The ring statistics - how to separate E- and B-modes of cosmic shear correlation functions on a finite interval

Aims. Cosmic shear, the distortion of images of distant sources by the tidal gravitational field of the large-scale matter distribution in the Universe, is one of the most powerful cosmological probes. The measured shear field may be due not only to the gravitational lensing effect, but may contain systematic effects from the measurement process or intrinsic alignment of galaxy shapes. One of the main probes for these systematics is the division of the shear field into E- and B-mode shear, where lensing only produces the former. As shown in a recent note, all currently used E-/B-mode separation methods for the shear correlation functions zeta(+/-) require them to be measured to arbitrarily small and/or large separations which is of course not feasible in practice. Methods. We derive second-order shear statistics which provide a clean separation into E- and B-modes from measurements of zeta(+/-)(theta) over a finite interval only. We call these new statistics the circle and ring statistics, respectively; the latter is obtained by an integral over the former. The mathematical properties of these new shear statistics are obtained, as well as specific expressions for applying them to observed data. Results. It is shown that an E-/B-mode separation can be performed on measurements of zeta(+/-) over a finite interval in angular separation, using the ring statistics. We furthermore generalize this result to derive the most general class of second-order shear statistics which provide a separation of E- and B-mode shear on a given angular interval theta(min) <= theta <= theta(min). In view of these generalization, we discuss the aperture dispersion and the shear dispersion and their relation to the shear correlation functions. Our results will be of practical use particularly for future cosmic shear surveys where highly precise measurements of the shear will become available and where control of systematics will be mandatory.