1 Introduction

Gravitational lensing by the large-scale structure (LSS) leads to the distortion of the images of distant galaxies, owing to the tidal gravitational field of the matter inhomogeneities. Following very early work on the study of light propagation in an inhomogeneous universe (e.g., Gunn 1967; Kantowski 1969), Blandford et al. (1991), Miralda-Escude (1991) and Kaiser (1992) have pointed out that the observation of this "cosmic shear'' effect immediately yields information about the statistical properties of the LSS and, thus, on cosmology. Non-linear evolution of the matter spectrum was taken into account in later analytical (e.g., Jain & Seljak 1997; Bernardeau et al. 1997; Kaiser 1998; Schneider et al. 1998, hereafter SvWJK) and numerical (e.g., van Waerbeke et al. 1999; Jain et al. 2000; White & Hu 2000) studies; see Mellier (1999) and Bartelmann & Schneider (2001; hereafter BS01) for recent reviews.

It was only in 2000 when four teams nearly simultaneously and independently announced the first detections of cosmic shear from wide-field imaging data (Bacon et al. 2000; Kaiser et al. 2000; van Waerbeke et al. 2000; Wittman et al. 2000). The detections reported in these papers (and in Maoli et al. 2001, using the VLT, and Rhodes et al. 2001, using HST images obtained with the WFPC2 camera) concerned various two-point statistics, like the shear dispersion in an aperture, or the shear correlation function. In van Waerbeke et al. (2001), the aforementioned statistics, as well as the aperture mass statistics (SvWJK), were inferred from the effective 6.5 square degrees of high-quality imaging data. Very recently, Hämmerle et al. (2002) reported on a cosmic shear detection using HST parallel images taken with the STIS instrument on an effective angular scale of $\sim$30''.

The shear field, originating from the inhomogeneous matter distribution, is a two-dimensional quantity, whereas the projected density field of the matter is a scalar field. The relation between the shear $\gamma(\vec\theta)=\gamma_1(\vec\theta)+{\rm i}\gamma_2(\vec\theta)$ and the projected matter density $\kappa(\vec\theta)$ is

$$% \gamma(\vec\theta)={1\over\pi}\int_{{\rm I\mathchoice{\kern... ...heta'\; {\cal D}(\vec\theta-\vec\theta')\kappa(\vec\theta')\;,$$ (1)

with the kernel

$$% {\cal D}(\vec\theta)={\theta_2^2-\theta_1^2-2{\rm i}\theta_1\theta_2 \over \left\vert \vec\theta \right\vert^4}\;;$$ (2)

here, $\kappa$ is the dimensionless surface mass density, i.e., the physical surface mass density divided by the "critical'' surface mass density, as usual in gravitational lensing; we follow the notation of BS01 in this paper. Since the two shear components originate from a single scalar field, they are related to each other; in particular, their partial derivatives should satisfy compatability relations, as we shall discuss in Sect.2 below. In analogy with the polarization of the CMB, a shear field satisfying these compatability relations is called an E-mode shear field.

Pen et al. (2002) pointed out that the cosmic shear data of van Waerbeke et al. (2001) contains not only an E-mode, but also a statistically significant B-mode contribution in addition. Such B-modes can be generated by effects unrelated to gravitational lensing, such as intrinsic alignment of galaxies (e.g., Heavens et al. 2000; Crittenden et al. 2001a; Croft & Metzler 2000; Catelan et al. 2000) or remaining systematics in the data reduction and analysis.

In this paper we show that a B-mode contribution to the cosmic shear is obtained by lensing itself. A B-mode is generated owing to the clustering properties of the faint galaxies from which the shear is measured. This spatial clustering implies an angular separation-dependent clustering in redshift, which is the origin not only of the B-mode of the shear, but also of an additional E-mode contribution.

The paper is organized as follows: in Sect. 2 we provide a tutorial description of the E/B-mode decomposition of a shear field. Most of the results there were derived before in Crittenden et al. (2001b, hereafter C01), but we formulate them in standard lensing notation, which will be needed for the later investigation. The calculation of two-point cosmic shear statistics in the presence of source clustering is presented in Sect. 3 where it is shown that this clustering produces a B-mode. Numerical and analytical estimates of the amplitude of this B-mode are provided in Sect. 4 and discussed in Sect. 5.