3 B-mode from source clustering

In the previous section we have presented the decomposition of a general shear field into E/B-modes. It is usually assumed that lensing alone yields a pure E-mode shear field, so that the detection of a B-mode in the van Waerbeke et al. (2001) data (see also Pen et al. 2002) was surprising and interpreted as being due to systematic errors or a signature of intrinsic alignment of sources. Here we show that lensing indeed does generate a B-mode component of the shear if the source galaxies from which the shear is measured are clustered.

3.1 Correlation functions and power spectra

Define the "equivalent'' surface mass density for a fixed source redshift, or comoving distance w,

$$% \kappa(\vec\theta,w)= \int_0^w {\rm d} w'\; F(w',w)\; \delta[f(w')\vec\theta,w']$$ (43)

where

$$% F(w',w)={3 H_0^2 \Omega_0\over 2 c^2}~{f(w') f(w-w')\over a(w') f(w)};$$ (44)

here, H₀ and $\Omega_0$ denote the Hubble constant and the density parameter, w is the comoving distance, f(w) is the comoving angular-diameter distance to comoving distance w, $\delta$ is the density contrast, and a(w)=(1+z)^-1 is the cosmic scale factor, defined such that a=1 today, again using the notation of BS01. Accordingly, we define the shear components

$$% \gamma_\alpha(\vec\theta,w)= \left( {{\cal D}_\alpha\over \pi} * \kappa \right)(\vec\theta,w),$$ (45)

where the operator ${\cal D}$ is defined in (1) and (2). Then, the shear correlation function for two sources at positions $\vec\theta_i$ and distances w_i becomes

$$\left\langle \gamma_\alpha(\vec\theta_1,w_1)~\gamma_\beta(\vec\theta_2,w_2) \right\rangle \left( {\cal D}_\alpha {\cal D}_\beta\over \pi^2 \right) * \int_0... ...elta[f(w_1')\vec\theta_1,w_1']\;\delta[f(w_2')\vec\theta_2,w_2'] \right\rangle,$$ =

where the first ${\cal D}$ operates on $\vec\theta_1$, and the second ${\cal D}$ on  $\vec\theta_2$. This equation is of the form (BS 2.78), and thus we obtain, using (BS 2.82),

 

$$% \left\langle \gamma_\alpha~\gamma_\beta \right\rangle ={{\cal D... ...left\lbrack {\rm i}f(w)\vec k\cdot (\vec\theta_1{-}\vec\theta_2) \right\rbrack,$$ (46)

where we have temporarily dropped the arguments of the shear correlator, $w_{1,2}=\min(w_1,w_2)$, and $P_\delta$ is the power spectrum of the density fluctuations which develops as a function of cosmic time (or as a function of comoving distance w). The upper limit of the integral expresses the fact that a correlated shear can only be generated by matter which is at smaller distance than both sources.

The operators ${\cal D}$ only act on the final term of (47) which can be evaluated using the Fourier transform of ${\cal D}$, as in (11),

$$D_\alpha {\cal D}_\beta * {\rm e}^{{\rm i}f\vec k\cdot(\vec\t... ... k) {\rm e}^{{\rm i}f\vec k\cdot(\vec\theta_1-\vec\theta_2)} ,$$

so that

$$% \left\langle \gamma_\alpha~\gamma_\beta \right\rangle ={9 H_0^4... ...ell)\over \pi^2}~ {\rm e}^{{\rm i}\vec \ell\cdot(\vec\theta_1{-}\vec\theta_2)}.$$ (47)

Evaluating the relevant combinations, one finds

$$\left( \left\langle \gamma_{\rm t}\gamma_{\rm t} \right\rangle +\... ...ell\over (2\pi)}~P_\delta\left( { \ell\over f(w)},w \right)~{J}_0(\ell\theta) ,$$

 

$$\left( \left\langle \gamma_{\rm t}\gamma_{\rm t} \right\rangle -\... ...ll\over (2\pi)}~P_\delta\left( { \ell\over f(w)},w \right)~{J}_4(\ell\theta) ~,$$

where $\theta=\left\vert \vec\theta_1-\vec\theta_2 \right\vert$ and R(w,w_i)=f(w_i-w)/f(w_i) is the ratio of the angular diameter distances of a source at w_i seen from the distance $w\le w_i$ and that seen from an observer at w=0.

When measuring cosmic shear from source ellipticities, the source galaxies have a broad distribution in redshift, unless information on the redshifts are available and taken into account. Hence, to calculate the observable shear correlation functions, the foregoing expressions need to be averaged over the source redshift distribution. Let $p(w_1,w_2;\theta)$ be the probability density for comoving distances of two sources separated by an angle $\theta$ on the sky; then we have for the observable correlation functions

$$% \xi_\pm(\theta)=\int_0^{w_{\rm H}} {\rm d} w_1\int_0^{w_{\r... ...\left\langle \gamma_\times \gamma_\times \right\rangle \right)$$ (48)

and $w_{\rm H}$ is the comoving distance to the horizon. By changing the order of integration according to

$$\int_0^{w_{\rm H}} {\rm d} w_1\int_0^{w_{\rm H}}{\rm d} w_2\i... ...} w\int_w^{w_{\rm H}} {\rm d} w_1\int_w^{w_{\rm H}}{\rm d} w_2$$

we obtain

 

$$% \xi_\pm(\theta) = {9 H_0^4 \Omega_0^2\over 4 c^4} \int_0^{w_{\r... ...,w \right) {J}_{0,4}(\ell\theta) \left\langle R(w,w_1)~R(w,w_2) \right\rangle ,$$ (49)

where the angular brackets denote the averaging of the angular-diameter distance ratios over the source distance distribution. We shall write the source redshift distribution as

$$% p(w_1,w_2;\theta)={\bar p(w_1)~\bar p(w_2)\left\lbrack 1+\d... ...w_1-w_2) g(w_1;\theta) \right\rbrack \over 1+\omega(\theta)} ,$$ (50)

where

$$% \omega(\theta)=\int_0^{w_{\rm H}}{\rm d} w\;\bar p^2(w)~g(w;\theta)$$ (51)

is the angular correlation function of the galaxies, and $\bar p(w)$describes their redshift distribution. The second term in (51) accounts for source clustering. In making this ansatz, we have accounted for the fact that redshift clustering occurs only over a very small interval in redshift over which all the other redshift-dependent functions occurring in (50) can be considered constant. Note that (51) is normalized,

$$\int{\rm d} w_1\int{\rm d} w_2 \; p(w_1,w_2;\theta) =1,$$

as required. Then, the average of the angular-diameter distance ratios becomes

$$% \left\langle R(w,w_1)~R(w,w_2) \right\rangle={\bar W^2(w)+V(w,\theta)\over 1+\omega(\theta)},$$ (52)

where

$$% \bar W(w)=\int_w^{w_{\rm H}}{\rm d} w_1\;\bar p(w_1) {f(w_1-w)\over f(w_1)},$$ (53)

$$% V(w,\theta)=\int_w^{w_{\rm H}}{\rm d} w_1\;\bar p^2(w_1) \left( f(w_1-w)\over f(w_1) \right)^2~g(w_1;\theta).$$ (54)

The correlation of sources thus yields an average of the angular-diameter distance ratios which is not simply the square of the mean distance ratio $\bar W$, but contains in addition a correlated part described by V and the normalization correction $1+\omega$. If the angular separation of the sources is large, the correlation in redshift is expected to be small; hence, for large separations one expects V and $\omega$ to vanish. The degree of redshift correlation depends on the angular separation considered; the fact that the mean of the product of the angular diameter distance ratio (53) depends on the separation $\theta$ is the cause for a B-mode contribution to the shear correlation function!

One can check that the correlated redshift probability distribution behaves as expected in some simple cases. For example, if $w\ll c/H_0$is very much smaller than the characteristic source distance w₀, one finds that

$$\left\langle R(w,w_1)~R(w,w_2) \right\rangle\approx 1;$$

in this case, the lensing strength of matter at distance w is basically independent of the exact source redshift, so that source redshift clustering is irrelevant for those lens redshifts. Another case of interest occurs when the selected sources come from a very narrow distance interval, of width $\Delta w$ centered on w₀; then, (52) yields the relation $\omega(\theta)\approx g(w_0;\theta)/\Delta w$, and

$$\left\langle R(w,w_1)~R(w,w_2) \right\rangle\approx\left\lbrack f(w_0-w)\over f(w_0) \right\rbrack^2~{\rm H}(w_0-w).$$

Hence, also in this case, $\left\langle RR \right\rangle$ does not depend on $\theta$, and therefore no B-mode contribution occurs - as noted before, if all sources are at the same redshift, one obtains a pure E-mode shear field.

We can now rewrite (50) in the form

$$% \xi_\pm(\theta)={1\over 1+\omega(\theta)} \int{{\rm d} \ell... ...ft\lbrack P_\kappa(\ell)+P_{\rm c}(\ell;\theta) \right\rbrack,$$ (55)

where the 0 (4) corresponds to $\xi _+$ ($\xi _-$),

$$% P_\kappa(\ell)= {9 H_0^4 \Omega_0^2\over 4 c^4} \int_0^{w_{... ...a^2(w)}~\bar W^2(w)~P_\delta\left( {\ell\over f(w)},w \right),$$ (56)

and

$$% P_{\rm c}(\ell;\theta)={9 H_0^4 \Omega_0^2\over 4 c^4} \int... ...2(w)} V(w,\theta) ~P_\delta\left( {\ell\over f(w)},w \right) .$$ (57)

The first term of (56) in the absence of source correlations (i.e., $\omega=0$) is the one usually derived in cosmic shear considerations; $P_\kappa$ is the power spectrum of the projected matter density, related to the three-dimensional power spectrum $P_\delta$ by a Limber-type equation (e.g., Kaiser 1992). The second term in (56) and the "normalization correction'' $1+\omega$comes about due to source correlations.

3.2 The E/B-mode decomposition

From the correlation functions (56), by writing

$$\frac{P_\kappa(\ell)+P_{\rm c}(\ell;\theta)}{1+\omega(\theta)} P_\kappa(\ell)+{\cal P}(\ell;\theta)$$ =

$$\equiv P_\kappa(\ell) +\frac{P_{\rm c}(\ell;\theta)-\omega(\theta)P_\kappa(\ell)}{1+\omega(\theta)},$$ (58)

we can derive the E- and B-mode power spectra, making use of (19),

$$% P_{\rm E}(\ell)=P_\kappa(\ell)+ P_{\rm cE}(\ell) ; \;\; P_{\rm B}(\ell)=P_{\rm cB}(\ell) ;$$ (59)

with

 

$$% P_{\rm cE,B}(\ell)= {1\over 2}\int{\rm d}\theta\;\theta \int{\r... ...\theta){J}_0(\ell'\theta) \pm{J}_4(\ell\theta){J}_4(\ell'\theta) \right\rbrack,$$ (60)

where the "+'' ("-'')-sign corresponds to the E- (B-) mode. If the ratio containing the power spectra did not depend on $\theta$, $P_{\rm B}$ would vanish identically, as it should. However, this term does depend on $\theta$ due to source correlations; therefore, without using redshift information, the presence of B-modes in cosmic shear observations is unavoidable.

Using the definitions of the E- and B-mode correlation function, we obtain

$$% \xi_{\rm E+}(\theta)=\int{{\rm d} \ell\;\ell\over (2\pi)}~P_\ka... ...) \left( {4\over {\vartheta}^2}-{12\theta^2\over {\vartheta}^4} \right)\Biggr],$$ (61)

$$\xi_{\rm B+}(\theta)=\int{{\rm d} \ell\;\ell\over (4\pi)} \Biggl[... ...) \left( {4\over {\vartheta}^2}-{12\theta^2\over {\vartheta}^4} \right)\Biggr],$$ (62)

$$\xi_{\rm E-}(\theta)=\int{{\rm d} \ell\;\ell\over (2\pi)}~P_\kapp... ...heta}) \left( {4\over \theta^2}-{12{\vartheta}^2\over \theta^4} \right)\Biggr],$$ (63)

$$\xi_{\rm B-}(\theta)=\int{{\rm d} \ell\;\ell\over (4\pi)} \Biggl[... ...}) \left( {4\over \theta^2}-{12{\vartheta}^2\over \theta^4} \right)\Biggr]\cdot$$ (64)