Introduction

A basic result of standard harmonic analysis on ℝ is that every 2π-periodic function can be expanded into an orthonormal basis of trigonometric polynomials of the type {exp(ikx): k = 0, ±1, ±2,…}. This fundamental fact is exploited in several branches of applied mathematics: in particular – as we shall mention more extensively in the subsequent chapters – it is a main staple in the analysis of stationary stochastic processes. A proof can be provided by elementary techniques, requiring little more than standard calculus – see for instance Rudin [174, p. 88]. It is also well-known (see e.g. [63]) that one can deduce a similar result for fields defined on the sphere S2, where the appropriate orthonormal basis is given by the class of spherical harmonics.

The aim of this introductory chapter is to put these two results into the broader framework of abstract harmonic analysis on compact groups.

At first sight, this choice may seem unduly complicated, but it is indeed necessary for the purposes we are aiming at. On one hand, our standpoint allows us to develop quite straightforwardly Fourier analysis on the sphere. On the other hand, group representation ideas are necessary to understand most of the material presented in the subsequent parts of the book, namely: the characterizations of isotropy, the explicit computation of higher order moments and angular polyspectra, the derivation of high-frequency asymptotic properties, and the discussion of spin ideas at the very end of the book.

Overview

The purpose of this monograph is to discuss recent developments in the analysis of isotropic spherical random fields, with a view towards applications in cosmology. We shall be concerned in particular with the interplay among three leading themes, namely:

• the connection between isotropy, representation of compact groups and spectral analysis for random fields, including the characterization of polyspectra and their statistical estimation;

• the interplay between Gaussianity, Gaussian subordination, nonlinear statistics, and recent developments in the methods of moments and diagram formulae to establish weak convergence results;

• the various facets of high-resolution asymptotics, including the high-frequency behaviour of Gaussian subordinated random fields and asymptotic statistics in the high-frequency sense.

These basic themes will be exploited in a number of different applications, some with a probabilistic flavour and others with a more statistical focus.

On the probabilistic side, we mention, for instance, a systematic study of the connections between Gaussianity, independence of Fourier coefficients, ergodicity and high-frequency asymptotics of Gaussian subordinated fields. We will also discuss at length the role of isotropy in constraining the behaviour of angular power spectra and polyspectra, thus providing a characterization of the dependence structure of random fields, as well as a sound mathematical background for establishing meaningful estimation procedures.

Among the statistical applications, we mention the estimation of angular power spectra and polyspectra, and their use to implement tests for Gaussianity, isotropy and asymmetry.

Introduction

In the previous chapters of this book, we provided a systematic analysis of scalar-valued spherical random fields, with special emphasis on applications in astrophysics and cosmology. These same fields of applications are now prompting stochastic models which are more sophisticated (and more intriguing) than ordinary, scalar-valued random fields. For instance, in areas as diverse as brain imaging and astrophysics more and more examples are emerging of random fields which do not take ordinary (scalar) values, but rather have a domain given by more sophisticated mathematical structures. In this final chapter, we shall be again concerned with astrophysical and cosmological applications, but several related issues can be found in other disciplines, see for instance [178] for related mathematical models in the field of brain mapping.

In particular, almost all mathematical statistics papers in the CMB area have been concerned with the temperature component of this radiation; however, most recent and forthcoming experiments (such as Planck, which was launched on May 14, 2009, or the projected mission CMBPOL) are focussing on a more complex and elusive feature, namely the so-called polarization of the Cosmic Microwave Background. The physical significance of the latter is explained for instance in [33, 181] and [108]; we do not enter into these motivations here, but we do stress how the analysis of such a feature is supposed to provide extremely rewarding physical informations.

Introduction

The primary goal of this monograph is to investigate the mathematical foundations of the analysis of spherical random fields, and because of this we neglect throughout this work a detailed discussion on practical data analysis. In this chapter, however, we do find it necessary to discuss some background issues on CMB data collection, as these topics are indeed useful for understanding the motivations of the techniques presented below.

As discussed earlier in the monograph, CMB (temperature) maps can be viewed as the single realization of an isotropic, scalar-valued spherical random field. Observations are provided by means of electromagnetic detectors (so-called radiometers and/or bolometers) which measure fluxes of incoming radiations (i.e. photons) on a range of different frequencies. For instance, the celebrated WMAP experiment is endowed with 16 detectors, centred at frequencies 40.7, 60.8 and 93.5 GHz, which are labelled the Q, V and W band, respectively. The ESA (European Space Agency) mission Planck (launched on May 14, 2009) is based upon 70 channels ranging from 30 GHz to 857 GHz. As the satellites scan the sky, observations are collected as a vector time series, the number of observations being in the order of 109 for WMAP and 5×1010 for Planck. A first issue then relates to the construction of spherical maps starting from the Time Ordered Data vector (TOD) provided by the satellite observations; this is the so-called map-making challenge, see for instance [109] and [45].

Introduction

As discussed in the previous chapters (see, in particular, Section 6.3), if an isotropic field is Gaussian, its dependence structure is completely identified by the angular correlation function and its harmonic transform, that is, the angular power spectrum. For non-Gaussian fields, the dependence structure becomes much richer, and higher order correlation functions are of interest. In turn, this leads to the analysis of so-called higher order angular power spectra, which we investigated in Chapter 6. Cumulant angular power spectra are identically zero for Gaussian fields, and hence they also provide natural tools to test for non-Gaussianity: this is a topic of the greatest importance in modern cosmological data analysis (see [51, 60]). Indeed, the validation of the Gaussian assumption is urged by the necessity to provide firm grounds to statistical inference on cosmological parameters, which is dominated by likelihood approaches. More importantly, tests for Gaussianity are needed to discriminate among competing scenarios for the physics of the primordial epochs: here, the currently favoured inflationary models predict (very close to) Gaussian CMB fluctuations, whereas other models yield different observational consequences (see [8, 18, 19, 20, 67, 184]). Tests for non-Gaussianity are also powerful tools to detect systematic effects in the outcome of the experiments. For these reasons, very many papers have focussed on testing for non-Gaussianity on CMB, some of them by means of topological properties of Gaussian fields, some others through spherical wavelets, or by harmonic space methods, see for instance [18, 35, 36, 37, 117, 119, 130, 164, 165, 175, 176, 198], and the references therein.

Several people have worked with the authors in the last years to develop the material covered in this monograph. In particular, following the order of the book, we wish to mention Ivan Nourdin and David Nualart for recent developments on the generalized method of moments for Gaussian subordinated processes, and relationships with Stein's method; Jean-Renaud Pycke for spectral representations of isotropic random fields; Paolo Baldi for the characterizations of spherical harmonic coefficients under isotropy; Paolo Baldi, Gerard Kerkyacharian and Dominique Picard for the stochastic analysis of standard needlets, and Xiaohong Lan for the needlets bispectrum; Daryl Geller (who introduced Mexican needlets with Azita Mayeli) for the extension of the needlet paradigm to random sections of spin fiber bundles, We learned a lot from discussions with Mauro Piccioni and lgor Wigman, who have also provided very useful comments on an earlier draft, as did PhD students Mirko D'Ovidio and Claudio Durastanti.

The material of this book is strongly motivated by Cosmological applications, and it has benefited enormously from a decade-long interaction of the first author with physicists providing insights. Suggestions, and applications to real data: we mention in particular (in alphabetic order) Amedeo Balbi, Paolo Cabella, Giancarlo de Gasperis, Frode Hansen, Michele Liguori, Sabino Matarrese, Paolo Natoli, Davide Pietrobon, Gianluca Polenta, Oystein Rudjord, Sandro Scodeller and Nicola Vittorio. Frode Hansen is to be thanked also for some insightful comments on the CMB description parts.