2 E/B-mode decomposition of a shear field

In this section we provide the basic relations for the decomposition of the shear field into E- and B-modes. Most of these relations have been obtained in C01; we shall write them here in standard lensing notation.

2.1 Motivation

If the shear field is obtained from a projected surface mass density $\kappa$ as in Eq. (1), then the gradient of the density field $\kappa$ is related to the first spatial derivatives of the shear components in the following way (Kaiser 1995):

$$% \nabla\kappa=\left( \begin{array}{c} \gamma_{1,1}+\gamma_{2... ...\gamma_{2,1}-\gamma_{1,2} \end{array} \right) \equiv {\vec u}.$$ (3)

The vector field $\vec u$ can be obtained from observations, e.g. in weak lensing cluster mass reconstructions, by obtaining a smoothed version of the shear field and then differentiating this numerically. Owing to noise, the resulting ("observed'') field $\vec u$will in general not be a gradient field. The non-gradient part of $\vec u$ is then a readily identifiable noise component and can be filtered out in the mass reconstruction. Seitz & Schneider (1996) provided a scheme for this noise filtering (see also Seitz & Schneider 2001 for a simpler though equivalent method), which was shown by Lombardi & Bertin (1998) to be an optimal reconstruction method.

If the shear field cannot be ascribed to a single geometrically thin gravitational lens, the non-gradient part of $\vec u$ is not necessarily due to noise. For example, if the galaxies have intrinsic alignments, this may induce a curl-part of $\vec u$. To project out the gradient and curl part of $\vec u$, we take a further derivative of $\vec u$, and define

$$% \nabla^2\kappa^{\rm E}=\nabla\cdot{\vec u}; \quad \nabla^2\kappa^{\rm B}=\nabla\times{\vec u}\equiv u_{2,1}-u_{1,2}.$$ (4)

Through these relations, $\kappa^{\rm E}$ and $\kappa^{\rm B}$ are not uniquely defined on a finite data field; as discussed in Seitz & Schneider (1996), a further condition is needed to specify the two modes uniquely. However, we shall not be concerned here with finite-field effects.

An alternative way to define $\kappa^{\rm E}$ and $\kappa^{\rm B}$ is through the Kaiser & Squires (1993) mass-reconstruction relation

$$% \kappa^{\rm E}(\vec\theta)+{\rm i}\kappa^{\rm B}(\vec\theta... ... {\cal D}^*({\vec\theta}-{\vec\theta'})~\gamma({\vec\theta'}),$$ (5)

which formally requires data on an infinite field; here, ${\cal D}^*$denotes the complex-conjugate of the complex kernel (2). If $\gamma$ is of the form (1) with a real field $\kappa$, then the result from (5) will be real, $\kappa^{\rm E}=\kappa$, $\kappa^{\rm B}=0$. In applications of the KS-formula (5) to observational data, where the recovered shear field necessarily is noisy, one usually takes the real part of the integral to obtain the projected mass density field. For a general shear field, the result from (5) will be complex, with the real part yielding the E-mode, and the imaginary part corresponding to the B-mode.

To simplify notation and calculations, it is convenient to express two-component quantities in terms of complex numbers. We define the E- and B-mode potentials $\psi^{\rm E}$ and $\psi^{\rm B}$ by

$$% \nabla^2\psi^{\rm E,B}=2\kappa^{\rm E,B},$$ (6)

and combine the two modes into the complex fields

$$% \kappa=\kappa^{\rm E}+{\rm i}\kappa^{\rm B}, \quad \psi=\psi^{\rm E}+{\rm i}\psi^{\rm B}.$$ (7)

The complex shear $\gamma=\gamma_1+{\rm i}\gamma_2$ is obtained from the potential $\psi$ by $\gamma={D}\psi$, where the differential operator ${D}=(\partial_{11}-\partial_{22})/2+{\rm i}\partial_{12}$; hence,

$$\gamma{=}\left\lbrack {1\over 2}\left( \psi^{\rm E}_{,11}{-}\... ...( \psi^{\rm B}_{,11}-\psi^{\rm B}_{,22} \right) \right\rbrack.$$

Inserting this into (3) yields

$$\vec u=\left(\begin{array}{c}{\kappa^{\rm E}_{,1}-\kappa^{\rm... ...{\kappa^{\rm E}_{,2}+\kappa^{\rm B}_{,1}} \end{array}\right).$$

Indeed, the shear field can be decomposed into E/B-modes, $\gamma=\gamma^{\rm E}+{\rm i}\gamma^{\rm B}$, with

$$\gamma^{\rm E}={{\cal D}\over \pi} * {\cal R}\hbox{e}\left\lbrack {{\cal D}^*\over \pi} * \gamma \right\rbrack,$$

$$\gamma^{\rm B}={{\cal D}\over \pi} * {\cal I}\hbox{m}\left\lbrack {{\cal D}^*\over \pi} * \gamma \right\rbrack,$$

where the operator ${\cal D}$ is defined in Eqs. (1), (2), and "*'' denotes complex conjugation. Thus, the two components can be obtained from the shear field by filtering, except for an additive constant.

2.2 Shear correlation functions and power spectra

The discussion above dealt with the shear field itself. In the application to cosmic shear, one usually does not investigate the shear of a $\kappa$-field itself, but its statistical properties. In this paper we shall concentrate solely on two-point statistical measures of the cosmic shear, and their decomposition into E- and B-modes.

Owing to statistical homogeneity and isotropy of the Universe, $\kappa^{\rm E,B}(\vec\theta)$ are homogeneous and isotropic random fields. Hence, in terms of their Fourier transforms

$$% \hat\kappa^{\rm E,B}(\vec\ell)=\int{\rm d}^2\theta~{\rm e}^{{\rm i}\vec\ell\cdot\vec\theta}~\kappa^{\rm E,B}(\vec\theta),$$ (8)

one defines the two power spectra $P_{\rm E}$, $P_{\rm B}$, and the cross power spectrum $P_{\rm EB}$ by

$$% \left\langle \hat\kappa^{\rm E}(\vec\ell)\hat\kappa^{\rm E*}(\vec\ell') \right\rangle (2\pi)^2~\delta_{\rm D}(\vec\ell-\vec\ell')~P_{\rm E}(\ell),$$ =

$$\left\langle \hat\kappa^{\rm B}(\vec\ell)\hat\kappa^{\rm B*}(\vec\ell') \right\rangle (2\pi)^2~\delta_{\rm D}(\vec\ell-\vec\ell')~P_{\rm B}(\ell),$$ =

$$\left\langle \hat\kappa^{\rm E}(\vec\ell)\hat\kappa^{\rm B*}(\vec\ell') \right\rangle (2\pi)^2~\delta_{\rm D}(\vec\ell-\vec\ell')~P_{\rm EB}(\ell),$$ = (9)

where $\delta_{\rm D}$ denotes Dirac's delta distribution. In terms of the complex field $\kappa$, we then have

$$\left\langle \hat\kappa(\vec\ell)\hat\kappa^{*}(\vec\ell') \right... ...(\vec\ell-\vec\ell')\left\lbrack P_{\rm E}(\ell)+P_{\rm B}(\ell) \right\rbrack,$$

$$\left\langle \hat\kappa(\vec\ell)\hat\kappa(\vec\ell') \right\ran... ...\lbrack P_{\rm E}(\ell)-P_{\rm B}(\ell)+2{\rm i}P_{\rm EB}(\ell) \right\rbrack.$$ (10)

The Fourier transform $\hat\gamma(\vec\ell)$ of the shear is related to $\hat\kappa(\vec\ell)$ through

$$% \hat\gamma(\vec\ell)=\left( \ell_1^2-\ell_2^2+2{\rm i}\ell_... ...\kappa(\vec\ell) ={\rm e}^{2{\rm i}\beta}\hat\kappa(\vec\ell),$$ (11)

where $\beta$ is the polar angle of $\vec\ell$. The correlators of the shear then become

$$% \left\langle \hat\gamma(\vec\ell)\hat\gamma^*(\vec\ell') \right... ...(\vec\ell-\vec\ell')\left\lbrack P_{\rm E}(\ell)+P_{\rm B}(\ell) \right\rbrack,$$

$$\left\langle \hat\gamma(\vec\ell)\hat\gamma(\vec\ell') \right\ran... ...\lbrack P_{\rm E}(\ell)-P_{\rm B}(\ell)+2{\rm i}P_{\rm EB}(\ell) \right\rbrack.$$ (12)

Next we define the correlation functions of the shear. This is done by considering pairs of positions $\vec{\vartheta}$ and $\vec\theta+\vec{\vartheta}$, and defining the tangential and cross-component of the shear at position $\vec{\vartheta}$ for this pair as

$$% \gamma_{\rm t}=-{\cal R}\hbox{e}\left( \gamma~{\rm e}^{-2{\... ...l I}\hbox{m}\left( \gamma~{\rm e}^{-2{\rm i}\varphi} \right) ,$$ (13)

respectively, where $\varphi$ is the polar angle of the separation vector $\vec \theta$. Then, the shear correlation functions are defined as

 

$$% \xi_+(\theta) \left\langle \gamma_{\rm t}\gamma_{\rm t} \right\rangle +\left\langle \gamma_\times \gamma_\times \right\rangle(\theta),$$ =

$$\xi_-(\theta) \left\langle \gamma_{\rm t}\gamma_{\rm t} \right\rangle -\left\langle \gamma_\times \gamma_\times \right\rangle(\theta),$$ =

$$\xi_\times(\theta) \left\langle \gamma_{\rm t}\gamma_{\times} \right\rangle(\theta).$$ = (14)

The shear correlation functions are most easily calculated by choosing $\vec\theta =(\theta,0)$, in which case $\gamma_{\rm t}=-\gamma_1$, $\gamma_\times=-\gamma_2$, and expressing the shear in terms of its Fourier modes,

  

$$% \left\langle \gamma(\vec 0)\gamma^*(\vec\theta) \right\rangle~ \xi_+(\theta)$$ =

$$\int{{\rm d}^2\ell\over (2\pi)^2} \int{{\rm d}^2\ell'\over (2\pi)... ...c\theta} \left\langle \hat\gamma(\vec\ell)\hat\gamma^*(\vec\ell') \right\rangle$$ =

$$\int_0^\infty {{\rm d}\ell~\ell\over 2\pi}~{J}_0(\ell\theta) \left\lbrack P_{\rm E}(\ell)+P_{\rm B}(\ell) \right\rbrack,$$ = (15)

$$\left\langle \gamma(\vec 0)\gamma(\vec\theta) \right\rangle~~ \xi_-(\theta)+2{\rm i}\xi_\times(\theta)$$ =

$$\int{{\rm d}^2\ell\over (2\pi)^2} \int{{\rm d}^2\ell'\over (2\pi)... ...vec\theta} \left\langle \hat\gamma(\vec\ell)\hat\gamma(\vec\ell') \right\rangle$$ =

$$\int_0^\infty{{\rm d}\ell~\ell\over 2\pi}{J}_4(\ell\theta) \left\lbrack P_{\rm E}(\ell){-}P_{\rm B}(\ell){+}2{\rm i}P_{\rm EB}(\ell) \right\rbrack.$$ = (16)

Making use of the orthogonality of Bessel functions,

$$% \int_0^\infty {\rm d}\theta\;\theta~{J}_\nu(s\theta)~{J}_\nu(t\theta) ={\delta_{\rm D}(s-t)\over t}\;,$$ (17)

we can invert the relations (15) and (16) and express the power spectra in terms of the correlation functions,

 

$$% P_{\rm E}(\ell)~ \pi\int_0^\infty{\rm d}\theta~\theta~ \left\lbrack \xi_+(\theta){J}_0(\ell\theta) +\xi_-(\theta){J}_4(\ell\theta) \right\rbrack,$$ =

$$P_{\rm B}(\ell)~ \pi\int_0^\infty{\rm d}\theta~\theta~ \left\lbrack \xi_+(\theta){J}_0(\ell\theta) -\xi_-(\theta){J}_4(\ell\theta) \right\rbrack,$$ =

$$P_{\rm EB}(\ell) 2\pi\int_0^\infty{\rm d}\theta~\theta~ \xi_\times(\theta){J}_4(\ell\theta).$$ = (18)

Hence, we have now expressed the various power spectra in terms of the directly observable correlation functions $\xi$. One notes that the correlation functions $\xi _+$ and $\xi _-$ depend on both, the E- and B-mode power spectra, whereas the cross-correlation $\xi_\times$depends on the cross-power $P_{\rm EB}$ only. It is obvious that the cross-power and its corresponding correlation function do not "mix in'' with the E- and B-mode; in addition, the cross-power vanishes if the shear field is statistically invariant under parity transformations, which leave $\gamma_{\rm t}$ unchanged, but transform $\gamma_\times \to -\gamma_\times$. One can therefore assume that in realistic cases, $\xi_\times\equiv 0\equiv P_{\rm EB}$. However, since cosmic shear is measured from finite data fields, cosmic variance may lead to a non-zero measurement of the cross-power; in fact, the measurement of the cross-power may serve as a lower limit on error bars of the other power spectra. For a determination of the power spectra, the expressions (19) require a measurement of the correlation functions over an infinite range in angle; whereas the correlation functions decrease with $\theta$ and become very small for large $\theta$, so that in effect the integrals can be replaced by ones over a finite range of integration, one might want to obtain more local decompositions into E- and B-modes.

2.3 E/B-mode correlation functions

We define the four correlation functions

 

$$% \xi_{\rm E,B+}(\theta) \int{{\rm d}\ell~\ell\over 2\pi}\;P_{\rm E,B}(\ell)~{J}_0(\theta\ell) ,$$ =

$$\xi_{\rm E,B-}(\theta) \int{{\rm d}\ell~\ell\over 2\pi}\;P_{\rm E,B}(\ell)~{J}_4(\theta\ell) ,$$ = (19)

which are defined such that in the absence of B-modes, $\xi_{\rm E\pm}\equiv \xi_\pm$; these four correlation functions have also been defined in C01, although only the "+'' ones were investigated in more detail there. Inserting (19) into the foregoing definitions, one obtains

 

$$% \xi_{\rm E+}(\theta)= {1\over 2}\int_0^\infty{\rm d}\ell~\ell\i... ...{J}_0({\vartheta}\ell) +\xi_-({\vartheta}){J}_4({\vartheta}\ell) \right\rbrack.$$ (20)

The $\ell$-integration can be carried out; consider the function

$$% G({\vartheta},\theta)=\int_0^\infty{\rm d} t\; t~{J}_0(t{\vartheta})~{J}_4(t\theta).$$ (21)

Making use of the recurrence relations for Bessel functions, one can express J₄ as

$${J}_4(x)={24\over x^2}~{J}_2(x)-{8\over x}{J}_1(x) +{J}_0(x).$$

By using Eq. (11.4.41) of Abramowitz & Stegun (1965), together with (18), one can perform the integration in (22) term by term to obtain

$$% G({\vartheta},\theta)=\left( {4\over\theta^2}-{12{\vartheta... ...\vartheta}) +{1\over\theta}\delta_{\rm D}(\theta-{\vartheta}),$$ (22)

where ${\rm H}(x)$ is the Heaviside step function. We also note the interesting property,

$$% \int_0^\infty{\rm d}{\vartheta}\;{\vartheta}~G({\vartheta},\theta)~G({\vartheta},\varphi) {\delta_{\rm D}(\theta-\varphi)\over \varphi}$$ =

$$\int_0^\infty{\rm d}{\vartheta}\;{\vartheta}~G(\theta,{\vartheta})~G(\varphi,{\vartheta}) ,$$ = (23)

which is readily shown using (18). Thus, (21) becomes

 

$$% \xi_{\rm E+}(\theta) {1\over 2}\left\lbrack \xi_+(\theta) +\int_0^\infty{\rm d}{\vartheta}~{\vartheta}~\xi_-({\vartheta})~G(\theta,{\vartheta}) \right\rbrack$$ =

$${1\over 2}\left\lbrack \xi_+(\theta){+}\xi_-(\theta) {+}\int_\the... ...vartheta})\left( 4{-}12{\theta^2\over {\vartheta}^2} \right) \right\rbrack\cdot$$ = (24)

We have obtained a combination of shear correlation functions which depends only on the E-modes; however, in order to obtain $\xi _{\rm E+}$ one would need to know $\xi _-$ for arbitrarily large separations. However,

 

$$% \xi_{\rm E-}(\theta) {1\over 2}\left\lbrack \xi_-(\theta) +\int_0^\infty{\rm d}{\vartheta}~{\vartheta}~\xi_+({\vartheta})~G({\vartheta},\theta) \right\rbrack$$ =

$${1\over 2}\left\lbrack \xi_-(\theta){+}\xi_+(\theta) {+}\int_0^\t... ...i_+({\vartheta}) \left( 4-12{{\vartheta}^2\over \theta^2} \right) \right\rbrack$$ = (25)

depends on the observable correlation functions $\xi_\pm$ over a finite range only and thus can be measured from finite data sets. Analogously, we find for the B-mode correlation functions

 

$$% \xi_{\rm B+}(\theta) {1\over 2}\left\lbrack \xi_+(\theta) -\int_0^\infty{\rm d}{\vartheta}~{\vartheta}~\xi_-({\vartheta})~G(\theta,{\vartheta}) \right\rbrack$$ =

$${1\over 2}\left\lbrack \xi_+(\theta){-}\xi_-(\theta) {-}\int_\the... ..._-({\vartheta}) \left( 4-12{\theta^2\over {\vartheta}^2} \right) \right\rbrack,$$ =

$$\xi_{\rm B-}(\theta) {1\over 2}\left\lbrack -\xi_-(\theta) +\int_0^\infty{\rm d}{\vartheta}~{\vartheta}~\xi_+({\vartheta})~G({\vartheta},\theta) \right\rbrack$$ =

$${1\over 2}\left\lbrack \xi_+(\theta){-}\xi_-(\theta) {+}\int_0^\t... ...\vartheta}) \left( 4-12{{\vartheta}^2\over \theta^2} \right) \right\rbrack\cdot$$ = (26)

We note that $\xi_{\rm E+}+\xi_{\rm B+}=\xi_+$, $\xi_{\rm E-}-\xi_{\rm B-}=\xi_-$. In order to calculate the E- and B-mode correlation functions, one needs to know either the observable correlation function $\xi _-$ to arbitrarily large, or $\xi _+$ to arbitrarily small separations. This is of course impossible, owing to the finite size of data fields on the one hand, and the impossibility to measure shapes of very close pairs of galaxies. In either case, the lack of measurements for large (or small) separations can be summarized in two constants: suppose that $\xi _+$ can be measured down to separations of $\theta_{\rm min}$; then, the integral in the "-'' modes over $\xi _+$can be split into one from 0 to $\theta_{\rm min}$, and one from $\theta_{\rm min}$ to $\theta$. The former one has the $\theta$-dependence $a/\theta^2-b/\theta^4$, where a, b are two constants, depending on $\xi _+$ for $\theta < \theta_{\rm min}$. The decline of this contribution, with leading order $\theta^{-2}$, shows that it has a small influence on the determination of the "-'' modes for $\theta\gg \theta_{\rm min}$; in addition, "reasonable guesses'' for a and b may be obtained by extrapolating the measured $\xi _+$ towards small angles. The same reasoning shows the analogous situation for the "+'' modes.

2.4 Aperture measures

One very convenient way to separate E- and B-modes is provided by the aperture mass: defining the tangential and cross component of the shear relative to the center of a circular aperture of angular radius $\theta$, and defining

 

$$% M_{\rm ap}(\theta) \int{\rm d}^2{\vartheta}\;Q(\vert\vec{\vartheta}\vert)~\gamma_{\rm t}(\vec{\vartheta}),$$ =

$$M_{\perp}(\theta)~ \int{\rm d}^2{\vartheta}\;Q(\vert\vec{\vartheta}\vert)~\gamma_{\times}(\vec{\vartheta}),$$ = (27)

it was shown in C01 that E-modes do not contribute to $M_\perp$, and B-modes do not contribute to $M_{\rm ap}$; in fact, this can be easily seen directly by inserting the Fourier transform of the shear (11) into (28). Here, $Q({\vartheta})$ is an axially-symmetric weight function which can be chosen arbitrarily. The integration range in the foregoing equations extends over the support of the weight function Q. The aperture mass was introduced by Schneider (1996) in an attempt to detect mass peaks from shear fields, and later used by SvWJK as a two- and three-point statistics for cosmic shear. $M_{\rm ap}$ can also be written as a filtered version of the surface mass density (with a different and compensated weight function which is related to Q), whereas $M_\perp$ has no direct physical interpretation; in the absence of B-modes, $M_\perp$ should vanish, and any non-vanishing signal is usually interpreted as being due to noise or remaining systematics, and thus as a convenient error estimate for  $M_{\rm ap}$. In SvWJK, a family of convenient weight functions Q was considered, the simplest of which is

$$Q({\vartheta})={6\over \pi\theta^2}~{{\vartheta}^2\over\theta... ...vartheta}^2\over\theta^2} \right)~{\rm H}(\theta-{\vartheta}),$$

where $\theta$ is the radius of the aperture. This form of the weight function shall be assumed in the following.

Using the complex number

$$M(\theta) M_{\rm ap}(\theta)+{\rm i}M_\perp(\theta)$$ =

$$-\int{\rm d}^2{\vartheta}\;Q(\vert\vec{\vartheta}\vert)~\gamma(\vec{\vartheta})~{\rm e}^{-2{\rm i}\varphi},$$ =

where $\varphi$ is the polar angle of $\vec{\vartheta}$, one finds that

 

$$% \left\langle M_{\rm ap}^2 \right\rangle{+}\left\langle M_\perp^2 \right\rangle \left\langle MM^* \right\rangle$$ =

$${1\over 2\pi}\int_0^\infty{\rm d}\ell~\ell~\left\lbrack P_{\rm E}(\ell){+}P_{\rm B}(\ell) \right\rbrack W(\theta\ell),$$ = (28)

with

$$% W(\eta):={576{J}_4^2(\eta)\over \eta^4},$$ (29)

which was derived by using the Fourier transform (11) of the shear, and the final steps are as in SvWJK. Similarly, one obtains

 

$$\left\langle M_{\rm ap}^2 \right\rangle \left\langle M_\perp^2 \right\rangle+2{\rm i}\left\langle M_{\rm ap}M_\perp \right\rangle =\left\langle MM \right\rangle$$ -

$$={1\over 2\pi}\int_0^\infty{\rm d}\ell\;\ell~\left\lbrack P_{\rm E}(\ell)-P_{\rm B}(\ell)+2{\rm i}P_{\rm EB}(\ell) \right\rbrack W(\theta\ell).$$ (30)

Combining the two previous equations, one thus gets

 

$$\left\langle M_{\rm ap}^2 \right\rangle(\theta) {1\over 2\pi}\int_0^\infty{\rm d}\ell\;\ell~P_{\rm E}(\ell) ~ W(\theta\ell),$$ =

$$\left\langle M_{\perp}^2 \right\rangle(\theta) {1\over 2\pi}\int_0^\infty{\rm d}\ell\;\ell~P_{\rm B}(\ell) ~ W(\theta\ell),$$ = (31)

so that these two-point statistics clearly separate E- and B-modes. In addition, they provide a highly localized measure of the corresponding power spectra, since the filter function $W(\eta)$ involved is quite narrow (see SvWJK); in fact, Bartelmann & Schneider (1999) have shown that replacing $W(\eta)$in (32) by $A \delta_{\rm D}(\eta-\eta_0)$, with $\eta_0\approx 4.25$, provides a fairly accurate approximation. Furthermore, (31) can be used to check whether the shear data contain a contribution from the cross power $P_{\rm EB}$.

The aperture measures can be obtained directly from the observational data by laying down a grid of points, at each of which $M_{\rm ap}$and $M_\perp$ are calculated from (28). However, obtaining the dispersion with this strategy turns out to be difficult in practice, since data fields usually contain holes and gaps, e.g. because of masking (for bright stars), bad columns etc. It is therefore interesting to calculate these dispersions directly in terms of the correlation functions, which can be done by inserting (19) into (32),

 

$$% \left\langle M_{\rm ap}^2 \right\rangle(\theta) {1\over 2}\int{{\rm d}{\vartheta}~{\vartheta}\over\theta^2} \left... ...ght)+\xi_-({\vartheta})~T_-\left( {\vartheta}\over\theta \right) \right\rbrack,$$ =

$$\left\langle M_{\perp}^2 \right\rangle(\theta)~ {1\over 2}\int{{\rm d}{\vartheta}~{\vartheta}\over\theta^2} \left... ...ght)+\xi_-({\vartheta})~T_-\left( {\vartheta}\over\theta \right) \right\rbrack,$$ = (32)

where we have defined the functions

 

$$T_+(x)=576\int_0^\infty{{\rm d} t\over t^3}~{ J}_0(x t)~\left\lbrack {J}_4(t) \right\rbrack^2 ,$$

$$T_-(x)=576\int_0^\infty{{\rm d} t\over t^3}~{J}_4(x t)~\left\lbrack {J}_4(t) \right\rbrack^2 .$$ (33)

The integration range in (33) formally extends from zero to infinity, but as we shall see shortly, the functions $T_\pm(x)$ vanish for x>2, so the integration range is $0\le {\vartheta}\le 2\theta$: for T_-(x), an analytic expression can be obtained, using Eq. (6.578.9) of Gradshteyn & Ryzhik (1980),

$$% T_-(x)={192\over 35\pi}x^3\left( 1-{x^2\over 4} \right)^{7/2}~{\rm H}(2-x),$$ (34)

so that T_-(x) vanishes for x>2. Furthermore, the two functions T₊ and T_-(x) are related: using (18), one finds that

$$\int_0^\infty{\rm d} x\;x~T_+(x)~{J}_0(t x) =W(t)= \int_0^\infty{\rm d} x\;x~T_-(x)~{J}_4(t x)$$

so that

$$\int_0^\infty{\rm d} t\;t~W(t)~{J}_4(x t) =\int_0^\infty {\rm d} y\;y~T_+(y)~G(y,x) ,$$

$$\int_0^\infty{\rm d} t\;t~W(t)~{J}_0(x t) =\int_0^\infty {\rm d} y\;y~T_-(y)~G(x,y) ;$$

using the latter expression, together with (35) and (23), one obtains

 

$$T_+(x)={6(2-15x^2)\over 5}\left\lbrack 1-{2\over\pi}\arcsin\left(... ...sqrt{4-x^2}\over 100\pi} \left( 120{+}2320x^2{-}754x^4{+}132 x^6{-}9x^8 \right)$$ (35)

for $x\le 2$, and T₊(x) vanishes for x>2. Hence, the integrals in (33) extend only over $0\le {\vartheta}\le 2\theta$, so that $\left\langle M_{\rm ap}^2 \right\rangle$ and $\left\langle M_\perp^2 \right\rangle$ can be obtained directly in terms of the observable correlation function $\xi_\pm$ over a finite interval. The two functions $T_\pm$ are plotted in Fig. 1.

Figure 1: The four functions defined in text.

2.5 Shear dispersion

Another cosmic shear statistics often employed is the shear dispersion in a circle of angular radius $\theta$. It is related to the power spectra by

$$% \left\langle \left\vert \bar\gamma \right\vert^2 \right\ran... ...\ell~\ell~ (P_{\rm E}+P_{\rm B})(\ell)~W_{\rm TH}(\ell\theta),$$ (36)

where

$$% W_{\rm TH}(\eta)={4 {J}_1^2(\eta)\over \eta^2}$$ (37)

is the top-hat filter function. In contrast to the aperture measures of the previous subsection, the shear dispersion (37) contains both modes; furthermore, the filter function $W_{\rm TH}(\eta)$ is much broader than $W(\eta)$ in (30), as demonstrated in SvWJK. It thus provides a much less localized measure of the power spectra than the aperture measures. On the other hand, this larger filter width implies that the signal of the shear dispersion is larger than that of the aperture measures, which explains why the first cosmic shear detections (van Waerbeke et al. 2000; Bacon et al. 2001; Kaiser et al. 2000) were obtained in terms of the shear dispersion.

As before, the shear dispersion can be obtained by calculating the mean shear in circles which are laid down on a grid of points, with the drawback of being affected by gaps in the data field. Alternatively, the shear dispersion can be obtained directly from the correlation function,

$$% \left\langle \left\vert \bar\gamma \right\vert^2 \right\ran... ...~ \xi_+({\vartheta})~S_+\left( {\vartheta}\over\theta \right),$$ (38)

where (van Waerbeke 2000)

$$% S_+(x)={1\over\pi}\left\lbrack 4\arccos\left( x\over 2 \right)-x\sqrt{4-x^2} \right\rbrack$$ (39)

for $x\le 2$, and zero otherwise. Hence, the integral in (39) extends only over the finite interval $0\le {\vartheta}\le 2\theta$, which makes this a convenient way to calculate the shear dispersion.

One can also define the shear dispersions of the E- and B-mode, according to

$$% \left\langle \left\vert \bar\gamma \right\vert^2 \right\ran... ...int{\rm d}\ell~\ell~ P_{\rm E,B}(\ell)~W_{\rm TH}(\ell\theta),$$ (40)

but they cannot be individually obtained from measuring the shear directly. Nevertheless, both of these dispersions can be obtained in terms of the correlation functions,

 

$$% \left\langle \left\vert \bar\gamma \right\vert^2 \right\rangle_... ... \pm\xi_-({\vartheta})~S_-\left( {\vartheta}\over\theta \right) \right\rbrack ,$$ (41)

which can be derived in close analogy to the derivation of (33), and the function S_- is related to S₊ in the same way as the corresponding T-functions,

$$\int_0^\infty {\rm d} y\;y~S_+(y)~G(y,x)$$ S_-(x) =

$${x\sqrt{4-x^2}(6-x^2)-8(3-x^2)\arcsin(x/2) \over \pi x^4}$$ = (42)

for $x\le 2$, and S_-(x)=4(x²-3)/x⁴ for x>2. Hence, the integrals in (42) do not cut off at finite separation, which was to be expected, since a constant shear cannot be uniquely assigned to an E- or B-mode, but contributes to $\left\langle \left\vert \bar\gamma \right\vert^2 \right\rangle\cdot$