1 Introduction

The weak gravitational lensing effect by the large-scale matter distribution of the Universe, called cosmic shear, has long been recognized as a unique tool to study the statistical properties of the cosmological (dark) matter distribution, without referring to luminous tracers of this distribution (Blandford et al. 1991; Miralda-Escude 1991; Kaiser 1992, 1998; Jain & Seljak 1997; Bernardeau et al. 1997; Schneider et al. 1998; van Waerbeke et al. 1999; Bartelmann & Schneider 1999; Jain et al. 2000; White & Hu 2000; see Mellier 1999 and Bartelmann & Schneider 2001 for recent reviews). Owing to the smallness of the effect, its actual measurement has only fairly recently been achieved, nearly simultaneously by several groups (Bacon et al. 2000; Kaiser et al. 2000; van Waerbeke et al. 2000; Wittman et al. 2000). This breakthrough became possible due to the usage of wide-field optical cameras and the development of special-purpose image analysis software specifically designed to measure the shape of very faint galaxies and to correct their ellipticity for effects of PSF smearing and anisotropy. By now, several additional cosmic shear measurements have been reported (Maoli et al. 2001; van Waerbeke et al. 2001; Rhodes et al. 2001; Bacon et al. 2002; Refregier et al. 2002; Hämmerle et al. 2002; Hoekstra et al. 2002), both from the ground and from HST imaging, partly with appreciably larger sky area than the original discovery papers.

In all of these papers, the cosmic shear signal detected was one related to the two-point correlation function of the shear, or some function of it, such as the shear dispersion or the aperture mass. To measure higher-order statistical properties of the cosmic shear, the quantity of data must be larger than for the second-order measures. It has been pointed out by a number of authors (e.g., Bernardeau et al. 1997; Jain & Seljak 1997; Schneider et al. 1998; van Waerbeke et al. 1999; Hamana et al. 2002) that the third-order statistics (e.g. the skewness) contains very valuable cosmological information, such as the density parameter $\Omega_{\rm m}$. In particular, the near-degeneracy between $\sigma_8$ and $\Omega_{\rm m}$ in two-point cosmic shear statistics (see, e.g., van Waerbeke et al. 2002) can be broken if the three-point statistics is employed. Encouragingly, Bernardeau et al. (2002a) have reported the detection of a third-order statistical signal in their cosmic shear survey.

Apart from the larger difficulty to obtain a measurement of the third-order statistics of the cosmic shear, there is also the problem of an appropriate statistical estimator for the third-order shear. Whereas for the second-order, the statistically independent shear measures are known, we are not in this position for the third-order shear statistics. We shall briefly summarize the situation for the two-point statistics, and explain why the three-point shear statistics is substantially more complicated in Sect. 2 below. In Sect. 3 we shall then define the components of the shear three-point correlation function (3PCF) and study their transformation behavior under spatial rotations. From that, we shall then find in Sect. 4 the natural components of the shear 3PCF, which can be considered analogous to the natural components of the two-point correlation function of the shear. These components are "natural'' in the sense that they have the simplest behavior under rotation transformations: each of the four complex natural components is just multiplied with a phase factor when an arbitrary rotation is applied, which in particular means that the moduli of these natural components are invariants under rotations. We shall discuss the importance of these natural components in Sect. 5, where we also outline the perspectives of future work that can be based on the use of these natural components. In an appendix, we shall consider the projection of the shear onto several particular reference points, defined by the various centers of a triangle.

We want to point out that all the relations derived in this papers are not confined only to cosmic shear. In fact, this paper investigates the three-point correlation function of a polar - a polar is a two-component quantity which transforms under a rotation of the coordinate frame by a phase factor ${\rm e}^{2{\rm i}\varphi}$. Alternatively, a polar can be viewed as the trace-free part of a symmetric $2\times 2$ matrix. In the case of cosmic shear, this matrix is the Hessian of the deflection potential. Another polar of great cosmological importance is the polarization of the cosmic microwave background. Hence, all the results presented below do equally well apply to the three-point correlation function of the CMB polarization.