B-modes in cosmic shear from source redshift clustering

Abstract

Weak gravitational lensing by the large scale structure can be used to probe the dark matter distribution in the Universe directly and thus to probe cosmological models. The recent detection of cosmic shear by several groups has demonstrated the feasibility of this new mode of observational cosmology. In the currently most extensive analysis of cosmic shear, it was found that the shear field contains unexpected modes, so-called B-modes, which are thought to be unaccountable for by lensing. B-modes can in principle be generated by an intrinsic alignment of galaxies from which the shear is measured, or may signify some remaining systematics in the data reduction and analysis. In this paper we show that B-modes in fact {\it are produced} by lensing itself. The effect comes about through the clustering of source galaxies, which in particular implies an angular separation-dependent clustering in redshift. After presenting the theory of the decomposition of a general shear field into E- and B-modes, we calculate their respective power spectra and correlation functions for a clustered source distribution. Numerical and analytical estimates of the relative strength of these two modes show that the resulting B-mode is very small on angular scales larger than a few arcminutes, but its relative contribution rises quickly towards smaller angular scales, with comparable power in both modes at a few arcseconds. The relevance of this effect with regard to the current cosmic shear surveys is discussed.

Figures (4)

• Figure 1: The four functions defined in text
• Figure 2: Dimensionless power spectra $\ell^2 P(\ell)$, as a function of of wavenumber $\ell$. The solid curve corresponds to the power spectrum $\ell^2 P_\kappa(\ell)$ that is the 'standard' power spectrum of the projected mass density. The dotted curve displays $\ell^2 P_{\rm c}(\ell, 1')$, and the two dashed curves correspond to the E- and B-mode power caused by the source clustering. Here, a $\Lambda$CDM model was used, with shape parameter $\Gamma=0.21$, normalization $\sigma_8=1$, and the source redshift distribution is characterized by $z_0=1$, yielding $\left\langle z \right\rangle\approx 1.5$. Other parameters for the model used here are mentioned in the text.
• Figure 3: For the same model as in Fig. 2, several correlation functions are plotted. The solid line shows $\xi_+(\theta)$; in fact, the correlation function $\xi_{\rm E+}$ cannot be distinguished from $\xi_+$ on the scale of this figure; their fractional difference is less than 1%, even on the smallest scale shown. The two B-mode correlation functions are shown as well as $\xi_-$ and $\xi_{E-}$. Note that the difference between the latter two is larger than that of the corresponding '+'-correlation functions.
• Figure 4: Aperture measures, for the same model as used in Fig. 2. Shown here is the dispersion of the aperture mass, $\left\langle M_{\rm ap}^2 \right\rangle$, the corresponding function in the absence of source correlations (noted by the subscript '0') and ${\left\langle M_\perp^2 \right\rangle}$, which is the aperture measure for the B-mode. As expected from the power spectra shown in Fig. 2, and the fact that the aperture measures are a filtered version of the power spectra with a very narrow filter function, the B-mode aperture measure is considerably smaller than $M_{\rm ap}$ itself.