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Department of Physics
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DDU Gorakhpur University
Gorakhpur
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Gorakhpur
STUDIES IN
GRAVITATIONAL WAVE
DATA ANALYSIS
A thesis submitted to the
DDU GORAKHPUR UNIVERSITY
for the degree of
Doctor of Philosophy
IN PHYSICS
by
SANJAY KUMAR SAHAY
Department of Physics
DDU Gorakhpur University
Gorakhpur - 273009
December 2001
T O MY FAT HER
LAT E SHRI KRISHN A SAHAY
Contents
Acknowledgement ix
Declaration xi
List of Publications xii
Preface xiii
1 Introduction: Gravitational Waves, Detectors and Sources 1
1.1 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Linear field approximation . . . . . . . . . . . . . . . . 5
1.1.2 Propagation of gravitational waves . . . . . . . . . . . 7
1.1.3 The effect of waves on free particles and its polarisation 9
1.1.4 Generation of gravitational wave . . . . . . . . . . . . 12
1.2 Gravitational wave detectors . . . . . . . . . . . . . . . . . . . 16
1.2.1 Bar detectors . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.2 Ground based laser interferometric detectors . . . . . . 23
1.2.3 Laser interferometric space antenna . . . . . . . . . . . 26
1.3 Sources of gravitational waves . . . . . . . . . . . . . . . . . . 28
1.3.1 Transient sources . . . . . . . . . . . . . . . . . . . . . 29
1.3.2 Continuous gravitational wave . . . . . . . . . . . . . . 31
1.3.3 Stochastic waves . . . . . . . . . . . . . . . . . . . . . 32
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2 Data analysis concept 35
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Gravitational wave antenna sensitivity . . . . . . . . . . . . . 36
2.3 Noises in the earth based interferometric detectors . . . . . . . 40
2.4 Matched filtering and optimal signal-to-noise ratio . . . . . . . 43
2.5 Computational costs . . . . . . . . . . . . . . . . . . . . . . . 48
2.6 Detection criteria . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 Data analysis of continuous gravitational wave: Fourier transform-I 53
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 The noise free response of detector: Beam pattern and amplitude modulation 55
3.3 Doppler shift and Frequency modulation . . . . . . . . . . . . 63
3.4 Fourier transform of the complete response . . . . . . . . . . . 72
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4 Data analysis of continuous gravitational wave: Fourier transform-II 83
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 Fourier transform for one year integration . . . . . . . . . . . 84
4.2.1 Frequency modulation . . . . . . . . . . . . . . . . . . 84
4.2.2 Complete response . . . . . . . . . . . . . . . . . . . . 89
4.3 Fourier transform for arbitrary observation time . . . . . . . . 96
4.4 Spin down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.5 N-component signal . . . . . . . . . . . . . . . . . . . . . . . . 109
4.6 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . 111
5 All sky search: Study of templates 113
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.2 Matched filter analysis: Templates . . . . . . . . . . . . . . . 115
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5.3 Number of Templates . . . . . . . . . . . . . . . . . . . . . . . 122
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Appendices 129
A Source code of the frequency modulated signal: FM.F 129
B Source code of the noise free complete response of the detector: CR.F133
C Source code for the estimation of templates for all sky search: TEM.F141
Bibliography 147
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List of Tables
1.1 The existing resonant bar detectors. . . . . . . . . . . . . . . . 22
1.2 Site and orientation of Earth based interferometric gravitational wave detectors. 24
2.1 Analytical fits to noise power spectral densities S (f) of ground based interferometers.
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5.1 Coefficients of the best fit graphs obtained for the number of templates.124
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List of Figures
1.1 Gravitational wave from binary. . . . . . . . . . . . . . . . . . 2
1.2 (a) A circle of free particles before a wave travelling in the z direction reaches them. (b)
1.3 An idealised gravitational wave detector with two identical masses connected with a spring.
1.4 A schematic of resonant bar detector. . . . . . . . . . . . . . . 21
1.5 A schematic of Earth based laser interferometric gravitational wave detector. 25
1.6 A schematic diagram of the laser interferometric space antenna. 27
3.1 Orientation of the detector. . . . . . . . . . . . . . . . . . . . 56
3.2 Earth frame (x′,y′,z′). . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Solar System Barycentre frame (X′,Y′,Z′). . . . . . . . . . . 56
3.4 FT of a FM signal of a source located at (π/36,π) with a resolution of 1.16 10−5. 69
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3.5 FT of a FM signal of a source located at (π/36,π) with a resolution of 10−6. 70
3.6 FT of a FM signal of a source located at (π/36,π) with a resolution of 10−7. 71
3.7 Power spectrum of a FM signal of a source located at (π/36,π) with a resolution of 10−
3.8 Power spectrum of the complete response of a Doppler modulated signal of a source located
3.9 Power spectrum of a Doppler modulated signal at frequencies f +2f of a source located
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3.10 Power spectrum of a Doppler modulated signal at frequencies f 2f of a source located
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3.11 Power spectrum of a Doppler modulated signal at frequencies f +f of a source located
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3.12 Power spectrum of a Doppler modulated signal at frequencies f f of a source located
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3.13 Power spectrum of a Doppler modulated signal at frequencies f of a source located at (
4.1 FT of a FM signal of a source located at (π/18,0) with a resolution of 3.17 10−8. 87
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