λ = 6 will cut away most of the
non-Gaussianity.
The sensitivity of Polynomial Search for the consistent filter criterion with T = 0.01,
102
5.8. Results of the simulations
FAP (%) λ {p1 , p2 (×10−2 )}filcount
1
10
50
1
10
50
6
6
6
3
3
3
3.25,
3.57,
3.90,
3.00,
3.26,
3.15,
9.98
9.90
10.4
17.7
11.3
11.7
{p1 , p2 (×10−2 )}fftcount
4.03,
4.81,
5.94,
3.60,
4.24,
5.14,
8.71
9.69
11.0
9.09
9.95
12.2
Table. 5.10: The parameters of the sensitivity model as a function of the false-alarm
probability and the significance level. These values are computed for
T = 0.01. The parameters
are given for the case of counting individ−2
ual filters ( p1 , p2 ×10
) and for counting FFTs with hits
filcount
( p1 , p2 ×10−2 fftcount ).
λ = 6, Cthr = 4.88 (FAP = 10%), counting FFTs and with Table 5.10 together with Eq.
(5.59) can be written as
p
−9.69×10−2
h = 0.14 × Nfft
× Sn .
(5.63)
Fig. 5.26 shows the sensitivity of the Polynomal Search for the Advanced Virgo sensitivity as input for different total observation times.
5.8
Results of the simulations
In section 5.1 two binary systems have been defined in order to test Polynomial Search.
A Virgo tool called SIESTA [70] has been used to generate the waveforms. SIESTA is
a data analysis tool which is used among other things to generate waveforms originating from a variety of sources. For the two binaries described in Table 5.1, the SIESTA
module MEBinaryPulsar was used.
Both binaries have been simulated at a sample frequency of 4000 Hz. For these simulations, the detector beam pattern function is taken to be unity and the source is taken to
be aligned with the detector (e.g. ψ = 0 χ = π/2). Figure 5.27 shows the periodograms
of both binaries in the absence of noise. Note that since the FFTs are overlapping and
thus plotting all the FFTs will complicate the periodograms, Fig. 5.27 only shows the
odd-numbered FFTs. From Fig. 5.27 it can be seen that the power of the binary will be
spread out over a significant number of frequency bins. This is due to the Doppler shifts
and is most pronounced in the periastron.
As stated before, the consistent filter argument requires an a priori threshold on the
correlation such that the number of filters to be checked for consistency is manageable.
In order to set this threshold, Fig. 5.28 shows the distribution of the values for the
correlation without noise (e.g. the correlation between filter and signal is given by ci in
Eq. (5.39)) for both binaries averaged over one period of the signal. From this figure
it can be seen that both the circular and the eccentric binary have the same shape for
103
h
Chapter 5. Polynomial Search
10-21
1200 seconds of data
1 day of data
30 days of data
1 year of data
10-22
10-23
10-24
10-25
10
102
103
f (Hz)
Figure. 5.26: The sensitivity of Polynomial Search as a function of frequency compared
to the Advanced Virgo design sensitivity. The sensitivity estimates have
been computed with the consistent filter criterion for T = 0.01, Cthr =
4.88 (F AP = 10 %), λ = 6 (a 6σ significance level), counting FFTs with
hits in them and ignoring the effects of the beam-pattern function of the
detector.
correlations 0 < c < 0.15. The eccentric binary appears to have less filters with correlation 0.15 < c < 0.7 with respect to the circular case which is due to the fact that the
higher order terms contribute more for the eccentric binary. Furthermore, it can be seen
that there is a sharp rise in the number of filters with correlation c > 0.01 with respect
to filters with c < 0.01 and then an exponential decay up to c ≈ 0.15. The filters with
c < 0.01 are the filters that have a large mismatch in f0 with respect to the original
signal. The fact that these filters give a low correlation can be explained by considering
the expression for the correlation in the time-domain Eq. (A.3). The large mismatch
in f0 will imply that either g(t) or s(t) will be a fast oscillating term with respect to
the other causing the integral to reduce to a small number. For example computing the
correlation between a filter with f0 of 1004 Hz to a signal with central frequency of 1000
Hz will yield a small number for the correlation (c < 0.01). When a filter has parameters
which cause the filter to intersect with the signal in the f-t plane, but the values of the
other parameters have the wrong sign, the correlation will still be small, however not as
small as in the previous case. These filters end up in the 0.01 < c < 0.15 range. Finally,
there is a sharp rise in the filter count with c ≈ 0.15. This is due to the fact that these
filters will have parameters with the correct sign and values which correspond to the
actual signal.
Figure 5.29 shows the hit maps for both binaries for c > 0.1 (grey) and c > 0.8
(black). These hit maps are made by drawing the frequency as a function of time for
each filter passing the threshold. When comparing this figure to Fig. 5.27 it can be
104
5.8. Results of the simulations
1003
10-1
1002
1006
1005
1004
1003
1002
1001
1000
999
998
997
996
10-2
1001
10-3
1000
10-4
999
10-5
998
10-6
997
996
0
5000
10000
15000
20000
time (s)
1
frequency (Hz)
1
frequency (Hz)
1004
10-7
10-1
10-2
10-3
10-4
10-5
10-6
0
5000
10000
15000
20000
time (s)
10-7
Counts
Figure. 5.27: The periodograms for both test binaries. The left panel shows the circular binary and the right panel shows the eccentric binary. The FFT time
is taken to be T = 1200 seconds and the gray scale shows the power normalized to the total power of the signal per FFT. The total observation
time shown in these plots is 6 hours. Only the odd-numbered FFTs are
plotted.
106
Circular binary
105
Eccentric binary
104
103
102
10
1
10-1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Correlation
Figure. 5.28: The correlation distributions, without noise, for the circular and the eccentric binary. Both distributions have been averaged over a single period
(11 FFTs for the circular and 35 FFTs for the eccentric one).
seen that the frequency resolution of Polynomial Search is far superior to simply dealing
with the power spectra in the form of periodograms. When raising the threshold, the
frequency evolution becomes clearer. In the eccentric case (right panel of Fig. 5.29) the
large frequency derivatives in the periastron of the signal lead to no hits with c > 0.8.
As discussed in section 5.4, the coherence time is chosen such that the filter bank does
105
Chapter 5. Polynomial Search
Figure. 5.29: Hit maps (without noise) for both binaries where the gray area indicates
hits for filters with c > 0.1 and the black area corresponds to hits with
c > 0.8.
not contain filters capturing these most extreme derivatives.
While it is desirable to keep the correlation threshold high in order to have a good
frequency resolution, the sensitivity to low SNR signals will decrease. As stated in section 5.7.4 by using the consistent filter criterion, the threshold can be lowered. When
lowering the threshold, the number of filters passing this threshold will increase roughly
exponentially as can be seen in Fig. 5.28. One of the drawbacks of employing the consistent filter criterion is that it is a combinatorial operation which is a CPU intensive
task. Figure 5.30 shows hit maps of before and after applying the consistency criterion
on the filters for both binaries for a total time of 6 hours. The initial threshold on the
correlations has been set to 0.15 which leads to an average number of hits per FFT to
be tested for consistency of about 5 × 103 . It can be seen from these hit maps that the
Figure. 5.30: The hit maps of the circular and eccentric test binary where the gray
area represents all the hits which passed the initial correlation threshold
of 0.15 and the black area represents the hits which remain after applying the consistent filter criterion with T = 0.01.
106
5.8. Results of the simulations
consistent filter criterion increases the frequency resolution as was predicted in section
5.7.4. Note that the actual gain in sensitivity lies in the possibility to lower the threshold when computing the consistent filter pairs, making the number count statistic more
sensitive to potential gravitational wave signals.
In order to test the sensitivity of the search, Gaussian white noise with different amplitudes has been added to simulated binaries and Polynomial Search is applied to these
data. Figure 5.31 shows the periodograms for both binaries for a noise amplitude in the
time domain of An = 50 × h0 and An = 250 × h0 , where h0 represents the amplitude of
the gravitational wave which is, according to Table 5.1, set to 10−27 .√When expressing
the
of the
√
√ density (ASD) Sn they become
√ noise levels in terms
√ amplitude spectral
Sn = 1.11 × h0 1/ Hz and Sn = 5.59 × h0 1/ Hz, respectively. From Fig. 5.31 it
1003
1002
105
1001
1000
999
104
998
997
996
0
5000
10000
15000
20000
time (s)
frequency (Hz)
1004
103
106
1003
1002
105
1001
1000
999
104
998
997
996
0
5000
10000
15000
20000
time (s)
103
1006
1005
1004
1003
1002
1001
1000
999
998
997
996
106
frequency (Hz)
106
1006
1005
1004
1003
1002
1001
1000
999
998
997
996
105
104
0
5000
10000
15000
20000
time (s)
103
106
frequency (Hz)
frequency (Hz)
1004
105
104
0
5000
10000
15000
20000
time (s)
103
Figure. 5.31: Periodograms normalized to the power spectral density of the noise.
The left and right top panels show the periodograms of the circular
and eccentric
binary, respectively
√ with a noise amplitude spectral den√
sity of Sn = 1.11 × h0 1/ Hz. The bottom left and right panels
show
but with an amplitude spectral density
√
√ the same periodograms,
of Sn = 5.59 × h0 1/ Hz.
can be seen that the spread of the power induced by the Keplerian Doppler shifts causes
the signal to be buried in the noise. It also shows that searches for gravitational waves
from pulsars in binary systems which rely on applying a threshold on the (normalized)
107
Chapter 5. Polynomial Search
power will be quite limited in sensitivity due to these Doppler shifts.
Figure 5.32 shows the hit maps
√ for both binaries with noise
√ and hit distributions
amplitude An = 250 × h0 or ASD Sn = 5.59 × h0 1/ Hz. The SNR is computed to be
6.20. The false-alarm probability has been set to 10 % corresponding to Cthr = 4.88 and
the tolerance for the consistent filter criterion has been set to T = 0.01. From Fig. 5.31
it can be seen that the noise amplitude is too high to resolve the signal from the power
spectrum. The hit maps shown in Fig. 5.32 however, show that Polynomial Search can
resolve the gravitational wave signal from the background with relative ease.
The hit distributions, shown in Fig. 5.32, are calculated by counting the number
of FFTs in which at least one filter will give a hit in a certain frequency band. The
hits themselves are counted by computing the frequency in the middle of the filter and
binning the result in 1 Hz bins. Polynomial Search has been applied to the data between 960 and 1040 Hz in blocks of 16 Hz. The hit maps are shown only for the interval
992 Hz < f < 1008 Hz but the FFT distribution is shown for the entire interval. From
these hit distributions it can be seen clearly that Polynomial Search can distinguish the
signal from the background as expected when considering the SNR.
In order to verify that the expected number of noise counts as described in section
5.7 is obtained, it is insightful to compare the mean and standard deviation of the noise
counts from Eqs. (5.48) and (5.49) to the observed noise counts in Fig. 5.32. Figure 5.33
shows the count distributions taken from the FFT distributions shown in Fig. 5.32. From
this figure the mean and variance of the noise distributions can be calculated by averaging the Gaussian-like distribution from approximately 0 to 20. The noise distributions
are independent of the signal and both plots are statistically independent realizations
of the noise. On average the number of noise hits taken from Fig. 5.33 is 8.2 ± 2.3
10%
hits/Hz. When taking Cthr
= 4.88 the number of noise hits is expected to be 7.1 ± 2.7
hits/Hz, implying an underestimate of the noise counts of about 12%. This discrepancy
is due to the fact that the threshold corresponding to the 10% false-alarm probability
has been obtained by linear interpolation of Fig. 5.21. Since these curves have a steep
slope, a small variation in threshold will have a large effect on the value of the falsealarm probability. Taking the slope of these curves into account, a 12% underestimate
of noise counts leads to a 0.3% overestimation of the threshold. In order to reach the
same 6σ significance, taking into account this underestimate, the threshold should be
put at λ ≈ 8.3. However, when comparing this value to the expectation value of the
renormalized number count shown in Fig. 5.24 the effect on the overall sensitivity is
minimal.
for both binaries with noise
Figure 5.34 shows the hit maps√and the hit distributions
√
amplitude An = 400 × h0 or ASD Sn = 8.94 × h0 1/ Hz. The SNR is computed to be
3.87. The signal cannot be recognized easily by eye in the hit maps for this particular
noise amplitude. However, when counting FFTs with hits in them, a clear excess is observed in the hit distribution.
From the noise counts and the estimated variance of the noise, the renormalized
number count can be calculated for both binaries as a function of frequency. The renormalized number count Ψ, as defined in section 5.7, will be the final search statistic of
Polynomial Search. Figure 5.35 shows ψ by using both the estimated noise count com-
108
5.8. Results of the simulations
#FFTs with hits
frequency (Hz)
1008
1006
1004
1002
1000
70
60
50
40
998
30
996
20
994
10
992
0
10000
20000
30000
0
960
40000
time (s)
#FFTs with hits
frequency (Hz)
1008
1006
1004
1002
1000
994
10
40000
time (s)
1000
1020
1040
frequency
40
20
30000
980
50
996
20000
1020
1040
frequency (Hz)
60
30
10000
1000
70
998
992
0
980
0
960
Figure. 5.32: The
√ distributions for both binaries with noise ASD of
√ hit maps and hit
Sn = 5.59 × h0 1/ Hz taken over a period of 12 h (72 FFTs). The top
and bottom left panels show the hit maps of the circular and eccentric
test binary, respectively. The hit maps have been made with T = 0.01
and Cthr = 4.88. The top and bottom right panels show the distributions of the FFTs in which one or more filters had a hit in a particular
frequency band for both binaries.
puted with Eq. (5.48) and the mean of the distributions shown in Fig. 5.33 together
with Eq. (5.56). It can be seen that the renormalized number count distribution for the
eccentric binary case is narrower compared to that of the circular binary. This seems
counter intuitive since the Doppler shifts for the eccentric binary are larger. However,
these large Doppler shifts only occur in the periastron, which was not covered completely
by the filter bank due to the choice of coherence time as described in section 5.4. Thus
this section of the orbit will not contribute much in Polynomial Search. On the other
hand since most of the time the eccentric binary has relatively low Doppler shifts, more
FFTs will be counted and Polynomial Search is actually more sensitive to the eccentric
case.
By setting the significance level λ = 6, a signal with SNR = 3.87 can be recovered,
but a signal with SNR = 3.44 will be missed depending on the accuracy of the estimate of the noise counts. These numbers can be compared to the predicted sensitivity
109
counts
counts
Chapter 5. Polynomial Search
10
8
10
8
6
6
4
4
2
2
0
0
10
20
30
40
50 60 70
#FFTs with hits
0
0
10
20
30
40
50 60 70
#FFTs with hits
Figure. 5.33: The renormalized number count distributions for the circular (left panel)
and eccentric case (right panel).
of the search for λ = 6 and FAP = 10 % shown in Fig. 5.25. Since the number of noise
counts has been underestimated, causing the false-alarm probability to be slightly off,
an accurate direct comparison of the SNR values with the theoretical sensitivity curve
is non-trivial. Taking the 11% underestimate into account, the computed SNR values
can be considered in agreement with those presented in Fig. 5.25.
110
5.8. Results of the simulations
frequency (Hz)
#FFTs with hits
1008
1006
1004
1002
1000
45
40
35
30
25
20
998
15
996
10
994
5
992
0
10000
20000
30000
0
960
40000
time (s)
frequency (Hz)
#FFTs with hits
1008
1006
1004
1002
1000
1000
1020
1040
frequency (Hz)
980
1000
1020
1040
frequency (Hz)
45
40
35
30
25
20
998
15
996
10
994
5
992
0
980
10000
20000
30000
40000
time (s)
0
960
Figure. 5.34: The
√ distributions for both binaries with noise ASD of
√ hit maps and hit
Sn = 8.94 × h0 1/ Hz taken over a period of 12 h (72 FFTs). The top
and bottom left panels show the hit maps of the circular and eccentric
test binary, respectively. The hit maps have been made for T = 0.01 and
Cthr = 4.88. The top and bottom right panels show the distributions of
the FFTs in which one or more filters had a hit in a particular frequency
band for both binaries.
111
Ψ
Ψ
Chapter 5. Polynomial Search
25
25
20
20
15
15
10
10
5
5
0
0
960
980
1000
960
1020
1040
frequency (Hz)
20
15
15
10
10
5
5
0
0
1000
1020
1040
frequency (Hz)
Ψ
980
960
980
1000
1020
1040
frequency (Hz)
Ψ
25
25
20
20
15
15
10
10
5
5
0
0
960
1020
1040
frequency(Hz)
25
20
960
1000
Ψ
Ψ
25
980
980
1000
1020
1040
frequency (Hz)
960
980
1000
1020
1040
frequency (Hz)
Figure. 5.35: The renormalized number count as a function of frequency for the circular (left panel) and eccentric binary (right panel). The first row of
panels shows the renormalized number count computed
with 5.59 × h
√0
√
(SNR = 6.20), the second row is computed with √
Sn = 8.94 × h0 1/ √Hz
(SNR = 3.87) and the final row is computed with Sn = 10.1×h0 1/ Hz
(SNR = 3.44). The hatched region represents the renormalized number
count computed from the observed number count from Fig. 5.33 and the
white filled region represents the renormalized number count when using the computed number count from Eq. (5.48). All number counts have
been computed running Polynomial Search with T = 0.01, Cthr = 4.88
and NFFTs = 72.
112
5.9. Discussion and outlook
5.9
Discussion and outlook
A data analysis method for an all-sky search for gravitational waves from neutron stars
in binary systems, called the ‘Polynomial Search’, has been presented. Polynomial Search
is designed for detecting gravitational wave candidates in data from interferometry experiments like Virgo and LIGO. This method employs a bank of polynomial phase templates, or filters, which approximate the phase of continuous gravitational wave signals
by a Taylor expansion. These filters, of order n, remain fully coherent for short stretches
of duration T if the higher order derivatives of the phase obey
dn+1 φ
<
dtn+1
which is equivalent to
dn f
1
<
dtn
2π
2π
Tcoh
n+1
2π
Tcoh
,
(5.64)
n+1
.
(5.65)
Note that as the filters are zero padded, the integration time T = 2Tcoh .
The data are split into short, overlapping stretches and the filter bank is applied to
each of these stretches by computing the correlation. A threshold is applied to the correlations, and the filters which do not match the signal will be discarded. The remaining
‘hits’ are collected and counted for each stretch of data. A significance is assigned to the
resulting number count statistic and another threshold on this significance is applied
to select the gravitational wave candidates. The details of Polynomial Search have been
presented and sensitivity estimates in terms of the minimal detectable gravitational wave
amplitudes are given.
A filter bank has been constructed that allows for the analysis of all binaries with
orbital parameters in between two extremal cases. These two extreme binaries are
1. A double neutron star (1.4 M ) system with a 2 hour circular orbit.
2. A 1.4 M neutron star orbiting a 14 M star in a 6 hour eccentric orbit with
eccentricity 0.6.
The integration time of the data stretches (T = 1200 s) has been chosen such that all
gravitational waves with frequency derivatives
2 df −3
< 4 × 10 Hz/s and d f < 2 × 10−5 Hz/s2
(5.66)
dt2 dt will be covered with optimal correlation as long as the third order terms obey
3 d f 3
−9
dt3 < 1.9 × 10 Hz/s .
(5.67)
Note that the range in df /dt and d2 f /dt2 can be compared to the allowed spin down
parameters in other all-sky searches for isolated neutron stars. For instance the Einstein
113
Chapter 5. Polynomial Search
Home search F-statistic search applied to S5 data [62] covered a single spin down parameter f˙ with ranges 3 × 10−12 f Hz/s < f˙ < 3 × 10−13 f Hz/s for f < 400 Hz and
−8 × 10−12 f Hz/s < f˙ < 8 × 10−13 f Hz/s for f > 400 Hz. The Hough and PowerFlux
[64] semi-coherent all-sky search have been done with −2.2 × 10−9 Hz/s < f˙ < 0 and
−1 × 10−8 Hz/s < f˙ < 0, respectively. This shows that Polynomial search can cover a
much larger parameter space in terms of spin down parameters than previous searches.
Even though Polynomial Search is designed to target binary systems, these limits
also include residual Doppler effects like the Earth’s motion (yearly and daily) as well
as other frequency modulations induced by glitches and spindown. For this filter bank,
when counting FFTs with hits and with a false alarm probability of 10% together with
a significance threshold of λ = 6 (e.g. a 6σ threshold on the significance), the estimated
sensitivity is
p
−5.21×10−2
h = 0.19 × Nfft
× Sn .
(5.68)
Furthermore, an addition to Polynomial Search called the ‘consistent filter criterion’ has
been shown to improve the sensitivity. This additional algorithm exploits the continuity
of the signal across the data stretch border to select consistent filter pairs. Two filters
form a consistent filter pair if the difference between their filter parameters is smaller
than a certain tolerance. Setting this tolerance to T = 0.01, together with the previous
values for the false alarm probability and significance, the sensitivity equals
p
−9.69×10−2
h = 0.14 × Nfft
× Sn .
(5.69)
The application of this consistent filter criterion yields an improvement of about a factor
of 2 (for a 1 year total observation time). The sensitivity estimates have been verified
by analyzing Gaussian noise with software injections.
The aforementioned sensitivities have all been computed assuming Gaussian and
stationary noise. However, Polynomial Search is to be applied to data from experiments
like LIGO and Virgo for which the noise in general cannot be considered Gaussian
distributed and stationary over large frequency bands. A reliable method to characterize
the power spectral density, Sn , of the noise has to be applied when analyzing real data5 .
In order to verify the sensitivity estimates given in this thesis for more realistic
data behavior it is necessary to redo the analysis in section 5.8 for the case of nonGaussian noise. The most accurate way of doing this is to inject simulated gravitational
wave signals into the data stream of the detectors. The issue with this method is that
simply using the two discussed cases of binaries (circular and eccentric) will not be
enough. In order to fully verify the sensitivity, a large series of binaries with different
values for the 12 + s Keplerian, orbital and gravitational wave parameters must be
constructed. Polynomial Search is then to be run on the combined data plus signal
stream. The computing power required may be problematic as the verification will take
more CPU time than the search itself. Also, due to the high dimensionality of the
parameter space the result can only be an average sensitivity of the search. A possible
way of implementing such a verification could be to inject a limited set of binaries with
5
The Ligo Analysis Library (LAL) [71] provides various methods to estimate the PSD and will
possibly be used in future developments of Polynomial Search.
114
5.9. Discussion and outlook
randomized parameters.
Another point of discussion is the fact that all tests have assumed that the detector
antenna pattern function can be considered constant for the duration of a single FFT. In
reality this is not the case and the sensitivity of Polynomial Search will be a function of
sky position of the source. Since Polynomial Search is designed to be an omni-directional
search, there is no a priori way to fold the detector beam pattern function into the
sensitivity. However, when the coherent time of the search is limited to T 6 h, it
is possible to take the beam pattern function into account by weighing the filter bank
with the sky and polarization-averaged beam pattern function. Since this sky-average
still depends on the detector position as a function of time, each time the filter bank
is applied to a stretch of data, it has to be weighed with a different sky-average. When
averaging the sky and polarization-average over a single day the mean value of the beam
pattern function is computed to be 0.42 and the variation is 0.16 [42]. This implies that
the reduction of the sensitivity of Polynomial Search will roughly be around this average
value. However, for a given polarization and sky position, the sensitivity can be easily
recomputed.
5.9.1
Follow-up and parameter estimation strategies
Polynomial Search, after setting a threshold on the significance, will give candidates in
the form of frequency intervals where the significance traverses this threshold. These
candidates can already be distinguished from the background with good sensitivity (depending on the significance threshold).
The sensitivity of Polynomial Search can be improved by adding a follow-up step,
where the coherence time of Polynomial Search is increased for the candidates. This
will require the filter bank to be recomputed and this new filter bank will then only be
applied to the frequency range of the √
candidates. Since increasing the overall sensitivity
of Polynomial Search will scale with T , an increase in coherence time will be a larger
gain in sensitivity than adding more FFTs to the original search. On the other hand
the limit on the coherence time will be dictated by the orbital period of the binary
(the Taylor expansion of the phase must remain valid during time T ). In addition, the
Earth’s rotational period will impose a limit on T of no more than about 6 h due to
the fact that the induced Doppler-shift of the gravitational wave signal is sinusoidal and
that the beam pattern function will vary too much. At maximum such a follow-up step
will gain about a factor of 4 in sensitivity.
When a gravitational wave candidate is found, the next step will be to derive an estimate of the physical parameters of this candidate. One approach could be to perform
a Bayesian parameter estimation method together with a Markov Chain Monte Carlo
technique. A similar strategy is used in the targeted isolated neutron star search [72, 73].
However, physical templates must be generated in order to compute the likelihood for a
particular parameter set which is a highly CPU intensive task. Since a template-based
all-sky search for gravitational waves from neutron stars in binary systems was shown
to be unfeasible, it would require further study to see if such templates could be used
for parameter estimation of the candidates. An alternative approach would be to fit the
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Chapter 5. Polynomial Search
phase model of the gravitational wave directly to the parameters of the hits of the candidates. This method could possibly be less CPU intensive, however such an approach
has never been attempted before in the LIGO-Virgo data analysis community.
Both of these methods and possible follow-up steps are outside the scope of this
thesis.
116
Chapter
6
Software and computing
6.1
The AnaPulsar framework
Polynomial Search has been implemented in a larger framework called AnaPulsar. This
is meant to be a multi-purpose environment in which different types of analyses can be
implemented. In order to keep the framework versatile and easily extendable by future
developers, an object-oriented (OO) approach has been chosen. AnaPulsar is entirely
written in C++ and includes classes for defining analysis pipelines, creating and handling
data containers, exception handling, IO and controlling the framework by parsing option
files. AnaPulsar uses the following external libraries:
ˆ BOOST v1.38.0 [74]: A general purpose collection of portable C++ source libraries.
ˆ FrLib v8.11 [75]: The official data format for various gravitational wave experiments, including Virgo and LIGO.
ˆ fftw3 v3.2.2 [76]: The ‘Fastest Fourier Transform in the West’ (FFTW) is a free
collection of fast C routines for computing the Discrete Fourier Transform.
ˆ ROOT v5.25.02 [77]: A set of OO frameworks with all the functionality needed to
handle and analyze large amounts of data in an efficient way.
The AnaPulsar framework has been developed with certain initial design choices:
ˆ A gravitational wave analysis consists of a series of operations on data elements.
ˆ Each data element has an associated data container.
ˆ Each analysis on a data container has an associated metadata container.
ˆ Each analysis has an input and output stream from which the data are read and
to which the data are written.
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Chapter 6. Software and computing
ˆ The framework should be able to run the different types of implemented analyses by supplying an option file in which the desired analyses together with their
parameters are specified.
ˆ Each class in this framework should inherit from the base class:
– Each derived object has an associated state which determines whether the
object is valid.
– Any derived object must have a method which prints information about its
class.
Figure 6.1 shows the inheritance graph of AnaPulsar. As can be seen, the framework
has been compartmentalized into pieces with similar behavioral properties. These commonalities have been captured into a series of (major) abstract base classes called Options, AnalysisBase, DataObj, Generator, GenericException, GenericOperation
and IOobj. Furthermore, some minor utility classes exist which do not have any obvious commonalities with other classes. These classes are called OptionFileReader,
LogStream, AnaPulsarTime and AnaPulsarTimeDuration. Appendix D gives a more
detailed view of the inheritance relationships between the classes and their data members and methods. The following text will briefly highlight the more important classes
of the framework. A more detailed description can be found in the Doxygen-generated
code manual from the source code [78].
The AnaPulsar framework is controlled by supplying a text file containing the desired parameters and (partial) analyses the user desires. These so-called option files have
a specific syntax which can be found in the code documentation. An option file is processed by the OptionFileReader. All Options-derived classes contain the information
parsed from the option file, to initialize the relevant analyses. An analysis requires one
or more Option-derived classes to be initialized (i.e. an SFTCreation-analysis requires
an FFTOptions class in order to be initialized).
All AnalysisBase-derived classes implement the various analyses in AnaPulsar.
These analysis classes are initialized by using the information from the option file. This
information has been converted by the OptionFileReader to a series of Option-derived
classes which are collected in one of the Analysis-option classes (i.e. SFTCreationOptions for creating FFTs. As can be seen in Fig. 6.1 there are currently three AnalysisBase-derived classes implemented. Table 6.1 gives a short description of each of these
classes.
All DataObj-derived classes contain various types of data. These types can be metadata, THitTuple classes and time and frequency domain data containers. These time
and frequency domain containers derive from the DataContainer class since the FrequencyDomainData and TimeDomainData both contain a pointer to a data block. Table
6.2 gives a short description of all the DataObj-derived classes.
All Generator-derived classes are used to generate DataObj-derived objects. Table
6.3 shows all the Generator-derived objects together with a short description.
All GenericOperation-derived classes are (complex) operations which are to be performed on the data. GenericOperation-derived classes are meant to operate on one sin-
118
6.1. The AnaPulsar framework
Figure. 6.1: The inheritance graph of AnaPulsar. The boxes represent the various
classes, the solid curves represent the inheritance relations and the dashed
curves represent template instantiations.
119
Chapter 6. Software and computing
This analysis takes the input data stream, divides it up in
chunks of a specified length and performs a fast fourier transform by using fftw3.
AvgPSDCreation:
This analysis takes the FFTs which have been created by
SFTCreation and estimates the PSD of the data by averaging
the FFTs per supplied frequency range.
PolynomialSearch: This analysis performs Polynomial Search on an SFT-by-SFT
basis by taking filters from the FilterGenerator and applying
them one-by-one to the SFT.
SFTCreation:
Table. 6.1: A short description of the AnalysisBase-derived classes currently implemented in AnaPulsar.
AvgPSDContainer: Contains the computed PSD from the data split into different
frequency ranges as supplied by the user.
THitTuple:
Contains the hits from Polynomial Search in a ROOT TNtuplederived object.
DataContainer:
Is the base class of the classes FrequencyDomainData, TimeDomainData, FrequencyDomainPolynomialFilter and TimeDomainPolynomialFilter which contain the associated time and
frequency domain data.
SFT_MetaData:
Contains the metadata associated with a single SFT resulting
from the SFTCreation-analysis.
AvgPSDMetaData:
Contains the metadata associated with the result of the AvgPSDCreation-analysis.
PSMetaData:
Contains the metadata associated with the THitTuple resulting
from the PolynomialSearch-analysis applied to a single SFT.
Table. 6.2: A short description of DataObj-derived objects.
gle DataObj-derived object at a time. The different operation-derived classes can be split
into two categories: the metadata operations and the operations which are performed
on the data. The metadata operation classes (AvgMetaDataOperation, PSMetaDataOperation and SFT_MetaDataOperation) are used to generate the metadata for each of
the AnalysisBase-derived classes discussed in Table 6.1. Table 6.4 shows the operation classes besides the metadata operation classes together with a short description.
Additional documentation of discussed classes as well as the remaining classes is available with the AnaPulsar source distribution.
The polynomial search algorithm up to the registration of the hits is implemented
in AnaPulsar. Further analysis on these hits is implemented in a series of PyROOT [77]
scripts. The counting of the hits as well as applying the consistent filter criterion are
considered part of the post processing and are not implemented directly in AnaPulsar.
Figure 6.2 shows the implementation of the first part of the polynomial search algorithm schematically. In this figure, the arrows with the solid line indicate the flow of the
120
6.1. The AnaPulsar framework
FakeDataGenerator: This class generates white, Gaussian noise with a certain mean
and variance supplied by the user in the time domain. Additionally it is possible to add a signal of the form
h(t) = As sin(2π(f0 t +
FilterGenerator:
α 2 β 3
t + t )) + φ0 ),
2
3
where As , f0 , α, β and φ0 are parameters specified in the
option file.
This class generates the filters used by Polynomial Search. It
is possible to either generate filters by using a rectangular, regular grid in parameter space or by using the step sizes defined
in section 5.5.
Table. 6.3: A short description of the Generator-derived objects.
This class computes the mean and variance of the
FFTs in the desired frequency range. This is done in
the AvgPSDCreation class by passing all the FFTs
through this operation twice (first to compute the
mean and the second time to compute the variance).
FFTFrequencyDomOperation: This class computes the FFT of frequency domain
data, wrapped in a FrequencyDomainData container
resulting in a TimeDomainData container.
FFTTimeDomOperation:
This class computes the FFT of a time series wrapped
in a TimeDomainData container resulting in a FrequencyDomainData container.
FilterApplicator:
This class uses the FilterGenerator to generate
polynomial filters and applies them to the FFT.
FilterSearchOperation:
This class is used by the PolynomialSearch analysis
class to perform Polynomial Search on all the data.
AvgPSDOperation:
Table. 6.4: A short description of the GenericOperation-derived objects.
data, the square boxes represent the different data containers where D(t) denotes the
time domain data, D(f ) the frequency domain data, ‘Sn (f )’ the estimated PSD, ‘Filter’
the polynomial filter, ‘Cmax ’ the maximum correlation for each filter-data-correlationcomputation and ‘Collection of Cthr ’ are the maximum correlations for all the filters and
for all the FFTs. The elliptical balloons and arrows with the dashed lines indicate the
different operations and the data stream they affect. SF T is the SFT creation operation
done by using SFTCreation, AV G the estimate of the PSD done by AvgPSDCreation,
P S Polynomial Search done by PolynomialSearch and F ILGEN the filter generator
done by FilterGenerator.
The polynomial search algorithm shown in Fig. 6.2 starts with the time domain data
stream from the detector. With the desired FFT length, supplied by the user, the FFTs
121
Chapter 6. Software and computing
Figure. 6.2: The polynomial search algorithm shown schematically. First, the data are
Fourier transformed and the frequency domain data, D(f ), are averaged
using the AV G operation resulting in the estimate of the PSD, Sn (f ).
Next, the individual chunks of frequency domain data are combined with
the estimated PSD and the filters from the filter generator in the PS operation. This is done by computing the correlation as in Eq. (5.13). Finally,
the maximum correlation is chosen per filter-data combination (Cmax )
and collected for each filter in the output of the algorithm (Collection of
Cmax ).
are generated from this data stream. These FFTs are then averaged in order to estimate
the PSD. The estimated PSD is then used to normalize the frequency domain data.
Then the filters are generated one-by-one and applied to the normalized data and the
maximum correlations are recorded.
Figure 6.3 shows the post processing part of Polynomial Search including the consistent filter criterion. Again, the data stream is indicated by the solid arrows, the square
boxes represent the different data containers and the oval shapes the different operations
on the data. The post processing analysis runs over all hits from all FFTs and starts
with the ‘Collection of Cmax ’ which is the output of the previous analysis. The hits are
generated by applying a threshold Cthr to the data. Next, the consistent filter pairs are
computed by using the tolerance supplied by the user, the consistent filter pairs are
counted and the number count statistic is generated. The number of noise hits are computed either from these number counts or from first principles as discussed in section
5.8 and from these values the significance is computed for each frequency bin. Finally a
threshold is applied to the significance. This yields a list of candidate gravitational wave
signals.
122
6.2. Computing costs
Figure. 6.3: The post processing algorithms to create candidates from the hits.
6.2
Computing costs
Polynomial Search on a single FFT can be split into five steps which are shown in Table
6.5 together with the scaling of the number of operations required for each step. Note
1)
2)
3)
4)
5)
Computation of the FFT of the data:
Computation of the filters:
Computation of the filter-data product:
FFT back to time-domain:
Computing the maximum correlation:
∝ ND log ND
∝ Nfil Nf log Nf
∝ Nfil? Nf
∝ Nfil? Nf log Nf
∝ Nf Nfil?
Table. 6.5: The scaling of the number of operations per separate step of the polynomial
search algorithm.
that the computation of the PSD has been omitted since the computing cost for this is
negligible compared to Polynomial Search itself. In Table 6.5 ND represents the number
of sample points in a single FFT of the data, Nfil the number of explicitly computed
filters (e.g. the number of α and β steps), Nf the number of sample points in the filter
and Nfil? the number filters including the f0 steps. Since all these operations are applied
to NFFTs FFTs, everything scales linearly with NFFT . The computation of the FFT of
a data block has to be done only once per entire filter bank. This implies that the
most computationally intensive part of the algorithm depicted in Fig. 6.2 lies in the P S
operation (steps 2 to 5).
123
Chapter 6. Software and computing
Increasing the total observation time (equivalent to increasing NFFT ) will increase
the number of operations in a linear way. Increasing the coherence time however, will
increase NF in a linear way and Nfil? in a non-linear way due to the way the step sizes
in parameter space are chosen. This implies that wanting to improve the sensitivity by
increasing the coherence time will run into computational limitations much sooner than
simply adding more FFTs.
The application of Polynomial Search with parameters from Table 5.6 took on average
≈ 1.2 h per FFT on an Intel(R) Xeon(R) CPU L5520 2.27GHz. As shown in the table
this filter bank covers a frequency range of 16 Hz, the filters are sampled at 32 Hz, the
data are sampled at 4000 Hz and the FFT length is 1200 s. A month of data can be
searched over a band from 10 Hz to 1 kHz in ≈ 36 CPU year on this particular CPU.
Of the algorithm depicted in Fig. 6.3 the most computational-intensive part is the
‘Consistent filter criterion’ step. When applying the consistent filter criterion the number
of operations per pair of FFTs will scale as
X
NFFT /2
Nopt ∝
Nhits,2i Nhits,2i+1 ,
(6.1)
i=0
where Nhits represents the number of hits per FFT given a certain threshold. The number
of operations grows rapidly for lower thresholds due to the fact that the false-alarm
probability as a function of the threshold approaches a step function (see Fig. 5.14).
When comparing the computational costs shown in Table 6.5 to Eq. (6.1) it can be seen
that the number of operations for applying the consistent filter criterion will be less than
that for the actual polynomial search. This has to do with the fact that in order to have
a good separation between signal and noise correlations it is required that Nfil? Nhit .
10%
When demanding consistent filter pairs with a threshold of Cthr
= 4.88, on average
32, 484 filters will pass the threshold per Hz per FFT. Since at fixed threshold the
number of hits registered does not vary more than 1%, the assumption is made that the
computing power scales linearly with the number of FFTs and 16 Hz frequency bands.
When running the consistent filter code on these hits on the same Intel(R) Xeon(R)
CPU L5520 2.27GHz CPU, the average CPU time is about 6.8 minutes per FFT pair.
Analyzing the entire frequency band (about 1 kHz) for one month of data requires about
10 days of CPU time. This is negligible compared to the CPU time required to perform
the initial filter search.
In order to perform Polynomial Search in a timely fashion, the framework is run on
the Grid [79]. Running analyses on the Grid requires tools for data and job-management
in the form of Python scripts. These have been included with the AnaPulsar sourcedistribution. When running the analysis on the Grid, the total number of CPUs needed
to run Polynomial Search including the consistent filter criterion in real-time is computed
waves from neutron stars in
to be about1 433. Hence an all-sky search for gravitational
2 df d
f
binary systems (with dt < 4 × 10−3 Hz/s and dt2 < 2 × 10−5 ) Hz/s2 can be performed
in a reasonable time with moderate computational resources.
1
Note that this an approximate number since it depends on various circumstances such as the precise
CPUs used, data transfer speeds, etc.
124
6.2. Computing costs
The computing requirements can be compared to the most sensitive all-sky search
for isolated pulsars, the Einstein@Home search [62, 80]. This search is a template-based
all-sky search using the F-statistic. When run on the S5 data set the search ran over 840
hours of LIGO data and for coherent stretches of 30 hours the search took about 1 × 104
CPU years. Even though this search’ sensitivity is roughly an order of magnitude better
than Polynomial Search, it is meant for isolated pulsars. Thus it covers a single spin
down parameter f˙ (comparable to the α-parameter in Polynomial Search) of −f /τ <
f˙ < 0.1f /τ where τ = 103 yr for f < 400 Hz and τ = 4 × 103 yr for f > 400 Hz.
125
Chapter 6. Software and computing
126
Chapter
7
Summary
General relativity predicts the emission of gravitational waves by any mass distribution
which has a time-dependent quadrupole (or higher order multipole) moment. Gravitational waves are perturbations of the background metric traveling at the speed of light.
A gravitational wave induces strain in spacetime which affects the relative distance
between two freely-falling test masses. Gravitational waves emitted from astrophysical
sources can be detected with modern day interferometric gravitational wave detectors.
One such detector is Virgo which is located in Italy near the city of Pisa. The Virgo
detector is a Michelson interferometer with 3 km long Fabry-Perot resonant cavities as
arms. The entire detection band runs from approximately 20 Hz to 10 kHz (sensitivities
about 10−20 Hz−1/2 and 10−21 Hz−1/2 respectively), and the best sensitivity of the Virgo
detector per 1-10-2009 was 6 × 10−23 Hz−1/2 for gravitational waves of 200 Hz.
Due to the losses in the mirrors of the arm cavities, the stored laser power will cause
these mirrors to heat up. This heating will cause variations in the optical path length
which change the properties of the Fabry-Perot cavities. This causes adverse effects in
the stable operation of the interferometer (e.g. loss of lock) and will limit the laser power
which can be injected in the interferometer.
Chapter 3 of this thesis describes a three dimensional finite element model of the
Virgo input mirror, including the recoil mass, which is used to simulate the heating
effects. It has been shown that in the steady state situation, the average temperature of
the mirrors increase by 8.52 mK. Furthermore, the simulation shows that two principal
eigenmodes of the input mirrosr (the drum and butterfly modes) increase in frequency
with 0.42 Hz/K and 0.29 Hz/K, respectively. These results can be used to monitor the
mirror temperature by extracting the resonance frequency of the modes from the detector output data. It has been shown that the mirrors have an even higher fractional
absorption value that expected. In the case of the West End input mirror the losses are
about an order of magnitude higher than previously thought. Also, the finite element
analysis shows that in the transient situation, the average temperature increase is governed by a time constant which is computed to be 4 hours. Finally, due to the shape of
the beam, the mirrors will heat up in a non-uniform way resulting in an effective radius
of curvature of 21 km.
One of the sources of gravitational waves are non-axisymmetric rotating neutron
127
Chapter 7. Summary
stars. Due to the high magnetic fields of these stars, cones of electromagnetic radiation
can be emitted from their magnetic poles. If such a neutron star is oriented favorably
with respect to Earth, these cones of radiation can be observed as pulses. These neutron stars are known as ‘pulsars’. From observing many of these pulsars and taking the
observational biases into account, about 105 active pulsars are expected to exist in our
Galaxy. Furthermore, it has been observed that the majority of the observed pulsars
with rotational frequencies above 20 Hz are in binary systems. Since the emission of
gravitational waves is not limited to just the magnetic poles, the unobserved neutron
stars could be seen as potentially observable gravitational wave sources. The amplitude
of the gravitational waves emitted by such systems is weak. However, they are also continuous in nature, meaning that observing such systems over longer stretches of time
will increase the signal-to-noise ratio with the square root of the observation time.
In order to take full advantage of the long integration times of a potential continuous
gravitational wave signal, despite the fact that the locations of the majority of the neutron stars in our Galaxy are unknown, so-called all-sky searches have been developed.
A fully-coherent search (i.e. the data are integrated over the entire observation time)
has been shown to be computationally limited. The most sensitive search for isolated
neutron stars required 104 CPU years and was limited to 30 hours of data. In order to
increase the observation time with limited computational resources, sub-optimal analysis methods have been developed. These so-called ‘semi-coherent’ analysis methods are
based on taking multiple short coherent stretches of data and combining them in an
incoherent way. What these searches lack in sensitivity, they partially make up in decreased computing requirements making it possible to consider longer stretches of data.
When attempting to apply such semi-coherent searches to signals from neutron stars in
binary systems, the initial coherent integration time is shorter by an order of magnitude
with respect to the isolated neutron star case. This short integration time will limit the
sensitivity of the semi-coherent analyses when applying them to binary systems.
A new data analysis algorithm, called ‘Polynomial Search’ has been developed and
is described in Chapter 5. Polynomial Search is designed to be an all-sky search for
gravitational waves from neutron stars in binary systems. Since Polynomial Search employs a bank of polynomial phase templates up to 4th order in time, the initial coherent
integration time can be increased by an order of magnitude with respect to applying
a traditional semi-coherent search to this signal. It has been shown that the increase
in coherence time is about 25 for neutron stars in a Keplerian orbit, where the orbital
period (P ) is larger than 2 hours for circular orbits (eccentricity e = 0). A filter bank has
been constructed which has been shown to cover all gravitational waveforms of which
the orbital parameters obey P > 2 hours for circular orbits and P > 6 hours for eccentric orbits with 0 < e < 0.6. These filters can be applied to the data with a coherence
time of 600 seconds (whereas the coherence time of a semi-coherent search for circular
orbits with P = 2 hours would be limited to 20 seconds). Furthermore, a strategy for
combining the results from the individual coherent stretches, called ‘the consistent filter
criterion’, has been developed. When applying Polynomial Search to white noise and to
simulated gravitational wave signals, the projected sensitivity of the new search method
has been presented.
128
Finally, in chapter 6 the developed framework is presented in which Polynomial
Search has been implemented. This so-called ‘AnaPulsar’ framework has been developed for implementing various types of gravitational wave data analysis algorithms.
Furthermore, the computing requirements of Polynomial Search have been discussed. It
has been shown that in order to perform Polynomial Search on all gravitational wave
signals originating from neutron stars in a Keplerian orbit with parameters P > 2 hours
and e = 0, or P > 6 hours and 0 < e < 0.6, for an analysis of 1 month of data
approximately 36 CPU years are required.
129
Chapter 7. Summary
130
Appendix
A
Fourier transform definitions
The Fourier transform of function s(t) and its inverse are defined as
Z ∞
F [s(t)] =
s(t)e2πif t dt = s̃(f )
Z−∞
∞
−1
F [s̃(f )] =
s̃(f )e−2πif t df = s(t),
(A.1)
(A.2)
−∞
where f represents the frequency in Hz in case t is measured in seconds, and s̃(f )
represents the Fourier transform of s(t). The correlation csg (τ ) between a function s(t)
and g(t) is defined as
Z
csg (τ ) ≡
∞
g(t)s(t + τ )dt,
(A.3)
−∞
where the correlation at a fixed lag, τ , measures the overlap of function g(t) with function
g(t + τ ). The parameter τ can be interpreted as a lag parameter with which the function
g is shifted in time.
Combining Eqs. (A.1), (A.2) and (A.3) results in the correlation theorem stating
that
csg (τ ) = F −1 [s̃(f )g̃(f )∗ ] .
(A.4)
When taking g(t) = s(t), the Wiener-Khinchin theorem [81] is obtained,
F(css (τ )) = |s̃(f )|2 ,
(A.5)
stating that the Fourier transform of the autocorrelation css is equal to the power spectrum of s(t). Using Eqs. (A.3), (A.4) and (A.5) leads to Parseval’s theorem
Z ∞
Z ∞
2
Ps ≡
|s(t)| dt =
|s̃(f )|2 df,
(A.6)
−∞
−∞
stating that the total power in s, Ps , is the same if computed in the frequency or the
time domain. When s(t) is a real function (e.g. s(t) represents the detector output),
then the total power in s becomes
Z ∞
Ps = 2
|s̃(f )|2 df.
(A.7)
0
131
Appendix A. Fourier transform definitions
The normalized power spectrum of s is then defined as
s̃(f )
ŝ(f ) = √ .
Ps
(A.8)
The spectral power of a function s between f and f + df can thus be written as
Ps (f ) = 2 |s̃(f )|2 ,
(A.9)
where Ps (f ) is the single sided power spectral density (PSD) of s.
If the function s(t) is not square integrable, which is the case for continuous signals,
then the power spectral density will become infinite. Consequently, a better measure
for the power spectral density is the power spectral density per unit time. The power
spectral density of random noise, Sn (f ) is defined as the ensemble average of many
possible realizations of the noise. When observing a noise source n(t) over a time period
T , the PSD is computed in Ref. [27] to be
Sn (f ) = 2
< |ñ(f )|2 >
,
T
(A.10)
where ñ is the Fourier transform of the noise source and the brackets denote the ensemble
average. In the case of Gaussian and white noise with mean µ = 0 and variance σ 2 , the
PSD reduces to
Sn (f ) = Sn = 2σ 2 .
(A.11)
Note that by defining the noise as white the PSD per definition does not depend on the
frequency.
When considering data as a series of discretely sampled points, the continuous signal
s(t) is replaced by a finite number of N samples. These N samples are the result of
sampling the signal over a time T with sample frequency1 fs . When defining
sk = s(tk ),
k = 0, 1, 2, ..., N − 1,
tk = k/fs ,
N = fs T,
the Fourier transforms defined in Eqs. (A.1) and (A.2) reduce to the Discrete Fourier
Transforms (DFT)
s̃n =
sk =
N
−1
X
sk e2πikn/N
k=0
N
−1
X
1
N
n=0
1
,
fs
s̃n e−2πikn/N
(A.12)
1
,
T
(A.13)
where fn and tk are the n-th and k-th frequency and time sample respectively. It has
been chosen to apply the normalization factor, 1/N , in the inverse Fourier transform.
1
When sampling data with a certain sample frequency, the Shannon-Nyquist theorem states that a
band limited analog signal can be perfectly reconstructed from an infinite sequence of samples when
the highest frequency component in the signal is at most fN = fsample /2, where fN is known as the
Nyquist frequency.
132
In practice, discrete Fourier transforms are computed by using a Fast Fourier Transform (FFT) algorithm [81]. Instead of requiring O(N 2 ) operations to compute a DFT,
the FFT algorithm enables the computation of a Fourier transform in O(N log2 N ) operations.
Furthermore, Eqs. (A.3) and (A.4) can be written in discrete form as
c(s, g)j =
N
−1
X
sj+k gk = F −1 [s̃k g̃k∗ ] ,
(A.14)
k=0
where j = τk fs . Now, the power spectral density per unit time is
Ss (fk ) =
2
|s̃k |2 ,
T
(A.15)
133
Appendix A. Fourier transform definitions
134
Appendix
B
The correlation statistic
When searching for a known signal in a noise background, a so-called matched filtering
approach is the most optimal [27]. A matched filter is obtained by correlating a signal
template to a data stream containing noise to detect the presence of the signal in the
data. The matched filter is the optimal linear filter for maximizing the signal to noise
ratio (SNR) in the presence of additive stochastic noise.
The detector output D(t) can be written as
D(t) = s(t) + n(t),
(B.1)
where s(t) and n(t) are the gravitational wave signal and the stochastic noise respectively. The correlation given in Eq. (A.4) between the detector output and a filter function F , is written in time and frequency domain for τ = 0 as
Z ∞
Z ∞
C=
D(t)F (t)dt =
D̃(f )F̃ ∗ (f )df.
(B.2)
−∞
−∞
When the shape of the gravitational wave, h(t) is known, the filter function that maximizes the signal-to-noise ratio for such a signal is known as the ‘optimal filter’. This
technique of matching the filter with the signal is called ‘matched filtering’.
The signal-to-noise ratio (in amplitude) is defined as S/N where S is the expected
value of C when the signal is present and N is the rms value of C when no signal is
present. The signal-to-noise ratio with filter function F is computed in Ref. [27] to be
R∞
s̃(f )F̃ ∗ (f )df
S
−∞
≡ R
(B.3)
1/2 ,
N
∞
2
1/2Sn (f )|F̃ (f )| df
−∞
where Sn is the single-sided power spectral density of the noise.
The optimal filter function is the function which maximizes Eq. (B.3). This maximization yields that the optimal filter has the form
F̃ (f ) = const.
s̃(f )
,
Sn (f )
(B.4)
135
Appendix B. The correlation statistic
where the constant is arbitrary since rescaling by an overall factor does not change the
signal-to-noise ratio. The signal-to-noise ratio for this optimal filter can be computed by
substituting Eq. (B.4) into Eq. (B.3) and yields.
2
Z ∞
S
|s̃(f )|2
=4
df.
(B.5)
N
Sn (f )
0
√
When taking a finite observation time T , the signal-to-noise ratio grows with T .
This can be seen by considering the fact that the power |s(f )|2 in Eq. (B.5) will grow
linearly with T while the power spectral density Sn will remain constant by construction
as defined in appendix A .
Polynomial Search is based on the assumption that shape of a GW signal can be
approximated by polynomial filters. These polynomial filters are per definition suboptimal thus the SNR shown in Eq. (B.5) does not apply. The SNR for Polynomial
Search can be computed from Eq. (B.3). Considering the fact that Polynomial Search
is applied to small frequency intervals of the data, the PSD in each interval can be
considered white (independent of frequency: Sn (f ) = Sn ). Under this assumption, the
denominator in Eq. (B.3) becomes
r
1/2 r
Z ∞
1
1
2
|F̃ (f )| df
=
Sn ×
Sn P F ,
2
2
−∞
where PF is defined in Eq. (A.6). With the notation defined in Eq. (A.8), Eq. (B.3)
becomes
R∞
s̃(f )F̂ ∗ (f )df
S
−∞
q
=
.
(B.6)
N
1
S
2
n
The numerator in Eq. (B.6) is the expectation value of C when signal is present. This
expectation value is obtained by computing the correlation of the data stream with the
filter as in Eq. (B.2). Furthermore, in order to accommodate for the sliding of the filter
function in time, the filter function is multiplied with a phase factor in the frequency
domain1 . Finally, defining a filter function parameter vector ξ and combining all the
aforementioned arguments yields the Polynomial Search correlation statistic, C(τ ; ξ), as
Z ∞
~
D̃(f )F̂ ∗ (f ; ξ)
~
q
C(τ ; ξ) =
e−2πif τ df.
(B.7)
1
−∞
S
2 n
Note that Eq. (B.7) is derived from Eq. (B.3) and thus is equivalent to a signal-to-noise
ratio. When considering the fact that the data are sampled during time T with sample
frequency fs , Eq. (B.7) becomes
Nf
1 X D̃(fk )F̂ ∗ (fk ) −2πifk τj
q
C(τj ) =
e
.
T k=0
1
S
2
(B.8)
n
In general one can show that F(g(t + τ )) = g̃(f )e−2iπf τ for any quadratically integrable function
g(t).
1
136
Here τj is the j th time sample, k denotes the frequency bin number, Nf = fs T /2 is the
number of frequency bins and fk is the k th frequency bin.
In order to distinguish between the correlation and the correlation statistic, the
correlation is denoted by a csg , where the sub indices denote the functions between which
the correlation is computed while the correlation statistic derived in Eq. (B.7) will be
~ However, since C(τ ; ξ)
~ can also be seen as a rescaled correlation, it
denoted by C(τ ; ξ).
will often be referred to as the correlation.
Assuming a coherent integration time T , Eq. (B.7) can be written in the time domain
by using the correlation theorem shown in Eqs. (A.3) and (A.4). The correlation statistic
in time-domain is
r Z T
2
~ =
~
C(τ ; ξ)
D(t)F̂ (t − τ ; ξ)dt.
(B.9)
Sn 0
And in the discrete case
r
~ =
C(τj ; ξ)
Ns
2 1 X
~
D(tk )F̂ (tk − τj ; ξ),
Sn fs k=0
When substituting Eq. (B.1) into Eq. (B.9), the correlation statistic becomes
r Z T
Z T
2
~
~
~
s(t)F̂ (t − τ ; ξ)dt +
n(t)F̂ (t − τ ; ξ)dt .
C(τ ; ξ) =
Sn
0
0
(B.10)
(B.11)
From Eq. (B.11) it can be seen that when the noise is Gaussian, C will also be Gaussian.
The mean and variance of C can be computed from Eq. (B.11). As the first term in Eq.
(B.11) does not contain any random variables (only n(t) is a random variable), the
variance can be computed to be
r Z T
2
~
Var(C) = Var
n(t)F̂ (t − τ ; ξ)dt
Sn 0
Z T
2
~
Var
n(t)F̂ (t − τ ; ξ)dt
=
Sn
0
Z T
2
~
=
Var(n(t))F̂ 2 (t − τ ; ξ)dt
Sn 0
2 2
=
σ = 1,
(B.12)
Sn
where the final equality arises from substituting the PSD of Gaussian noise (Eq. (A.11)).
When computing the mean of Eq. (B.11), it is important to realize that Mean(n(t)) =
0 so that
r Z T
2
~
Mean(C) = Mean
s(t)F̂ (t − τ ; ξ)dt
Sn 0
r Z T
2
~
=
s(t)F̂ (t − τ ; ξ)dt.
(B.13)
Sn 0
137
Appendix B. The correlation statistic
Polynomial Search operates under the assumption that the signal s(t) can be approximated by a polynomial during time T . Furthermore, T is assumed to be short enough
that amplitude modulation due to the beam-pattern functions do not affect the signal shape during the coherent integration time (e.g. the beam-pattern functions can be
taken constant). With these assumptions, the signal is of the form
s(t) = Asig sin (Φ(t)) ,
(B.14)
where Asig is the GW amplitude arriving at the detector (i.e. as given in Eq. (1.15))
times a constant due to the beam-pattern function and the phase Φ(t) is the Dopplershifted phase of the GW signal. With the signal shape shown in Eq. (B.14), the total
power in the signal, Ps can be computed to be
Z T
Z T
1 2
T
2
2 T
cos (2Φ(t)) dt ≈ A2sig ,
Ps =
s (t)dt = Asig − Asig
(B.15)
2
2
2
0
0
where the assumption that the integral over cos(2Φ(t)) term can be neglected as it is
a rapidly oscillating term and will be small compared
√ to the Asig T /2 which will grown
linearly with T . With the notation ŝ(t) = s(t)/ Ps (Eq. (A.8)), Eq. (B.13) can be
rewritten as
r Z T
T
~ = SNR × csF (τ ),
Mean (C) = Asig
ŝ(t)F̂ (t − τ ; ξ)dt
(B.16)
Sn 0
r
where
SNR ≡ Asig
T
Sn
(B.17)
and csF (τ ) is the correlation between normalized filter function F̂ and signal ŝ. Note
that Eq. (B.16) can be rewritten as
h
i
−1
∗
Mean (C) = SNR × ci = SNR × F
ŝ(f )F̂ (f ) .
(B.18)
Note that the functions ŝ and F̂ are periodic functions of time. This implies that −1 <
cŝF̂ (τ ) < 1; the correlation will be +1 if F̂ matches ŝ during T and −1 if F̂ matches ŝ
but is π out of phase during T . Furthermore, if ξ~ is such that the filter does not match
the signal, the correlation will lie in between these extremal values.
Note that the data are sampled with sample frequency fs,data . Since this means that
each second, fs,data samples are taken, the PSD of a Gaussian white noise process with
time-domain amplitude becomes
2A2noise
(B.19)
Sn =
fs
Inserting this expression in Eq. (B.17) yields
SNR =
Asig √
√ N,
Anoise 2
where N = T fs are the number of samples.
138
(B.20)
Appendix
C
Stepsize plots
In this appendix the slices of parameter space are shown for the remaining data sets
described in Table 5.4.
×10
1
-3
β (Hz/s^2) = 1e-06
0
600
×10-3
1
600.2
600.4
β (Hz/s^2) = 1.75e-06
0
600
×10
1
-3
600.2
600.4
β (Hz/s^2) = 2.5e-06
0
600
600.2
600.4
1
×10-3
1
0.5
0
0
600
1
×10-3
1
0.5
0
0
600
1
×10
1
0.5
0
-3
0
600
β (Hz/s^2) = 1.25e-06
600.2
600.4
β (Hz/s^2) = 2e-06
600.2
600.4
β (Hz/s^2) = 2.75e-06
600.2
600.4
1
×10-3
1
0.5
0
0
1
×10-3
1
0.5
0
600.2
600.4
β (Hz/s^2) = 2.25e-06
0
1
0.5
600
1
×10
1
0.5
0
-3
0
1
0.5
600
0
β (Hz/s^2) = 1.5e-06
600.2
600.4
β (Hz/s^2) = 3e-06
0
1
0.5
600
600.2
600.4
0
Figure. C.1: Step size plots for data set 2 (Table 5.6). The plots show α versus f0 for
different values for β and the color codes shows the correlation, normalized to unity.
139
Appendix C. Stepsize plots
3
×10-6
α (Hz/s) = -0.00016
3
0.5
2
0
1
1
3
0.5
2
0
1
1
3
0.5
2
0
1
1
3
2
1
3
600
×10-6
600.2
600.4
α (Hz/s) = -1e-05
2
1
600
×10
-6
3
600.2
600.4
3
600
×10-6
600.2
600.4
f0 (Hz) = 599.98
0.5
2
600
×10-6
600
×10
-6
α (Hz/s) = 0.00014
2
1
×10-6
1
600
×10-6
α (Hz/s) = -0.00011
600.2
600.4
α (Hz/s) = 4e-05
600.2
600.4
α (Hz/s) = 0.00019
600.2
600.4
f0 (Hz) = 599.985
0
1
3
0
×10-6
0.001
f0 (Hz) = 599.995
1
0.5
2
0
×10
-6
3
0.001
f0 (Hz) = 600.01
3
0
×10-6
0.5
2
0
0.001
0
1
1
3
0.5
2
0
1
1
3
0.5
2
0
1
1
3
1
600
×10-6
600.2
0
3
×10
0.001
f0 (Hz) = 600.015
α (Hz/s) = 9e-05
0.5
2
600
×10
600.2
600.4
α (Hz/s) = 0.00024
0
0.001
0
1
0.5
600
×10-6
600.2
600.4
f0 (Hz) = 599.99
0
1
0.5
2
0
1
3
0
×10-6
0.001
f0 (Hz) = 600.005
1
0.5
2
0
1
0
3
×10
0.001
f0 (Hz) = 600.02
1
0.5
2
0
0
1
1
0.5
-6
1
0
600.4
0
1
1
0.5
-6
0.5
0
1
2
0.001
f0 (Hz) = 600.0
2
-6
1
0.5
α (Hz/s) = -6e-05
0
1
0
1
3
0.5
2
×10-6
1
1
0
0.001
Figure. C.2: Step size plots for data set 2 (Table 5.6). The top nine plots show β versus f0 for different values for α. The bottom nine plots showα versus β
for different values for f0 . The color codes shows the correlation, normalized to unity.
140
×10-3
1.1
β (Hz/s^2) = 1e-06
0.5
1
0.9
599.9
600
×10-3
1.1
0
600.1
β (Hz/s^2) = 1.75e-06
1
0.5
1
0.9
599.9
600
×10
1.1
-3
0
600.1
β (Hz/s^2) = 2.5e-06
600
×10-3
1.1
0
600.1
β (Hz/s^2) = 1.25e-06
0.9
599.9
600
×10-3
1.1
0
600.1
β (Hz/s^2) = 2e-06
1
0.5
1
0.9
599.9
600
×10
1.1
0
600.1
β (Hz/s^2) = 2.75e-06
600
×10-3
1.1
0
600.1
β (Hz/s^2) = 1.5e-06
1
0.5
1
0.9
599.9
600
×10-3
1.1
0
600.1
β (Hz/s^2) = 2.25e-06
0.9
599.9
600
×10
1.1
0
600.1
β (Hz/s^2) = 3e-06
1
0.5
1
0.9
599.9
1
0.5
1
-3
1
0.5
1
0.9
599.9
1
0.5
1
-3
1
0.5
1
0.9
599.9
1
600
0
600.1
Figure. C.3: Step size plots for data set 3 (Table 5.6). The plots show α versus f0 for
different values for β and the color codes shows the correlation, normalized to unity.
141
Appendix C. Stepsize plots
3
×10-6
α (Hz/s) = 0.0009
2
1
3
600
×10-6
600.2
600.4
α (Hz/s) = 0.000975
2
1
600
×10
-6
3
600.2
600.4
α (Hz/s) = 0.00105
2
1
3
600
×10-6
600.2
600.4
f0 (Hz) = 599.64
3
0.5
2
0
1
1
3
0.5
2
0
1
1
3
0.5
2
0
1
1
3
600
×10-6
600
×10
-6
0.5
2
×10-6
1
600
×10-6
α (Hz/s) = 0.000925
600.2
600.4
α (Hz/s) = 0.001
600.2
600.4
α (Hz/s) = 0.001075
600.2
600.4
f0 (Hz) = 599.73
0
1
0.0009
3
0.001
×10-6
0.0011
f0 (Hz) = 599.91
1
0.5
2
0.001
×10
-6
3
0.0011
f0 (Hz) = 600.18
3
0.001
×10-6
0.5
2
0.001
0.0011
0
1
1
3
0.5
2
0
1
1
3
0.5
2
0
1
1
3
1
600
×10-6
600.2
600.4
α (Hz/s) = 0.001025
0.001
3
×10
0.0011
f0 (Hz) = 600.27
600
×10
0.5
2
600.2
600.4
α (Hz/s) = 0.0011
600
×10-6
600.2
600.4
f0 (Hz) = 599.82
0.001
0.0011
0
1
0
1
0.5
2
0
1
0.0009
3
0.001
×10-6
0.0011
f0 (Hz) = 600.09
1
0.5
2
0
1
0.0009
0.001
3
×10
0.0011
f0 (Hz) = 600.36
1
0.5
2
0
1
0.0009
1
0.5
-6
1
0
0.5
0
1
0.0009
1
0.5
-6
0.5
0
1
0.0009
2
0.0011
f0 (Hz) = 600.0
2
-6
1
0.5
α (Hz/s) = 0.00095
0
1
0.0009
0
1
0.0009
3
0.5
2
×10-6
1
0
1
0.0009
0.001
0.0011
Figure. C.4: Step size plots for data set 3 (Table 5.6). The top nine plots show β versus f0 for different values for α. The bottom nine plots showα versus β
for different values for f0 . The color codes shows the correlation, normalized to unity.
142
1.2
×10-3
β (Hz/s^2) = -1e-07
1
1.1
1.2
×10-3
β (Hz/s^2) = -5e-08
1.1
1
1.2
600.5
×10-3
β (Hz/s^2) = 5e-08
0
0.9
1
1.2
1.1
0.9
600
600.5
×10-3
β (Hz/s^2) = 1e-07
×10
-3
1.2
1
1.2
β (Hz/s^2) = 2e-07
0.9
1
1.2
600
×10
600.5
β (Hz/s^2) = 2.5e-07
0.9
1
1.2
600
×10
-3
600.5
β (Hz/s^2) = 3e-07
0.9
1
0
1
1.1
0.5
1
0
β (Hz/s^2) = 1.5e-07
0.5
0
1.1
600.5
×10-3
0
1
0.5
600
600.5
1.1
-3
1
600
0.5
0
1.1
0.9
0.9
1
600.5
1
0.5
0
1.1
600
β (Hz/s^2) = 0.0
1
0.5
1
×10-3
0.5
1
600
1.2
1.1
0.5
0.9
1
0.5
1
600
600.5
0
0.9
600
600.5
0
Figure. C.5: Step size plots for data set 4 (Table 5.6). The plots show α versus f0 for
different values for β and the color codes shows the correlation, normalized to unity.
143
Appendix C. Stepsize plots
0.4
×10-6
α (Hz/s) = 0.0008
1
0.2
0.4
×10-6
α (Hz/s) = 0.00085
0.2
0
0.4
600.5
×10-6
α (Hz/s) = 0.00095
0
-0.2
1
0.4
0.2
-0.2
600
600.5
×10-6
α (Hz/s) = 0.001
×10
-6
0.4
α (Hz/s) = 0.0011
-0.2
1
0.4
600
×10
600.5
α (Hz/s) = 0.00115
×10-6
f0 (Hz) = 599.28
0.2
1
0.4
-0.2
1
0.4
0.5
1
×10-6
1.2
f0 (Hz) = 599.82
0.2
0.5
0
600
600.5
1
×10
-6
1.2
f0 (Hz) = 600.36
×10-6
f0 (Hz) = 599.46
-0.2
1
0.4
0.5
1
×10-6
1.2
f0 (Hz) = 600.0
0.2
0.2
0.5
0
-0.2
0.8
1.2
600.5
×10-6
f0 (Hz) = 599.64
0.2
0
1
0.5
1
0.4
1
×10-6
1.2
f0 (Hz) = 600.18
0.2
0 -3
×10
1
0.5
0
1
×10
1.2
f0 (Hz) = 600.54
0.2
0 -3
×10 -0.2
0.8
0 -3
×10 -0.2
0.8
1
×10
-6
1
0.5
0
1
600
0 -3
×10 -0.2
0.8
0.5
-6
0.4
1
0
0 -3
×10 -0.2
0.8
1
α (Hz/s) = 0.0012
0
0.5
0
0
-0.2
0.8
600.5
0
0.2
0.4
1
0.2
0 -3
×10 -0.2
0.8
1
×10
-6
0
-0.2
0.8
600
0.5
0
0
0.4
-0.2
0
600.5
α (Hz/s) = 0.00105
0.5
0
0.2
600
×10-6
0
0
-6
0
0.4
0.4
600.5
0.2
0.5
0.4
1
600
0.5
0
0.2
-0.2
-0.2
0
600.5
1
0.5
0
0.2
600
α (Hz/s) = 0.0009
0
0.5
0
×10-6
0.5
0
600
0.4
0.2
0.5
-0.2
1
0.4
1.2
f0 (Hz) = 600.72
0.2
0 -3
×10
1
0.5
0
1
1.2
0 -3
×10 -0.2
0.8
1
1.2
0 -3
×10
Figure. C.6: Step size plots for data set 4 (Table 5.6). The top nine plots show β versus f0 for different values for α. The bottom nine plots showα versus β
for different values for f0 . The color codes shows the correlation, normalized to unity.
144
Appendix
D
AnaPulsar inheritance diagrams
In this appendix the inheritance and contents of the classes of the AnaPulsar framework
will be shown in more detail. The boxes in Figs. D.1 to D.8 have the following meaning:
ˆ A filled gray box represents the struct or class for which the graph is generated.
ˆ A box with a black border denotes a documented struct or class. This documentation is available with the source code distribution.
ˆ A box with a grey border denotes an undocumented struct or class.
Each class is denoted by the class name, a list of class data members and a list of class
methods. The − sign represents a private data member or method, the + sign represents
a public data member or method and the # sign represents a protected data member
or method.
The arrows have the following meaning:
ˆ A solid arrow is used to visualize a public inheritance relation between two classes.
ˆ A dashed arrow denotes a relation between a template instance and the template
class it was instantiated from. The arrow is labeled with the template parameters
of the instance.
145
Appendix D. AnaPulsar inheritance diagrams
Figure. D.1: The inheritance plot of the AnaPulsarBase class with all its derived
classes, data members and methods.
146
Figure. D.2: The inheritance plots of the AnaPulsarTime (left) and AnaPulsarTimeDuration (right) classes with all their derived classes, data members and
methods.
147
Appendix D. AnaPulsar inheritance diagrams
Figure. D.3: The inheritance plot of the DatObj class with all its derived classes, data
members and methods.
148
Figure. D.4: The inheritance plots of the DataStream (left) and Generator classes
with all their derived classes, data members and methods.
149
Appendix D. AnaPulsar inheritance diagrams
Figure. D.5: The inheritance plot of the GenericException class with all its derived
classes, data members and methods.
150
Figure. D.6: The inheritance plot of the IOobj class with all its derived classes, data
members and methods.
151
Appendix D. AnaPulsar inheritance diagrams
Figure. D.7: The inheritance plot of the OptionFileReader class with all its derived
classes, data members and methods.
152
Figure. D.8: The inheritance plot of the Options class with all its derived classes, data
members and methods.
153
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160
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De algemene relativiteitstheorie voorspelt dat een willekeurige massadistributie met een
tijdsafhankelijk quadrupool (of hoger) moment gravitatiegolven zal uitzenden. Deze gravitatiegolven zijn variaties op de achtergrondmetriek en planten zich voort met de lichtsnelheid. Een gravitatiegolf vervormt de ruimtetijd dusdanig dat de relatieve afstand
tussen twee testmassa’s tijdsafhankelijk wordt. Gravitatiegolven die worden uitgezonden
door astrofysische bronnen kunnen gedetecteerd worden met interferometrische gravitatiegolf detectoren. Eén van deze detectoren, Virgo genaamd, is gebouwd in Italië in de
buurt van Pisa. Virgo is een zogenaamde Michelson interferometer met twee Fabry-Perot
resonante armen van elk 3 km lang. De detectieband loopt van 10 Hz tot 10 kHz waarbij
de gevoeligheid tussen de 10−20 Hz−1/2 tot 10−21 Hz−1/2 ligt voor beide uitersten. The
beste gevoeligheid van de Virgo detector gemeten op 1-10-2009 was 6 × 10−23 Hz−1/2
voor gravitatiegolven van 200 Hz.
Omdat de zogenaamde ‘Input’ spiegels van de armen van de Michelson interferometer niet perfect reflecterend zijn, zal de laserbundel de spiegels opwarmen. Door
deze opwarming zullen de optische eigenschappen van spiegels veranderen, hetgeen de
eigenschappen van de Fabry-Perot armen beı̈nvloedt. Deze thermische effecten hebben
negatieve gevolgen voor de stabiele werking van de interferometer.
Hoofdstuk 3 van dit proefschrift beschrijft een volledig drie-dimensionaal eindige elementen model van de Virgo ‘Input’ spiegels samen met hun reactie massa’s. Dit model
wordt gebruikt om een analyse te doen naar de opwarming van de spiegels. Deze analyse
laat zien dat in de evenwichtssituatie de gemiddelde temperatuur met 8.52 mK stijgt.
Verder laat de analyse zien dat de frequentie van de twee prominente resonanties van
de input spiegels (de zogenaamde ‘drum’ en ‘butterfly’ modes) met respectievelijk 0.42
Hz/K en 0.29 Hz/K toenemen. Dit resultaat kan gebruikt worden om de spiegeltemperatuur te meten door de frequentie van de beide resonanties uit de data te extrapoleren.
Het is gebleken dat de spiegels een grotere fractie van het laserlicht absorberen dan
aanvankelijk werd voorspeld. In het geval van de ‘West End’ inputspiegel is gebleken
dat de absorptie een orde van grootte hoger is dan voorheen gedacht. In het geval
van een tijdsafhankelijke analyse is gevonden dat de gemiddelde temperatuursverhoging
bepaald wordt door een tijdconstante van ongeveer 4 uur. Tot slot is berekend dat het
intensiteitsprofiel van de laser leidt tot een niet-uniforme opwarming van de spiegel.
De opgewarmde spiegel heeft in de evenwichtssituatie een berekende kromtestraal van
161
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ongeveer 21 km.
Een van de bronnen van gravitatiegolven zijn niet axiaal-symmetrisch, roterende
neutronensterren. Doordat deze sterren een sterk magnetisch veld hebben, kunnen ze
bundels van elektromagnetische straling vanaf beide polen uitzenden. Als een neutronenster zó georiënteerd is ten opzichte van de Aarde dat de bundel de Aarde kruist, dan
wordt dit geobserveerd als een puls. Deze neutronensterren staan bekend als ‘pulsars’.
Door veel van deze pulsars te observeren en te compenseren voor het feit dat slechts een
deel van de neutronensterren als pulsars gezien kunnen worden, wordt geschat dat er
zich ongeveer 105 actieve pulsars in ons melkwegstelsel bevinden. Verder tonen waarnemingen aan dat het merendeel van de pulsars met rotatiefrequentie boven de 20 Hz
zich in binaire systemen bevindt. Omdat de emissie van gravitatiegolven niet beperkt is
tot de magnetische polen van een neutronenster kunnen de onbekende neutronensterren
gezien worden als potentiële bronnen van gravitatiegolven. De amplitude van de gravitatiegolven die uitgezonden worden door neutronensterren is zwak. Aan de andere kant
zijn de golven continue, dus als zulke systemen over langere tijd geobserveerd worden
zal de signaal-ruis verhouding toenemen met de wortel van de observatietijd.
Om volledig gebruik te maken van het continue karakter van deze gravitatiegolven en
zo de signaal-ruis verhouding te verbeteren, is het noodzakelijk om de precieze golfvorm
te kennen. Dit heeft als vereiste dat de positie van de neutronenster bekend is. Aangezien
we de meeste neutronensterren in ons melkwegstelsel niet hebben waargenomen, is het
niet mogelijk om gebruik te maken van een lange integratietijd. Er zijn zogenaamde ‘allsky’ analysemethodes ontwikkeld die in meer of mindere mate de relatie tussen observatietijd en signaal-ruis verhouding gebruiken. Het is gebleken dat een volledig coherente
analyse (d.w.z. data worden geı̈ntegreerd over de volledige observatietijd) computationeel gelimiteerd is. De meest gevoelige blinde analyse voor geı̈soleerde neutronensterren
kostte 104 CPU jaren aan computertijd en bestreek slechts 30 uur aan data. Om toch
de observatietijd te kunnen verlengen gegeven de gelimiteerde hoeveelheid aanwezige
computertijd, zijn er verschillende sub-optimale analyses ontwikkeld. Deze zogenaamde
semi-coherente analysemethodes zijn gebaseerd op het analyseren van meerdere korte
stukjes data die naderhand op een incoherente manier weer gecombineerd worden. Wat
deze individuele, korte stukjes, data missen aan gevoeligheid wordt deels gecompenseerd
door het feit dat het computationeel mogelijk is om langere stukken data te analyseren.
Het blijkt dat als zulke semi-coherente analyses toegepast worden op gesimuleerde gravitatiegolven afkomstig van neutronensterren in binaire systemen, de lengte van de korte
stukjes data met een orde van grootte gereduceerd moet worden. Deze korte integratietijd zal de gevoeligheid van de semi-coherente analyses extreem benadelen als ze zonder
meer toegepast worden op binaire systemen.
Een nieuwe analysemethode genaamd ‘Polynomial Search’ is ontwikkeld en omschreven in hoofdstuk 5. Polynomial Search is ontwikkeld als een ‘all-sky’ analyse voor
gravitatiegolven die worden uitgezonden door niet axiaal-symmetrisch roterende neutronensterren die zich in binaire systemen bevinden. Polynomial Search maakt gebruik van
een collectie van test-golfvormen gebaseerd op een polynomiaal verloop van de fase als
functie van de tijd. Omdat deze test-golfvormen (ook wel filters genoemd) tot de vierde
orde in tijd gaan, is het mogelijk om de initiële coherente tijd een orde van grootte langer
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Samenvatting
te maken vergeleken met het toepassen van traditionele semi-coherente analysemethodes op dit soort systemen. Voor een neutronenster in een circulaire, Kepleriaanse, baan
(e = 0) met een periode (P ) van 2 uur is gebleken dat Polynomial Search de coherente
tijd ongeveer een factor 25 kan vergroten. Aan de hand van de verwachtte golfvormen
is er een geschikte collectie filters samengesteld. Alle golfvormen uit deze filtercollectie
beschrijven neutronensterren in Kepleriaanse banen met parameters P > 2 uur en e = 0
zowel als P > 6 uur en e < 0.6. Deze filters kunnen op de data toegepast worden met een
coherente tijd van 600 seconden (ter vergelijking: een traditionele semi-coherente analyse
toegepast op uitsluitend een circulaire baan van P = 2 uur kan maximaal 20 seconden
coherent blijven). Naast de collectie van filters is er ook een strategie ontwikkeld voor
het combineren van de resultaten van de verschillende stukjes coherente analyse. Dit
wordt het ‘consistente filter criterium’ genoemd. Door Polynomial Search op witte ruis
samen met gesimuleerde gravitatiegolf signalen toe te passen, is de voorspelde gevoeligheid berekend. Deze gevoeligheid samen met de volledige beschrijving van Polynomial
Search is gepresenteerd in dit proefschrift.
Tot slot wordt in hoofdstuk 6 het ontwikkelde ‘framework’ waarin Polynomial Search
is geı̈mplementeerd gepresenteerd. Dit framework, genaamd ‘AnaPulsar’, is ontwikkeld
om verschillende soorten gravitatiegolf analysealgoritmes te implementeren. Ook zijn de
computationele eisen voor Polynomial Search besproken. Het is gebleken dat om Polynomial Search met de voorheen genoemde filtercollectie toe te passen op een maand data,
er ongeveer 36 CPU jaren aan rekenkracht nodig is.
163
Samenvatting
164
Acknowledgements
The people without whom I could not have finished this thesis are too many to list here.
Still, I would like to thank a few people in particular.
First of all I would like to thank my promotors Jo van den Brand and Frank Linde for
giving me the opportunity to do this PhD research at Nikhef. As well Henk-Jan Bulten,
Thomas Bauer and Maurik Holtrop for their help with starting a brand new analysis
from scratch in a new field. Also I would like to thank as Eric Hennes for helping me
understand the intricacies of finite element analysis and for letting me give that first
talk at Virgo.
Also, I would like to thank the members of the ever growing gravitational waves group
for their support, help and companionship: Mark, Mathieu, David, Tjonnie, Michalis,
Walter, Salvatore and Chris. May you guys hunt down all unexplained noise resonances
and find many gravitational waves.
An honorable mention goes to my office mates past, present and future: Fabian,
Martin, Paul, Rob, Reinier and of course Gideon. You guys have always been a source
of entertainment and discussions on a variety of subjects. In particular I would like to
thank Gideon for all discussions on anything from thermodynamics to general relativity
to field theory to advanced trigonometry and of course the odd bad-taste joke.
I must of course mention my fellow brothers-in-arms with whom I have had the
pleasure of ‘putting on the penguin’ in the past: Aras, Patrick, Gabriel G., Gordon and
Alex. This last time I will finally be standing in the middle to hopefully join the ranks
of learned and productive members of society.
Of course, without my paranymphs Fabian and Chiara I could never have never
brought this scientific endeavor to the proper conclusion. I know that I will have little
to fear with both of you at my side during the defense. Special thanks to Chiara and
Hegoi for all the work they put into designing the cover of this thesis.
Then, I would like to thank some of the people inside and outside of Nikhef with
whom I was lucky enough to have spent various coffee breaks, ‘borrels’, PhD defenses,
conferences, dinners, regular Saturday nights and many other social occasions. In no
particular order: Hegoi, Giada, Gabriel Y., Eleonora, Ermes, Gianni, Gossie, Aart, Surya,
Monique, Solkin, Kazu, Joana, Daniel, Duncan and many others. Thanks, all of you for
making these past few years unforgettable!
En uiteraard ook de mensen van #WERK en omstanders: Lamp, Monn, DQ, Harvey,
165
Acknowledgements
Oel, Xik, Wok, Wies, Jona, Kees, Marjolein, Wouter, Maarten, Leon, Mark en Jasper.
Bedankt voor de ruige midweken, gave vakanties in de bergen, de bekende dinsdagavond
kansloosheid en uiteraard de jaarlijkse Hemelvaart traditie. Dat we deze activiteiten nog
lang vol mogen houden.
Tot slot mijn familie: ma, pa en Andi ik dank jullie vanuit de bodem van mijn
hart voor alle ondersteuning, interesse en luisterende oren die jullie mij gegeven hebben.
Zonder jullie zou ik nooit de energie hebben gehad om dit proefschrift tot een goed einde
te brengen.
Sipho van der Putten
Amsterdam, Decmber 2011
166