Advanced methods for cosmic microwave background data analysis: the big N^3 and how to beat it

Abstract

In this talk I propose the first fast methods which can analyze CMB data taking into account correlated noise, arbitrary beam shapes, non-uniform distribution of integration time on the sky, and partial sky coverage, without the need for approximations. These ring torus methods work by performing the analysis in the time ordered domain (TOD) rather than on the sky map of fluctuations. They take advantage of the simplicity of noise correlations in the TOD as well as certain properties of the group of rotations SO(3). These properties single out a family of scanning strategies as favorable, namely those which scan on rings and have the geometry of an n-torus. This family includes the strategies due to TOPHAT, MAP and Planck. I first develop the tools to model the time ordered signal, using Fast Fourier Transform methods for convolution of two arbitrary functions on the sphere (Wandelt and Górski 2000). Then I apply these ideas to show that in the case of a 2-torus one can reduce the time taken for CMB power spectrum analysis from an unfeasible order N^3 to order N^2, where N ~ 10^5-10^8 is the number of resolution elements (Wandelt and Hansen, in preparation).