Efficient analysis algorithms for Gravitational Waves and Cosmology O T: Gravitational Waves from Inspiraling Binaries and Cosmological Ramifications B Sanjit Mitra AT F D Doctor of Philosophy ( in Physics ) University of Pune S: Prof. Sanjeev Dhurandhar C-: Prof. Tarun Souradeep Inter-University Centre for Astronomy and Astrophysics Post Bag 4, Ganeshkhind, Pune 411 007, INDIA DECEMBER 2006 Dedicated to Nature of which I am a part iv v Contents Dedication iii Acknowledgment xiii List of Publications xv Declaration xvii Abstract xix 1 Introduction 1 1.1 Basics ..................................... 1 1.1.1 Gravitational Waves (GW) ..................... 1 1.1.2 Cosmic Microwave Background (CMB) ............. 2 1.2 Search for GW from Inspiraling Binaries: Chebyshev Interpolation . 3 1.3 Search for GW Background: Radiometer Analysis ........... 5 1.4 Deconvolution of Sky-Maps ........................ 5 1.5 Non-circular Beam Correction to CMB Power Spectrum ........ 6 1.6 Organization of the Thesis ......................... 7 2 Introduction to Gravitational Waves (GW) 9 2.1 Gravitational Waves ............................ 10 2.1.1 General Relativity: Einstein’s Equation ............. 10 2.1.2 Weak field limit: Linearized Theory ............... 10 2.1.3 Plane Polarized Monochromatic Gravitational Waves ..... 14 2.2 Detection ................................... 14 2.3 Sources and Analysis Strategies ...................... 18 2.4 Summary and Conclusion ......................... 20 vi CONTENTS 3 Search for GW from Inspiraling Binaries: Chebyshev Interpolation 21 3.1 Chirp Signal ................................. 23 3.2 Dense Search ................................. 25 3.2.1 Matched Filtering .......................... 25 3.2.2 Template placement ........................ 26 3.2.3 Choice of parameter ........................ 29 3.3 Efficient Search Algorithms ........................ 30 3.4 Interpolated Search ............................. 32 3.4.1 Interpolation of the Wiener filter output ............. 32 3.4.2 Search strategy ........................... 36 3.4.3 Template placement ........................ 38 3.5 Comparison between Dense and Interpolated search: ROC curves .. 42 3.6 Summary and Conclusion ......................... 44 4 Search for GW Background (GWB): Radiometer Analysis 49 4.1 Stochastic Gravitational Wave Background (GWB) ........... 50 4.2 Sources of GWB ............................... 54 4.3 Detection of GWB: Radiometer Analysis ................. 56 4.3.1 Cross-correlation statistic ..................... 57 4.3.2 Correlation between GW Strains ................. 62 4.3.3 Detector Noise ........................... 64 4.3.4 Optimal Filter ............................ 65 4.3.5 Observed Point Estimate and SNR ................ 69 4.3.6 Summary .............................. 71 4.4 Applications ................................. 73 4.4.1 All-sky isotropic background ................... 74 4.4.2 Directed Search ........................... 76 4.5 Summary and Conclusion ......................... 79 5 Introduction to the Cosmic Microwave Background (CMB) 81 5.1 Origin ..................................... 82 5.2 CMB Anisotropy .............................. 84 5.3 Angular Power Spectrum: Cosmic Variance ............... 86 5.4 Experiments ................................. 89 5.5 Summary and Conclusion ......................... 90 CONTENTS vii 6 Beams and Deconvolution in CMB and GWB Mapmaking: Analysis and For- malism 91 6.1 Beam Function ................................ 92 6.2 Observed CMB Data ............................ 92 6.3 Observed (Directed) GW Radiometer Point Estimate .......... 95 6.4 Maximum Likelihood (ML) skymap Estimation ............. 101 6.5 ML Estimation of CMB Anisotropy Map ................. 103 6.6 ML Estimation of GWB Anisotropy .................... 104 6.6.1 Basic Analysis of Skymap ..................... 104 6.6.2 Extensions of the simple MapMaking .............. 105 6.6.3 Multipole moments from the directed search .......... 107 6.6.4 Multipole moments from isotropic all-sky search ........ 108 6.7 Summary and Conclusions ........................ 111 7 Numerical Implementation of ML Mapmaking for GWB 113 7.1 Preparation of Simulated Data ....................... 114 7.2 Preparation of the “Dirty” Maps ..................... 116 7.3 Computation of the Beam Matrix ..................... 117 7.4 Deconvolution: The “Clean” Maps .................... 120 7.5 Results and Comparisons ......................... 123 7.6 Summary and Conclusion ......................... 125 8 Non-circular Beam Correction to CMB Power Spectrum: Perturbative Analysis 127 8.1 Pseudo-Cl Approach to Non-circular Beam Correction ......... 128 8.2 Window functions of CMB experiments: a brief primer ........ 131 8.2.1 Window function for circular beams ............... 133 8.2.2 Window function for non-circular beams ............ 134 8.3 Bias Matrix ................................. 138 8.3.1 Circular Symmetric Beam ..................... 140 8.3.2 Non-circular Beam ......................... 141 8.4 Error-Covariance Matrix .......................... 148 8.4.1 Circular Beam ............................ 151 8.4.2 Non-circular Beam ......................... 151 8.5 Discussion and Conclusion ........................ 155 viii CONTENTS 9 Non-circular Beam Correction to CMB Power Spectrum: Complete Analysis Framework 159 9.1 Analytical Framework ........................... 160 9.1.1 Bias in the pseudo-Cl estimator .................. 160 9.1.2 Evaluation of the Bias Matrix ................... 162 9.1.3 Checking different limits ...................... 166 9.2 Implementation ............................... 167 9.3 Discussions and Conclusion ........................ 171 10 Conclusion 173 A Properties of the Chebyshev Polynomials 177 B Directed GW Radiometer Beam: Stationary Phase Approximation 179 B.1 Beam Pattern: SPA Trajectory ....................... 180 B.1.1 The case of small ∆Ω ........................ 181 B.1.2 General SPA solution ........................ 183 B.2 Evaluation of the Beam Function on SPA trajectory ........... 184 C Cosmic Variance 187 D Elliptical Gaussian fit to the WMAP beam maps 189 E Derivations for perturbative analysis of beam correction to pseudo-Cl es- timator 195 F Derivations for general analysis of beam correction to pseudo-Cl estimator201 F.1 Useful formulae ............................... 201 F.2 Expansion of Wigner-D Function ..................... 202 F.3 Evaluation of Jll00l0 using sinusoidal expansion of Wigner-d .... 205 nm00mm0 F.4 Evaluation of Jll00l0 using Clebsch-Gordon series and sinusoidal nm00mm0 expansion of Wigner-d ........................... 207 F.5 The full sky and circular beam limit ................... 209 F.6 The circular beam limit with cut sky ................... 211 F.7 The full sky limit with non-circular beam ................ 213 G Fast computation of non-circular beam correction to CMB power spectrum215 ix List of Tables 3.1 Comparison between dense and interpolated search .......... 44 8.1 Definition of parameters that quantify non-circularity ......... 138 B.1 Coordinate of the LIGO detectors ..................... 182 D.1 Ellipticities of WMAP beams ....................... 193 x LIST OF TABLES xi List of Figures 2.1 Effect of GW on matter: Detection scheme ................ 15 2.2 GW Detectors ................................ 16 2.3 LIGO-I sensitivity curves .......................... 17 3.1 Illustration of Matched Filtering ...................... 27 3.2 Ambiguity function ............................. 30 3.3 Illustration of Interpolated Search ..................... 37 3.4 Variation of bias with no. of interpolated search templates ...... 40 3.5 Variation of Match with signal location .................. 41 3.6 Comparison: False Alarm & False Dismissal probability ........ 45 3.7 Comparison: ROC Curves ......................... 46 4.1 Inflationary GWB spectra and detectability using the advanced LIGO detectors ................................... 55 4.2 GWB spectra and detectability landscape ................ 56 4.3 Time independent overlap reduction function .............. 75 4.4 Geometry of an elementary radiometer ................. 77 4.5 A test sky map made using the GWB radiometer ............ 78 5.1 Evolution of anisotropies in the universe ................ 83 5.2 Observed CMB frequency spectrum by COBE FIRAS ......... 84 5.3 WMAP CMB anisotropy sky ........................ 85 5.4 Observed CMB Power Spectrum ..................... 88 5.5 CMB Experiments .............................. 89 6.1 WMAP image planes ............................ 94 6.2 Numerical and Theoretical GW radiometer beam patterns ...... 97 6.3 Contour plots of GW radiometer beam patterns ............ 98 xii LIST OF FIGURES 7.1 A typical GW radiometer Beam Matrix .................. 120 7.2 Deconvolved (simulated) GWB skymaps ................ 126 8.1 Illustration of beam rotation ........................ 134 8.2 WMAP Q1 Beam .............................. 135 8.3 Bias matrix without beam rotation .................... 145 8.4 Characteristics of the bias matrix elements (no beam rotation) .... 146 8.5 Effect of beam rotation on bias matrix .................. 147 8.6 Cl estimation error due to non-circular beams .............. 149 8.7 Cl estimation error published in the WMAP 3yr results [96] ...... 150 8.8 Covariance of Cl’s due to beam non-circularity ............. 154 9.1 The original Kp2 mask ........................... 170 9.2 Comparison of mask transforms ..................... 170 A.1 Plots of the first few orders of Chebyshev