The full sky cosmic microwave background polarization field can be decomposed into “electric” (E) and “magnetic” (B) components that are signatures of distinct physical processes. We give a general construction that achieves separation of E and B modes on arbitrary sections of the sky at the expense of increasing the noise. When E modes are present on all scales the separation of all of the B signal is no longer possible: there are inevitably ambiguous modes that cannot be separated. We discuss the practicality of performing $E/B$ decomposition on large scales with realistic nonsymmetric sky cuts, and show that separation on large scales is possible by retaining only the well supported modes. The large scale modes potentially contain a great deal of useful information, and $E/B$ separation at the level of the map is essential for clean detection of B without confusion from cosmic variance due to the E signal. We give simple matrix manipulations for creating pure E and B maps of the large scale signal for general sky cuts. We demonstrate that the method works well in a realistic case and give estimates of the performance with data from the Planck satellite. In the Appendix we discuss the simple analytic case of an azimuthally symmetric cut, and show that exact $E/B$ separation is possible on an azimuthally symmetric cut with a finite number of nonintersecting circular cuts around foreground sources.

• Received 30 May 2003

DOI:https://doi.org/10.1103/PhysRevD.68.083509