Bayesian Analysis of the Power Spectrum of the Cosmic Microwave Background

Jeff Jewell
Data Understanding Systems Group, JPL

There is a wealth of cosmological information encoded in the spatial power spectrum of temperature anisotropies of the cosmic microwave background. The sky, when viewed in the microwave, is very uniform, with a nearly perfect blackbody spectrum at 2.7 degrees. Very small amplitude brightness fluctuations (to one part in a million!!) were detected by the COBE satellite, and have now been mapped by ground, balloon, and satellite instruments to a spatial resolution smaller than 1 degree. These brightness fluctuations trace small density perturbations in the early universe (roughly 300,000 years after the Big Bang), which later grow through gravitational instability to the large-scale structure seen in redshift surveys. The details of the physics in the early universe leaves a telltale signature on the statistical structure of hot and cold spots, with more details of the physics encoded at sub-angular degree spatial scales. With the push to map the microwave sky at higher spatial resolution has come a flood of data, with maps containing millions of pixels observed at several different frequencies (from 30 to 900 GHz), all with slightly different resolutions and noise properties. The resulting analysis challenge is to estimate, and quantify our uncertainty in, the spatial power spectrum of the cosmic microwave background given the complexities of "missing data", foreground emission, and complicated instrumental noise. In this talk, I will discuss a Bayesian formulation of this problem, discuss a Gibbs sampling approach to numerically sampling from the Bayesian posterior, and the application of this approach to the first-year data from the Wilkinson Microwave Anisotropy Probe. I will also comment on recent algorithmic developments for this approach to be tractable for the even more massive data set to be retuned from the Planck satellite.