Applications of Wavelets to the Analysis of Cosmic Microwave Background Maps

We consider wavelets as a tool to perform a variety of tasks in the context of analyzing cosmic microwave background (CMB) maps. Using Spherical Haar Wavelets we define a position and angular-scale-dependent measure of power that can be used to assess the existence of spatial structure. We apply planar Daubechies wavelets for the identification and removal of points sources from small sections of sky maps. Our technique can successfully identify virtually all point sources which are above 3 sigma and more than 80% of those above 1 sigma. We discuss the trade-offs between the levels of correct and false detections. We denoise and compress a 100,000 pixel CMB map by a factor of about 10 in 5 seconds achieving a noise reduction of about 35%. In contrast to Wiener filtering the compression process is model independent and very fast. We discuss the usefulness of wavelets for power spectrum and cosmological parameter estimation. We conclude that at present wavelet functions are most suitable for identifying localized sources.