Random fields on the sphere : representation, limit theorems and cosmological applications

Cover......Page 1
Title......Page 4
Copyright......Page 5
Contents......Page 8
Dedication......Page 6
Preface......Page 12
1.1 Overview......Page 14
1.2 Cosmological motivations......Page 15
1.3 Mathematical framework......Page 18
1.4 Plan of the book......Page 20
2.1 Introduction......Page 25
2.2 Preliminary remarks......Page 26
2.3.1 First definitions and examples......Page 28
2.3.2 Cosets and quotients......Page 30
2.3.3 Actions......Page 31
2.4.1 Basic definitions......Page 32
2.4.2 Group representations and Schur Lemma......Page 35
2.4.3 Direct sum and tensor product representations......Page 43
2.4.4 Orthogonality relations......Page 44
2.5 The Peter-Weyl Theorem......Page 52
3.1 Introduction......Page 58
3.2.1 Euler angles for SU(2)......Page 59
3.2.2 Euler angles for SO(3)......Page 60
3.3.1 A family of unitary representations of SU(2)......Page 64
3.3.2 Expressions in terms of Euler angles and irreducibility......Page 68
3.3.3 Further properties......Page 73
3.3.4 The dual of SO(3)......Page 74
3.4.1 Spherical harmonics and Wigner’s Dl matrices......Page 76
3.4.2 Some properties of spherical harmonics......Page 78
3.4.3 An alternative characterization of spherical harmonics......Page 83
3.5.1 Clebsch-Gordan matrices......Page 90
3.5.2 Integrals of multiple spherical harmonics......Page 94
3.5.3 Wigner 3 j coefficients......Page 96
4.1 Introduction......Page 98
4.2 Brownian motion and stochastic calculus......Page 99
4.3 Moments, cumulants and diagram formulae......Page 107
4.4.1 Real kernels......Page 112
4.4.2 Further results on complex kernels......Page 116
4.5.1 From diagrams to graphs......Page 117
4.5.2 Further notation......Page 119
4.5.3 First example: sums of squares......Page 120
4.5.4 Cliques and Wigner 6 j coefficients......Page 122
4.5.5 Rule n. 1: loops are zero......Page 123
4.5.7 Rule n. 3: 2-loops can be cut, and leave a factor......Page 124
4.5.8 Rule n. 4: three-loops can be cut, and leave a clique......Page 125
5.2.1 General statements......Page 127
5.2.2 Decompositions on the sphere......Page 134
5.3 Weakly stationary random fields in Rm......Page 139
5.4 Stationarity and weak isotropy in R3......Page 143
6.1 Introduction......Page 147
6.2 First example: the cyclic group......Page 148
6.3 The spherical harmonics coefficients......Page 150
6.4 Group representations and polyspectra......Page 160
6.5.1 Spectra of strongly isotropic fields......Page 164
6.5.2 The structure of Δl1...ln......Page 165
6.6 Reduced polyspectra of arbitrary orders......Page 169
6.7 Some examples......Page 173
7.1 Introduction......Page 182
7.2 First example: the circle......Page 184
7.3 Preliminaries on Gaussian-subordinated fields......Page 186
7.4.1 Hermite subordination......Page 189
7.5.1 Convolutions on SO (3)......Page 194
7.5.2 The cases q = 2 and q = 3......Page 197
7.5.3 The case of a general q: results and conjectures......Page 198
7.6.2 Other convolutions and random walks on group duals......Page 201
7.7 Application: algebraic/exponential dualities......Page 203
7.7.1 The exponential case......Page 204
7.7.2 Regularly varying functions......Page 205
8.1 Introduction......Page 207
8.2 Angular power spectrum estimation......Page 209
8.3.1 Dealing with instrumental noise......Page 214
8.3.2 First remarks on missing observations......Page 217
8.4 Asymptotics in the non-Gaussian case......Page 218
8.5.1 The class D and main results......Page 223
8.5.2 Proof of Theorem 8.16......Page 225
8.6 Discussion......Page 232
9.1 Introduction......Page 236
9.2.1 Preliminary considerations......Page 237
9.2.2 Definitions......Page 239
9.3 A central limit theorem......Page 243
9.4 Limit theorems under random normalizations......Page 247
9.5 Testing for non-Gaussianity......Page 251
10.1 Introduction......Page 258
10.2.1 Definition of spherical needlets......Page 261
10.2.2 Numerical recipes......Page 263
10.3.1 Tight frames and reconstruction formulae......Page 265
10.3.2 Localization properties......Page 269
10.4 Stochastic properties of needlet coefficients......Page 270
10.5 Missing observations......Page 272
10.6 Mexican needlets......Page 274
11.2 A general convergence result......Page 278
11.3 Estimation of the angular power spectrum......Page 285
11.4 A functional central limit theorem......Page 286
11.5.1 The needlets bispectrum......Page 288
11.5.2 Behaviour under non-Gaussianity......Page 291
12.1 Introduction......Page 295
12.2 Motivations......Page 297
12.3 Geometric background......Page 299
12.4 Spin needlets and spin random fields......Page 303
12.5 Spin needlets spectral estimator......Page 307
12.6 Detection of asymmetries......Page 315
12.7 Estimation with noise......Page 318
13.1.1 Jacobi polynomials......Page 325
13.1.2 Legendre polynomials......Page 327
13.1.3 Associated Legendre functions......Page 328
13.2 Spherical harmonics and their analytic properties......Page 330
13.2.1 Power series representations......Page 331
13.2.2 Differential and integral relationships......Page 332
13.3 The proof of needlets\' localization.......Page 333
References......Page 339
Index......Page 351