Real-space computation of E/B-mode maps. Part I. Formalism, compact kernels, and polarized filaments

We derive full-sky, real-space operators that convert between polarization Stokes Q/U parameters and the coordinate-independent scalar E/B modes that are widely used in Cosmic Microwave Background (CMB) and cosmic shear analysis. We also derive real space operators that decompose the measured Stokes parameters into those corresponding to E-modes and B-modes respectively, without ever evaluating the scalar fields themselves. We cast the standard CMB polarization analysis operators in a matrix-vector notation which elucidates these derivations. For all these real space operators we show that the kernels split naturally into angular and radial parts and we show explicitly how the radial extent of these kernels depends on the targeted band-limit. We show that the kernels can be interpreted either as a complex convolving beam or as a Green's function when they are expressed in terms of the forward or inverse rotation Euler angles. We show that an arbitrary radial function can produce E/B-like maps, provided it vanishes at the origin and the antipodal point. These maps are simply filtered versions of the standard E/B maps. We can recover the standard power spectrum of the polarized CMB sky by correcting the power spectra of these maps with a simple window function, which we show how to derive for any radial dependence. For these reasons we can compute E/B maps in real space with a compactly-supported kernel, an approach that can guarantee the avoidance of known foreground regions and could be employed in a massively-parallel scheme at high-resolution. We show that the spin raising and lowering operators partial derivative(2)/partial derivative(2) are special cases of these generalized radial functions, and present their band limited versions. The spatial structure of the real space operators provides great intuition for the E/B structure of polarized, filamentary galactic foregrounds. We predict a non-zero B-mode signature that is expected from polarized filaments in the sky. This paper is the first part in a series of papers that explore real-space computation of polarization modes and their applications.