Spatial Statistics of Cosmic Microwave Background Maps

摘要

Spatial statistics in the Cosmic Microwave Background (CMB) maps are characterized by N-point correlation functions and their corresponding poly-spectra. I focus on basic statistical properties (noise, bias, optimality), and computational issues regarding two- and three-point functions, or angular power spectrum and bi-spectrum. Optimal estimators scale as D 3 for even the two-point statistics, where D is the number of data elements Even naive estimators scale scale DN for N-point functions. I show that these daunting computational challenges can be met for present and future megapixel CMB maps with considerations about symmetries, multi-resolution techniques, and Monte Carlo methods and careful balancing of optimality, and resolution against computational resources. Once estimated, the interpretation of higher order correlation functions presents unique difficulties due to the large number of configurations: e.g., X2 fitting of parameters becomes non-trivial because of the large size of the corresponding covariance matrices. I show that False Discovery Rate based methods can be used for massive hypothesis testing, and I present techniques which help diagnosing and inverting covariance matrices obtained from Monte Carlo simulations.