5 Discussion

We have shown that the clustering of galaxies from which the shear is measured leads to the presence of a B-mode in the cosmic shear field, in addition to providing an additional component to the E-mode. The reason for this effect in essence is the angular separation-dependent redshift correlation of galaxies, which causes the mean of the product of the angular-diameter distance ratio along two lines-of-sight not to factorize, but to depend on $\theta$. For a fiducial model considered in detail, the B-mode contribution amounts to more than $\sim$$2\%$ on angular scales below 1' (or $\ell \ga 2.16 \times 10^4$), and its relative importance quickly rises towards smaller angles. On substantially larger angular scales, however, the B-mode contribution is small. Furthermore, the additional E-mode contribution is very similar in size to the B-mode power, which will allow an approximate correction of the measured E-mode for this additional term.

From an observational point-of-view, the most easily accessible quantities are the shear correlation functions $\xi_\pm$, as one can easily deal with gaps in the data field. In Sect. 2 we have given explicit relations regarding how other two-point statistics of the shear can be calculated in terms of the shear correlation function. The finite support of the functions $T_\pm$ indicates that the aperture measures are more easily obtained from observational data than either the E- and B-mode correlation functions, or the E- and B-mode shear dispersions. Therefore, the aperture measures are the preferred method to check for the presence of a B-mode contribution in the shear data.

We have varied some of the model parameters; in particular, we have considered the case of lower mean source redshift (corresponding to a brighter flux threshold), and simultaneously increasing A(1'), such that the clustering length r₀ stays about the same. In this case we found a very similar ratio between the B- and E-mode power spectra as for the example considered in Sect. 4. We consider it unlikely that the observed B-mode in the present day data sets is due to the source clustering effect. The B-mode found in van Waerbeke et al.(2001) and Pen et al. (2002) can actually be used to search quantitatively for residual systematics. Its detection in van Waerbeke et al.(2001) was done by obtaining $M_{\rm ap}$ and $M_\perp$by laying down a grid of circular apertures on the data field. A more accurate measurement of $\left\langle M_{\rm ap}^2 \right\rangle$ and $\left\langle M_\perp^2 \right\rangle$ has been obtained from the same data by Pen et al. (2002), by calculating them from the observed correlation functions $\xi_\pm$, as in (33). In fact a subsequent analysis revealed that the B-mode measured in these data were essentially residual systematics caused by an overcorrection of the PSF, and can be corrected for (van Waerbeke et al. 2002). In this latter analysis, no significant B-modes are detected at small angular scales, but on scales above $\sim$10', slightly significant values of $M_\perp$ are detected; the effect discussed in this paper can certainly not account for them.

The effect considered here seems to have been overlooked hitherto. Bernardeau (1998) considered the effects of source clustering on cosmic shear statistics and concluded that this source clustering can strongly affect the skewness and kurtosis of the cosmic shear, but to first order leaves the shear dispersion (and thus the power spectrum) unaffected. Hamana et al. (2002) studied this effect with ray tracing simulations, again concentrating on the skewness. Most of the other ray tracing simulations of weak lensing (e.g., van Waerbeke et al. 1999; Jain et al. 2000) assumed all sources to be at the same redshift, in which case the additional power discussed here does not occur. Lombardi et al. (2002) calculated the effect of source clustering on the noise of weak lensing mass maps, showing that it can provide a significant noise contribution in the inner regions of clusters.

It must be pointed out that the effect considered here is unrelated to other lensing effects which in principle could generate a B-mode, such as lens-lens coupling or the break-down of the Born approximation (see Bernardeau et al. 1997 and SvWJK for a discussion of these two effects on the skewness). Numerical estimates (e.g., Jain et al. 2000) show that these latter two effects are very weak. Bertin & Lombardi (2001) considered the situation of lensing by two mass concentrations along the line-of-sight, where a B-mode is generated by a strong lens-lens coupling, but the fraction of lines-of-sight where this occurs is tiny. Another effect which could in principle generate a B-mode from lensing is the fact that the observable is not the shear itself, but the reduced shear (Schneider & Seitz 1995). In the appendix we show that this effect is indeed negligible.

Like the intrinsic alignment of galaxies, which can yield a spurious contribution to the measured cosmic shear, the source clustering effect can in principle be avoided if redshift estimates of the source galaxies are available. In that case, by estimating the shear correlation function, pairs of galaxies with a large likelihood to be at the same distance can be neglected. In contrast to the intrinsic correlation of galaxies, the B-mode from source clustering appears to be fairly insensitive to the redshift distribution of the source galaxies, provided the clustering length is kept fixed.

Acknowledgements

We thank L. J. King for useful comments on the 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 and by the German Ministry for Science and Education (BMBF) through the DLR under the project 50 OR 0106.