Nonparametric Inference for the Cosmic Microwave Background

The Cosmic Microwave Background (CMB), which permeates the entire Universe,is the radiation left over from just 380,000 years after the Big Bang. On verylarge scales, the CMB radiation field is smooth and isotropic, but theexistence of structure in the Universe - stars, galaxies, clusters of galaxies- suggests that the field should fluctuate on smaller scales. Recentobservations, from the Cosmic Microwave Background Explorer to the WilkinsonMicrowave Anisotropy Project, have strikingly confirmed this prediction. CMBfluctuations provide clues to the Universe's structure and composition shortlyafter the Big Bang that are critical for testing cosmological models. Forexample, CMB data can be used to determine what portion of the Universe iscomposed of ordinary matter versus the mysterious dark matter and dark energy.To this end, cosmologists usually summarize the fluctuations by the powerspectrum, which gives the variance as a function of angular frequency. Thespectrum's shape, and in particular the location and height of its peaks,relates directly to the parameters in the cosmological models. Thus, a criticalstatistical question is how accurately can these peaks be estimated. We userecently developed techniques to construct a nonparametric confidence set forthe unknown CMB spectrum. Our estimated spectrum, based on minimal assumptions,closely matches the model-based estimates used by cosmologists, but we can makea wide range of additional inferences. We apply these techniques to testvarious models and to extract confidence intervals on cosmological parametersof interest. Our analysis shows that, even without parametric assumptions, thefirst peak is resolved accurately with current data but that the second andthird peaks are not.Comment: Invited review for "Statistical Science". Accepted for publication in Feburary 2004 journa