2 Motivation

For the two-point cosmic shear statistics, the basic quantities are the two-point correlation functions. Given a pair of points, $\vec{X}_i$, and the Cartesian components of the shear $\gamma_\mu(\vec{X}_i)$there (i=1,2, $\mu=1,2$), one projects the shear along the direction $\varphi$ connecting these two points by defining the tangential and cross component, $\gamma_{\rm t}$ and $\gamma_\times$ by $\gamma_{{\rm t}i} + {\rm i}\gamma_{\times i} = -[\gamma_1(\vec{X}_i)+{\rm i}\gamma_2(\vec{X}_i)]~{\rm e}^{-2{\rm i}\varphi}$. Then, one forms the correlation functions $\xi_{\rm tt}(\theta)=\left\langle \gamma_{{\rm t}1}\gamma_{{\rm t}2} \right\rangle$ and  $\xi_{\times\times}(\theta)=\left\langle \gamma_{\times 1}\gamma_{\times 2} \right\rangle$, where the average is an ensemble average over all pairs of points with separation $\theta$. Even more useful are the linear combinations $\xi_\pm(\theta)=\xi_{\rm tt}(\theta) \pm\xi_{\times\times}(\theta)$. Another combination which one may be tempted to take is $\left\langle \gamma_{{\rm t}1}\gamma_{\times 2} \right\rangle$, but this changes sign under parity transformations and thus should vanish. All other two-point statistical measures of the cosmic shear, such as the shear dispersion in a circle or the aperture mass dispersion, can be expressed as integrals over these two correlation functions (e.g., Crittenden et al. 2002; Schneider et al. 2002a). From a practical point of view, the determination of the shear correlation functions is also most convenient, as they can be measured also in data fields of complicated geometry (as normally data fields are, due to masking). The two correlation functions $\xi_\pm$ can be expressed readily in terms of the power spectrum of the mass distribution in the Universe.

Compared with this situation, a proper measure of the three-point shear statistics is much more difficult to define. First we note that the two-point function $\xi_+=\left\langle \gamma \gamma^* \right\rangle$ (where "*'' denotes complex conjugation) can be defined without any reference direction; this is not the case for any three-point function of the shear, since with three two-component quantities alone, no tri-linear scalar can be formed. Hence, one needs to project the shear components. In contrast to the case of two points, where there is a unique choice of the reference direction, this is no longer true for a triangle: there is not a single "natural direction'' defined in a triangle; in fact, there are several of those (see the appendix). Therefore, it is not a priory clear how to define "useful'' components of the three-point correlation function. One might ask, for example, whether there are similar "invariant'' combinations of the components of the shear 3PCF as there are for the two-point function ($\xi_\pm$).

Given these difficulties, it is not surprising that the work on the three-point statistics of cosmic shear has been relatively sparse. Bernardeau et al. (1997) and van Waerbeke et al. (1999) consider the 3PCF of the surface mass density as reconstructed from the shear measurements. Whereas possible in principle, the fact that real data sets have gaps and holes makes the reconstructed mass map susceptible to systematics due to the geometry. Schneider et al. (1998) suggested to use the aperture mass (Schneider 1996) as a cosmic shear statistics for which the third-order moment is readily calculated directly from the shear data. However, as is the case for the shear dispersion, one needs to cover the data field with (circular) apertures which presents again a problem in case of gaps in the data. Recognizing this, Bernardeau et al. (2002b) defined a particular component (or, more precisely, a particular linear combination of components) of the 3PCF that is readily measured from observational data, calculated its expectation value from numerical ray-tracing simulations of Jain et al. (2000) and successfully applied it to the VIRMOS-DESCART survey in Bernardeau et al. (2002a).

In addition to the component of the three-point shear correlator considered by Bernardeau (2002b), the other components (there are a total of 8) may contain equally valuable information about the bispectrum of the mass distribution. In addition, it is easily seen that all measures of the third-order shear statistics can be expressed as integrals over the shear 3PCF. They are easiest to determine from real data and shall therefore be considered as the basic quantities. In this paper we will derive the "natural'' components of the shear 3PCF, by considering their transformation behavior under rotations.