Daryl Geller

Spin needlets for cosmic microwave background polarization data analysis

Scalar wavelets have been used extensively in the analysis of cosmic microwave background (CMB) temperature maps. Spin needlets are a new form of (spin) wavelets which were introduced in the mathematical literature by Geller and Marinucci (2008) as a tool for the analysis of spin random fields. Here we adopt the spin needlet approach for the analysis of CMB polarization measurements. The outcome of experiments measuring the polarization of the CMB are maps of the Stokes Q and U parameters which are spin two quantities. Here we discuss how to transform these spin two maps into spin two needlet coefficients and outline briefly how these coefficients can be used in the analysis of CMB polarization data. We review the most important properties of spin needlets, such as localization in pixel and harmonic space and asymptotic uncorrelation. We discuss several statistical applications, including the relation of angular power spectra to the needlet coefficients, testing for non-Gaussianity on polarization data, and reconstruction of the E and B scalar maps.

Spin needlets spectral estimation

We consider the statistical analysis of random sections of a spin fibre bundle over the sphere. These may be thought of as random fields that at each point p in

Introducing Mexican Needlets for CMB Analysis: Issues for Practical Applications and Comparison with Standard Needlets

S^2

Adaptive nonparametric regression on spin fiber bundles

take as a value a curve (e.g. an ellipse) living in the tangent plane at that point

Cubature Formulas and Discrete Fourier Transform on Compact Manifolds

T_{p}S^2

Wavelets on Manifolds and Statistical Applications to Cosmology

, rather than a number as in ordinary situations. The analysis of such fields is strongly motivated by applications, for instance polarization experiments in Cosmology. To investigate such fields, spin needlets were recently introduced by Geller and Marinucci (2008) and Geller et al. (2008). We consider the use of spin needlets for spin angular power spectrum estimation, in the presence of noise and missing observations, and we provide Central Limit Theorem results, in the high frequency sense; we discuss also tests for bias and asymmetries with an asymptotic justification.

BAND-LIMITED LOCALIZED PARSEVAL FRAMES AND BESOV SPACES ON COMPACT HOMOGENEOUS MANIFOLDS

Over the last few years, needlets have emerged as a useful tool for the analysis of cosmic microwave background (CMB) data. Our aim in this paper is first to introduce into the CMB literature a different form of needlets, known as Mexican needlets, first discussed in the mathematical literature by Geller & Mayeli. We then proceed with an extensive study of the properties of both standard and Mexican needlets; these properties depend on some parameters which can be tuned in order to optimize the performance for a given application. Our second aim in this paper is then to give practical advice on how to adjust these parameters for WMAP and Planck data in order to achieve the best properties for a given problem in CMB data analysis. In particular, we investigate localization properties in real and harmonic space and propose a recipe for quantifying the influence of galactic and point-source masks on the needlet coefficients. We also show that for certain parameter values, the Mexican needlets provide a close approximation to the Spherical Mexican Hat Wavelets (whence their name), with some advantages concerning their numerical implementation and derivation of their statistical properties.

Nearly tight frames and space-frequency analysis on compact manifolds

The construction of adaptive nonparametric procedures by means of wavelet thresholding techniques is now a classical topic in modern mathematical statistics. In this paper, we extend this framework to the analysis of nonparametric regression on sections of spin fiber bundles defined on the sphere. This can be viewed as a regression problem where the function to be estimated takes as its values algebraic curves (for instance, ellipses) rather than scalars, as usual. The problem is motivated by many important astrophysical applications, concerning, for instance, the analysis of the weak gravitational lensing effect, i.e. the distortion effect of gravity on the images of distant galaxies. We propose a thresholding procedure based upon the (mixed) spin needlets construction recently advocated by Geller and Marinucci (2008, 2010) and Geller et al. (2008, 2009), and we investigate their rates of convergence and their adaptive properties over spin Besov balls.

Continuous Wavelets on Compact Manifolds

The goal of this chapter is to describe essentially optimal cubature formulas on compact Riemannian manifolds which are exact on spaces of band-limited functions.

Spin Wavelets on the Sphere

We outline how many of the pioneering ideas that were so effective in developing wavelet theory on the real line, can be adapted to the manifold setting. In this setting, however, arguments using Fourier series are replaced by methods from modern harmonic analysis, pseudodifferential operators and PDE. We explain how to construct nearly tight frames on any smooth, compact Riemannian manifold, which are well-localized both in frequency (as measured by the Laplace–Beltrami operator) and in space. These frames can be used to characterize Besov spaces on the manifold for the full range of indices, in analogy to the Frazier–Jawerth result on the real line. We explain how our methods can be extended beyond the study of functions, to the wavelet analysis of sections of a particular line bundle on the sphere, which is important for the analysis of the polarization of CMB(cosmic microwave background radiation). The wavelet approach to CMB has been advocated by many people, including our frequent collaborators, the statistician Domenico Marinucci, and the physicist Frode Hansen, who earlier used spherical needlets to study CMB temperature.