Geometric Methods for Cosmological Data on the Sphere

This review is devoted to recent developments in the statistical analysis of spherical data, strongly motivated by applications in cosmology. We start from a brief discussion of cosmological questions and motivations, arguing that most cosmological observables are spherical random fields. Then, we introduce some mathematical background on spherical random fields, including spectral representations and the construction of needlet and wavelet frames.We then focus on some specific issues, including tools and algorithms for map reconstruction (i.e., separating the different physical components that contribute to the observed field), geometric tools for testing the assumptions of Gaussianity and isotropy, and multiple testing methods to detect contamination in the field due to point sources. Although these tools are introduced in the cosmological context, they can be applied to other situations dealing with spherical data. Finally, we discuss more recent and challenging issues, such as the analysis of polarization data, which can be viewed as realizations of random fields taking values in spin fiber bundles.