Appendix A: B-mode from reduced shear?

The shear is not directly an observable, but is estimated from the image ellipticities of distant galaxies. The expectation value of the image ellipticity, however, is not the shear, but the reduced shear $g=\gamma/(1-\kappa)$. Hence, the correlation of the observed ellipticities is the correlation of the reduced shear, not the shear itself. In cosmic shear, $\left\vert \kappa \right\vert\ll 1$ nearly everywhere, and so the difference between shear and reduced shear shall not play a big role. However, at least a priori, this effect cannot be neglected, as seen from the following argument:

The skewness $S_3=\left\langle X^3 \right\rangle/\left\langle X^2 \right\rangle^2$, where X is a measure of shear (such as $M_{\rm ap}$, or the reconstructed $\kappa$) has been calculated by van Waerbeke et al. (2001) to be of order a few hundred. On a scale of about one arcminute, $\left\langle X^2 \right\rangle\sim 5\times 10^{-4}$, so that $\left\langle X^3 \right\rangle\sim 0.1 \left\langle X^2 \right\rangle$, taking $S_3\sim 200$ for the top-hat smoothed $\kappa$. The difference between the correlation functions involving g and those involving $\gamma$ is in principle of the same order-of-magnitude as $\left\langle X^3 \right\rangle$ and thus can be present at the level of a few percent, and there is no reason why it should not contain a B-mode contribution.

We define the correlation functions

$$% \xi^g_\pm(\theta)=\left\langle g_{\rm t}(\vec 0)g_{\rm t}(\... ...\langle g_\times (\vec 0)g_{\times}(\vec \theta) \right\rangle$$ (A.1)

and choose $\vec\theta =(\theta,0)$, so that $g_{\rm t}=-g_1$, $g_\times=-g_2$. Using the approximation $g\approx\gamma(1+\kappa)$, valid for $\vert\kappa\vert\ll 1$, we obtain

$$% \xi^g_\pm(\theta)=\xi_\pm(\theta)+\Delta\xi_\pm(\theta)\;,$$ (A.2)

where

$$% \Delta\xi_\pm(\theta)= \left\langle \gamma_1(\vec 0)\gamma_1(\v... ...\left\lbrack \kappa(\vec 0)+\kappa(\vec\theta) \right\rbrack \right\rangle\cdot$$ (A.3)

Replacing the shear and convergence by their Fourier transforms, this becomes

 

$$% \Delta\xi_\pm(\theta)=\int{{\rm d}^2\ell_1\over (2\pi)^2} \int{... ...heta} \right) \cos\left\lbrack 2\left( \beta_1\mp\beta_2 \right) \right\rbrack,$$ (A.4)

where, as before, $\beta_i$ is the polar angle of $\vec \ell_i$. The triple correlator vanishes unless the sum of the wave-vectors equals zero; one defines the bispectrum by

$$% \left\langle \hat\kappa(\vec\ell_1)\hat\kappa(\vec\ell_2)\h... ...vec\ell_2{+}\vec\ell_3)~ b(\vec \ell_1,\vec\ell_2,\vec\ell_3).$$ (A.5)

Performing the $\ell_3$-integration in (A.4) yields

 

$$% \Delta\xi_\pm(\theta)=\int{{\rm d}^2\ell_1\over (2\pi)^2} \int{... ...heta} \right) \cos\left\lbrack 2\left( \beta_1\mp\beta_2 \right) \right\rbrack.$$ (A.6)

The function $b(\vec\ell_1,\vec\ell_2,-\vec\ell_1-\vec\ell_2)$ has three independent arguments, namely the moduli $\ell_1$ and $\ell_2$, and the angle $\phi=\beta_1-\beta_2$ between the two $\vec\ell$-vectors. We therefore write $b(\vec\ell_1,\vec\ell_2,-\vec\ell_1-\vec\ell_2) =\tilde b(\ell_1,\ell_2,\phi)$, make use of the symmetry in the integrand of (A.6), and replace the $\beta_1$-integration by one over $\phi$:

$$% \Delta\xi_+(\theta) 2\int_0^\infty{{\rm d}\ell_1~\ell_1\over (2\pi)^2} \int_0^\infty{... ..._0^{2\pi}{\rm d}\beta_2\; {\rm e}^{-{\rm i}\ell_2\theta\cos\beta_2}~\cos(2\phi)$$

$${1\over \pi} \int_0^\infty{{\rm d}\ell_1~\ell_1\over (2\pi)}~{J}_... ...ver (2\pi)} \int_0^{2\pi}{\rm d}\phi\;\cos(2\phi)~\tilde b(\ell_1,\ell_2,\phi);$$ (A.7)

analogously, one obtains

$$% \Delta\xi_-(\theta)= {1\over \pi} \int_0^\infty{{\rm d}\ell_1~\... ...ver (2\pi)} \int_0^{2\pi}{\rm d}\phi\;\cos(2\phi)~\tilde b(\ell_1,\ell_2,\phi).$$ (A.8)

Inserting these expressions into (19) and making use of (18), one immediately sees that the reduced shear does not yield any B-mode contribution, and that the correlation functions for the reduced shear are

$$% \xi^g_\pm(\theta)=\int{{\rm d}\ell~\ell\over 2\pi} {J}_{0,4... ...theta)\left\lbrack P_\kappa(\ell)+P^{(3)}(\ell) \right\rbrack,$$ (A.9)

where

$$% P^{(3)}(\ell)~=~2\int{{\rm d}^2\ell'\over(2\pi)^2} \left\lb... ...^2}-1 \right\rbrack b(\vec\ell,\vec\ell',-\vec\ell-\vec\ell').$$ (A.10)

Thus, considering the reduced shear yields an additional E-mode power to the one obtained from considering the shear itself.