The three-point correlation function of cosmic shear - I. The natural components

The three-point correlation function of cosmic shear, the weak distortion of the images of distant galaxies by the gravitational field of the inhomogeneous matter distribution in the Universe, is studied here. Previous work on three-point statistics of cosmic shear has mainly concentrated on the convergence, or on aperture measures of the shear. However, as has become clear recently for the two-point statistics of cosmic shear, the basic quantity that should be used is the correlation function: first, it is much easier to measure from observational data, since it is immune against complicated geometries of data fields (which contain gaps and holes, e. g. due to masking); second, all other (linear) two-point statistics can be expressed as integrals over the correlation function. The situation is the same for the three-point statistics. However, in contrast to the two-point correlation function, the invariants (with respect to rotations) of the shear three-point correlation function have not been employed yet. Here we consider the transformation properties of the shear three-point correlation function under rotations. We show that there are four complex linear combinations of components of the three-point correlation function, which we shall call natural components, since they are multiplied just by a phase factor for arbitrary rotations, but do not mix. In particular, their moduli are invariant under rotations and thus (non-linear) invariants of the three-point correlation function. In terms of these natural components, the invariance of the statistical properties of the shear field under parity transformations are easily obtained. Our results do not apply only to cosmic shear, but also to other quantities with the same mathematical properties that of a polar. For example, practically every relation derived here applies also to the polarization of the cosmic microwave background radiation.