The three-point correlation function of cosmic shear

II. Relation to the bispectrum of the projected mass density and generalized third-order aperture measures

¹ Institut f. Astrophysik u. Extr. Forschung, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany e-mail: peter@astro.uni-bonn.de
² European Southern Observatory, Karl-Schwarzschild-Str. 2, 85741 Garching, Germany

Received: 21 August 2003
Accepted: 25 September 2004

Abstract

Cosmic shear, the distortion of images of high-redshift sources by the intervening inhomogeneous matter distribution in the Universe, has become one of the essential tools for observational cosmology since it was first measured in 2000. Since then, several surveys have been conducted and analyzed in terms of second-order shear statistics. Current surveys are on the verge of providing useful measurements of third-order shear statistics, and ongoing and future surveys will provide accurate measurements of the shear three-point correlation function which contains essential information about the non-Gaussian properties of the cosmic matter distribution.
We study the relation of the three-point cosmic shear statistics to the third-order statistical properties of the underlying convergence, expressed in terms of its bispectrum. Explicit relations for the natural components of the shear three-point correlation function (which we defined in an earlier paper) in terms of the bispectrum are derived. The behavior of the correlation function under parity transformation is obtained and found to agree with previous results. We find that in contrast to the two-point shear correlation function, the three-point function at a given angular scale θ is not affected by power in the bispectrum on much larger scales. These relations are then inverted to obtain the bispectrum in terms of the three-point shear correlator; two different expressions, corresponding to different natural components of the shear correlator, are obtained and can be used to separate E and B-mode shear contributions. These relations allow us to explicitly show that correlations containing an odd power of B-mode shear vanish for parity-symmetric fields. Generalizing a recent result by Jarvis et al., we derive expressions for the third-order aperture measures, employing multiple angular scales, in terms of the (natural components of the) three-point shear correlator and show that they contain essentially all the information about the underlying bispectrum. We discuss the many useful features these (generalized) aperture measures have that make them convenient for future analyses of the skewness of the cosmic shear field (and any other polar field, such as the polarization of the Cosmic Microwave Background).