5 Discussion and outlook

The natural components of the shear 3PCF provide a generalization of the corresponding natural components $\xi_\pm$ of the two-point correlation function. Given that upcoming cosmic shear surveys will be substantially larger than the current ones, it is obvious that the 3PCF will be measurable in the future with a similar accuracy as the two-point correlation function in current surveys; in fact, a first significant measurement of the 3PCF has been reported in Bernardeau et al. (2002a). Therefore, it is worth to explore the dependence of the 3PCF on various parameters, as has been done for the two-point function. In particular, the detailed study of the interrelations between $\xi_\pm$ and the underlying power spectrum, as presented in Crittenden et al. (2002) and Schneider et al. (2002a), shall be generalized to the 3PCF. Of course, this will be substantially more difficult from a technical point of view, and will be deferred to future work. Nevertheless, we can outline a few aspects of what can be expected from such work.

In close analogy to the two-point correlation function, one can expect that the natural components can be more easily calculated from the bispectrum of the underlying mass distribution than the individual components of the 3PCF. As is the case for $\xi_\pm$, one can expect that $\Gamma^{(0)}$ probes the bispectrum in a different way than the $\Gamma^{(k)}$, $1\le k\le 3$. In particular, the different natural components will have a different dependence on cosmological parameters.

The natural components of the 3PCF are not independent of each other; provided that the shear indeed is due to a surface mass density field, there should be integral relations which interrelate them. Again one should note the analogy with the two-point correlation function, where $\xi_+$ can be obtained as an integral over $\xi_-$ and vice versa (Crittenden et al. 2002; Schneider et al. 2002a). The interrelations between the components of the 3PCF provide a redundancy which can be profitably combined to reduce the noise in real measurements.

If the shear is not solely due to a surface mass distribution, the shear field may contain a B-mode contribution (e.g., from intrinsic alignments of the galaxies from which the shear is measured). In the case of the two-point correlation function, the presence of a B-mode can be probed from integral relations between the two correlation functions $\xi_\pm$ (Crittenden et al. 2002; Schneider et al. 2002a); it is expected that similarly in the case of the 3PCF, the integral relations between the natural components will be modified in the presence of a B-mode.

All linear three-point statistics of the shear can be expressed in terms of the shear 3PCF. This is obvious from the fact that the bispectrum of the surface mass density can be expressed in terms of the 3PCF; on the other hand, all linear three-point statistics are linearly related to the bispectrum, and can therefore be expressed directly in terms of the 3PCF. In particular, the third-order aperture mass statistics (Schneider et al. 1998; van Waerbeke et al. 1999) can be expressed as an integral over the 3PCF. It remains to be seen whether the third-order aperture mass statistics is as useful for a separation of the shear field into E- and B-modes as it is the case for the two-point statistics (Crittenden et al. 2002; Schneider et al. 2002a; for applications in cosmic shear surveys, see e.g. Pen et al. 2002; Hoekstra et al. 2002).

Concerning practical measurements of the 3PCF, the procedure is straightforward: from a given catalog of galaxy images with position vectors $\vec{\theta}_i$ and measured ellipticities ${\epsilon}_i$, triplets are selected. For each such triplet, the sides of the corresponding triangle can be calculated, and for practical reasons the largest side be called x₃. The three points are then labeled $\vec{X}_l$, $1\le l\le 3$ in the (unique) way such that the orientation of the points is as described at the beginning of Sect. 3, and the longest side connects $\vec{X}_1$ and $\vec{X}_2$. Since the connecting vectors $\vec{x}_l$ need to be calculated anyway, it is simplest to project the ellipticities along the sides of the triangle, or, with opposite sign, towards the orthocenter, without having to calculate any trigonometric function. The eight (= $2\times 2\times 2$) triple products of the projected ellipticities are calculated and summed up in eight three-dimensional bins. Those are conveniently labeled by a scale x₃, and two shape parameters, q₁=x₁/x₃, q₂=x₂/x₃, so that the grid of bins runs as $0\le q_{1,2}\le 1$, $0<x_3<\infty$. After summing up over all triplets of points, for each of the eight components the sums in the bins are divided by the number of triplets contributing, which then yields an estimate of the 3PCF. Those can then be combined into the natural components for further analysis.

For a quantitative analysis of the three-point statistics of cosmic shear in relation to predictions of cosmological models, one can either use integrated properties of the shear 3PCF - such as the aperture mass or the integral of $\Gamma^{(\rm B)}$ over an elliptical region as done in Bernardeau et al. (2002b), or consider the (more noisy, but much more numerous) estimates of the natural components of the 3PCF directly. The first method yields one-dimensional functions of the three-point shear statistics depending on a scale parameter, and are thus very convenient for graphical displays. In contrast, using the full shear 3PCF employs multi-dimensional data which is difficult to display, but contains all the information about third-order statistics from the data. Hence, even though the signal-to-noise in each bin of the 3PCF can be small, its overall information content cannot be smaller than that of any of the integrated quantities, and should therefore be employed in extracting cosmological information from the measurements. To obtain a reliable figure-of-merit for the comparison of the observational results with model predictions, one needs to know the covariance matrix of the shear 3PCF - this, however, is a 64-components quantity depending on 6 arguments, and will be very difficult to obtain analytically (see Schneider et al. 2002b for the difficulties of obtaining the covariance of the two-point correlation functions). Corresponding covariances of integrated quantities may be slightly easier obtainable. On the other hand, it is quite likely that figures-of-merit will have to rely heavily on future ray-tracing simulations through N-body-generated cosmological mass distributions, as in, e.g., van Waerbeke et al. (1999), Jain et al. (2000) and Bernardeau et al. (2002a).

After this paper was finished, two recent preprints were posted on the Web which discussed very similar issues (Zaldarriaga & Scoccimarro 2002; Takada & Jain 2002b). In the latter one, the components $\gamma_{\mu\nu\lambda}$ of the correlation function were calculated from ray-tracing simulations; in particular they verified that all of the eight components of the 3PCF are non-zero in general and thus contain cosmological information.

Acknowledgements

We are grateful to Matthias Bartelmann for very useful comments on this manuscript. This work was supported by the TMR Network "Gravitational Lensing: New Constraints on Cosmology and the Distribution of Dark Matter'' of the EC under contract No. ERBFMRX-CT97-0172, by the German Ministry for Science and Education (BMBF) through the DLR under the project 50 OR 0106, and the Deutsche Forschungsgemeinschaft under the project SCHN 342/3-1.