4 Invariant components of the shear three-point correlation function

The normal way to choose "good'' components of a multi-index quantity like $\gamma_{\mu\nu\lambda}$ is to look for its behavior under coordinate transformations. Recall the situation for the shear two-point correlation function: the correlator $\left\langle \gamma_\mu \gamma_\nu \right\rangle(\vec{x})$ of the Cartesian components of the shear contains two terms, one which is independent on the phase $\varphi$ of $\vec{x}$, the other behaving as $\cos(4\varphi)$. The coefficients of these two terms are $\xi_+$ and $\xi_-$, respectively, which are the invariants, and therefore the natural choice for the components of the two-point correlation function of the shear.

Similarly, we can consider the behavior of the 3PCF under rotations. Given that no linear scalar can be built from the Cartesian components of the 3PCF alone we cannot expect to find a combination of components which is invariant under general rotations. However, as we shall see, there are linear combinations of the components of the 3PCF which have a simple behavior under rotations, namely multiplication by a phase factor. Owing to this simple transformation behavior, we shall term them "natural'' components.

Let now $\gamma_{\mu\nu\lambda}$ be the components of the 3PCF of the shear measured with respect to one particular choice of direction(s); choosing different projection directions, which differ from the old ones by $\zeta_i$, the 3PCF becomes

 

$$% \gamma'_{\mu\nu\lambda} \equiv \left\langle \gamma'_\mu(\vec{X}_1)\gamma'_\nu(\vec{X}_2)\gamma'_\lambda(\vec{X}_3) \right\rangle$$

$$R_{\mu\alpha}(2\zeta_1)~R_{\nu\beta}(2\zeta_2)~R_{\lambda\gamma}(2\zeta_3) \gamma_{\alpha\beta\gamma};$$ = (16)

note that this expression differs from Eq. (5) by a minus sign since in Eq. (5) the transition from Cartesian to tangential and cross components was included as well. To investigate more closely the action of the rotation operator in Eq. (16), we transform this equation into one which appears more familiar: We define the eight-component quantity

$$\begin{eqnarray*}\Gamma:=(\gamma_{{\rm ttt}},\gamma_{{\rm tt}\times}, \gamma_{\r... ...es}, \gamma_{\rm\times\times t},\gamma_{\rm\times\times \times}) \end{eqnarray*}$$

and analogously $\Gamma'$ for the transformed components; then, the rotation described in Eq. (16) can be written as $\Gamma'=R \Gamma$, where R is a $8\times 8$ matrix. The components of R are triple products of trigonometric functions. Since it describes a rotation, R is unitary and one expects that the eigenvalues of R have an absolute value of unity. They can in fact be obtained as

 

$$% \lambda^{(0)}_{1,2} \exp\left( \pm 2{\rm i}[\zeta_1+\zeta_2+\zeta_3] \right); \;\; \lambda^{(1)}_{1,2} = \exp\left( \pm 2{\rm i}[-\zeta_1+\zeta_2+\zeta_3] \right);$$ =

$$\lambda^{(2)}_{1,2} \exp\left( \pm 2{\rm i}[\zeta_1-\zeta_2+\zeta_3] \right); \;\; \lambda^{(3)}_{1,2} = \exp\left( \pm 2{\rm i}[\zeta_1+\zeta_2-\zeta_3] \right).$$ = (17)

Note that this result is not very surprising: either the eigenvalues are $\pm 1$, or they have to appear in the above form, i.e., they have to occur as pairs of complex conjugate numbers with absolute value of unity. The dependence of the eigenvalues on the rotation angles is in fact a natural one.

4.1 The natural components of the 3PCF

We see that under the general rotation described by Eq. (16), the eigenvalue +1 does not occur; in other words, there is no (linear) combination of the components of the 3PCF that is invariant under the transformation (16). However, for some special rotations, the eigenvalue +1 occurs. As a first example, we consider the case $\zeta_1+\zeta_2+\zeta_3=0$; as shown in Sect. 3.2, this case actually is encountered for transformations of the shear components between the orthocenter, the incenter and the circumcenter. Then, the two eigenvalues $\lambda^{(0)}_{1,2}$ are +1, and the corresponding eigenvectors can be found to be

E⁽⁰⁾₂ = (0, 1, 1, 0, 1, 0, 0, -1). (18)

Hence, under rotations of this kind, we expect that the two combinations

$$% \Gamma^{(0)}_1 = \gamma_{{\rm ttt}}-\gamma_{\rm t\times \times}... ...\gamma_{\rm t\times t} + \gamma_{{\rm\times tt}}-\gamma_{\rm\times\times\times}$$ (19)

of the components of the 3PCF are both invariant. It will turn out to be very useful to combine these two invariants into a single complex quantity, $\Gamma^{(0)}=\Gamma^{(0)}_1+{\rm i} \Gamma^{(0)}_2$. This quantity, however, can also be written in a different form, namely

 

$$% \Gamma^{(0)}=\left\langle \gamma(\vec{X}_1)\gamma(\vec{X}_2)\gamma(\vec{X}_3) \right\rangle,$$ (20)

where we again consider $\gamma=\gamma_{\rm t}+{\rm i}\gamma_\times$as a complex quantity. Written in this form, it is obvious that a rotation with $\zeta_1+\zeta_2+\zeta_3=0$ keeps $\Gamma^{(0)}$ invariant.

Figure 6: The centers of the escribed circles are intersections between one interior angle bisector and two exterior angle bisectors. These three escribed circles are tangent to one side of the triangle and the extensions of the two other sides.

Next we consider the case that $-\zeta_1+\zeta_2+\zeta_3=0$. Such a rotation occurs if one transforms the shear components from the incenter (or the out- or orthocenter) to the center of one of the escribed circles (see Fig. 6 for an explanation); in this case, $\lambda^{(1)}_{1,2}=1$, and from the corresponding eigenvectors one can again construct two combinations $\Gamma^{(1)}_1$, $\Gamma^{(1)}_2$, of the components of the 3PCF that stay invariant under such a rotation, and they can as well be combined into a single complex quantity. The same procedure can then be repeated for rotations with $\zeta_1-\zeta_2+\zeta_3=0$ and those with $\zeta_1+\zeta_2-\zeta_3=0$; these correspond, e.g., to the transformation from one of the aforementioned centers to the centers of the other two escribed circles. The corresponding invariants read:

 

$$% \Gamma^{(1)} \gamma_{{\rm ttt}}-\gamma_{\rm t\times \times}+\gamma_{{\rm\times... ...\times t} -\gamma_{{\rm\times tt}}+\gamma_{\rm\times\times\times} \right\rbrack$$ =

$$\left\langle \gamma^*(\vec{X}_1)\gamma(\vec{X}_2)\gamma(\vec{X}_3) \right\rangle;$$ =

$$\Gamma^{(2)} \gamma_{{\rm ttt}}+\gamma_{\rm t\times \times}-\gamma_{{\rm\times... ...\times t} +\gamma_{{\rm\times tt}}+\gamma_{\rm\times\times\times} \right\rbrack$$ =

$$\left\langle \gamma(\vec{X}_1)\gamma^*(\vec{X}_2)\gamma(\vec{X}_3) \right\rangle;$$ =

$$\Gamma^{(3)} \gamma_{{\rm ttt}}+\gamma_{\rm t\times \times}+\gamma_{\times{\rm... ...\times t} +\gamma_{{\rm\times tt}}+\gamma_{\rm\times\times\times} \right\rbrack$$ =

$$\left\langle \gamma(\vec{X}_1)\gamma(\vec{X}_2)\gamma^*(\vec{X}_3) \right\rangle\cdot$$ = (21)

Now, each of these $\Gamma^{(\alpha)}$ is invariant only under special rotations, as the derivation above has shown. But the remarkable finding here is that, under a general rotation, the different $\Gamma^{(\alpha)}$ do not mix, but they are just multiplied by a phase factor; indeed, since the rotation (16) implies that $\gamma(\vec{X}_1)\to\gamma'(\vec{X}_1)=\gamma(\vec{X}_1)~{\rm e}^{-2{\rm i}\zeta_1}$, and similarly for the other two points, we see from the final expression in Eqs. (20) and (21) that the transformed invariants read

 

$$\left(\Gamma^{(0)}\right)'=\exp\left( - 2{\rm i}[\zeta_1+\zeta_2+\zeta_3] \right)\Gamma^{(0)} =\lambda^{(0)}_2~\Gamma^{(0)},\;$$

$${\rm and}\; \left(\Gamma^{(\alpha)}\right)'=\lambda^{(\alpha)}_2~\Gamma^{(\alpha)},$$ (22)

where the $\lambda^{(\alpha)}_2$ have been defined in Eq. (17). The transformation (22) is indeed remarkable, though in hindsight not all that surprising. The fact that the $\Gamma^{(\alpha)}$ only transform amongst themselves justifies that they are called natural components of the three-point correlation function of the shear. Note in particular that Eq. (22) implies that the four absolute values $\left\vert \Gamma^{(\alpha)} \right\vert$ are independent under general rotations; they are therefore the (non-linear) invariants of the shear 3PCF.

Of course, if desired, the original components of the shear 3PCF can be obtained from the natural components; the inverse of Eq. (21) reads:

$$\gamma_{{\rm ttt}} = \left( \Gamma^{(1)}_1+\Gamma^{(2)}_1 +\Gamma^{(3)}_1+\Gamma^{(0)}_1 \right)/4;$$

$$\gamma_{\rm t\times \times} = \left( -\Gamma^{(1)}_1+\Gamma^{(2)}_1 +\Gamma^{(3)}_1-\Gamma^{(0)}_1 \right)/4;$$

$$\gamma_{{\rm\times t}\times} = \left( \Gamma^{(1)}_1-\Gamma^{(2)}_1 +\Gamma^{(3)}_1-\Gamma^{(0)}_1 \right)/4;$$

$$\gamma_{{\times}{\times}\rm t} = \left( \Gamma^{(1)}_1+\Gamma^{(2)}_1 -\Gamma^{(3)}_1-\Gamma^{(0)}_1 \right)/4;$$

$$\gamma_{{\rm tt}\times} = \left( \Gamma^{(1)}_2+\Gamma^{(2)}_2 -\Gamma^{(3)}_2+\Gamma^{(0)}_2 \right)/4;$$

$$\gamma_{\rm t\times t} = \left( \Gamma^{(1)}_2-\Gamma^{(2)}_2 +\Gamma^{(3)}_2+\Gamma^{(0)}_2 \right)/4;$$

$$\gamma_{{\rm\times tt}} = \left( -\Gamma^{(1)}_2+\Gamma^{(2)}_2 +\Gamma^{(3)}_2+\Gamma^{(0)}_2 \right)/4;$$

$$\gamma_{\rm\times\times\times} = \left( \Gamma^{(1)}_2+\Gamma^{(2)}_2 +\Gamma^{(3)}_2-\Gamma^{(0)}_2 \right)/4.$$ (23)

4.2 Parity transformation

A parity transformation is obtained by mirroring a given set of three points with respect to any straight line. Such a transformation has the following effects: if in the original triangle the three points $\vec{X}_1$, $\vec{X}_2$, $\vec{X}_3$ are ordered counterclockwise with respect to, say, the center of the incircle, in the transformed triangle they will be ordered clockwise. As a result, under a parity transformation we need to replace every angle with its opposite.

In order to understand the behavior of the 3PCF under parity, let us suppose that the triangle is flipped along a line perpendicular to the side $\vec{x}_3$ (since the 3PCF is invariant upon rotation, the direction used for the flipping is irrelevant). We first observe that this mirror symmetry is equivalent to interchanging the points $\vec{X}_1$ and $\vec{X}_2$, or the sides x₁ and x₂. Second, such a flipping of orientation keeps the tangential component of the shear invariant (if the direction relative to which the shear components are measured are as well subject to the mirror transformation - like it happens when shear components are defined relative to one of the centers of the triangle), but changes the sign of the cross-component; hence, this transformation implies $P \gamma =\gamma^*$. Together, these two effects therefore imply that under a parity transformation,

 

$$P \left\lbrack \gamma_{\mu\nu\lambda}(x_1,x_2,x_3) \right\rbrack =\Pi \gamma_{\nu\mu\lambda}(x_2,x_1,x_3),$$

$${\rm where}\quad \Pi=(-1)^{\nu+\mu+\lambda+1},$$ (24)

the parity, is +1 if all of the indices of $\gamma$ are ${\rm t}$'s, or two $\times$'s occur, otherwise it is negative. Components of the 3PCF with $\Pi=+1$ are called even, those of negative parity odd components. The action of the parity transformation is most easily expressed in terms of the natural components of the 3PCF; as can be seen from Eq. (21), the real part of each of the $\Gamma^{(\alpha)}$ is composed of even components, the imaginary part of odd components. Therefore,

$$% P \left\lbrack \Gamma^{(0)}(x_1,x_2,x_3) \right\rbrack \left( \Gamma^{(0)} \right)^*(x_1,x_3,x_2) = \left( \Gamma^{(0)} \right)^*(x_2,x_1,x_3)$$ =

$$\left( \Gamma^{(0)} \right)^*(x_3,x_2,x_1),$$ =

$$P \left\lbrack \Gamma^{(1)}(x_1,x_2,x_3) \right\rbrack \left( \Gamma^{(1)} \right)^*(x_1,x_3,x_2) = \left( \Gamma^{(2)} \right)^*(x_2,x_1,x_3)$$ =

$$\left( \Gamma^{(3)} \right)^*(x_3,x_2,x_1),$$ =

$$P \left\lbrack \Gamma^{(2)}(x_1,x_2,x_3) \right\rbrack \left( \Gamma^{(1)} \right)^*(x_2,x_1,x_3) = \left( \Gamma^{(2)} \right)^*(x_3,x_2,x_1)$$ =

$$\left( \Gamma^{(3)} \right)^*(x_1,x_3,x_2),$$ =

$$P \left\lbrack \Gamma^{(3)}(x_1,x_2,x_3) \right\rbrack \left( \Gamma^{(1)} \right)^*(x_3,x_2,x_1)= \left( \Gamma^{(2)} \right)^*(x_1,x_3,x_2)$$ =

$$\left( \Gamma^{(3)} \right)^*(x_2,x_1,x_3).$$ = (25)

Each of these parity transformations contains a permutation of the arguments with negative signature. One can also consider cyclic permutations of the arguments; since

$$% \gamma_{\mu\nu\lambda}(x_1,x_2,x_3) =\gamma_{\nu\lambda\mu}(x_2,x_3,x_1) =\gamma_{\lambda\mu\nu}(x_3,x_1,x_2),$$ (26)

one obtains for the natural components

 

$$% \Gamma^{(0)}(x_1,x_2,x_3) = \Gamma^{(0)}(x_2,x_3,x_1)=\Gamma^{(0)}(x_3,x_1,x_2) ,$$

$$\Gamma^{(1)}(x_1,x_2,x_3) = \Gamma^{(3)}(x_2,x_3,x_1)=\Gamma^{(2)}(x_3,x_1,x_2),$$

$$\Gamma^{(2)}(x_1,x_2,x_3) = \Gamma^{(1)}(x_2,x_3,x_1)=\Gamma^{(3)}(x_3,x_1,x_2),$$

$$\Gamma^{(3)}(x_1,x_2,x_3) = \Gamma^{(2)}(x_2,x_3,x_1)=\Gamma^{(1)}(x_3,x_1,x_2) .$$ (27)

Hence, $\Gamma^{(0)}$ is invariant under cyclic permutations of the arguments, whereas the other three natural components of the shear 3PCF transform into each other under such permutations. The relations (27) imply that only one of the three functions $\Gamma^{(k)}(x_1,x_2,x_3)$, $1\le k\le 3$, is independent, the other two can be obtained by cyclic permutations of the arguments.

The parity transformations have an immediate consequence for triangles where two sides are equal, say x₁=x₂; namely, from Eq. (24) one finds that

$$\gamma_{\rm tt\times}(x_1,x_1,x_3)= 0=\gamma_{\times\times\times}(x_1,x_1,x_3);$$

$$\gamma_{\rm t\times t}(x_1,x_1,x_3)= -\gamma_{\rm\times tt}(x_1,x_1,x_3).$$ (28)

This implies that for x₁=x₂, $\Gamma^{(0)}$ and $\Gamma^{(3)}$have no imaginary part, and those of $\Gamma^{(1)}$ and $\Gamma^{(2)}$have equal magnitude but opposite sign. Furthermore, for equilateral triangles, all odd components of the shear 3PCF vanish, in which case the natural components become purely real.

4.3 Two-point correlation function revisited

The foregoing formalism can of course also be applied to the two-point correlation function. In that case, the eigenvalues are $\lambda^{(0,1)}_{1,2}= {\rm e}^{\pm 2{\rm i}(\zeta_1\pm\zeta_2)}$, and the invariant combinations are $\Gamma^{(0)}=\gamma_{\rm tt}-\gamma_{\times\times} +{\rm i}\left( \gamma_{\rm t\times} +\gamma_{\rm\times t} \right)$ and $\Gamma^{(1)}=\gamma_{\rm tt}+\gamma_{\times\times} +{\rm i}\left( \gamma_{\rm t\times} -\gamma_{\rm\times t} \right)$. The two-point correlator is, however, special in the sense that after a parity transformation, the two points can be brought back into the old positions with a rotation. This then implies that the imaginary components of the $\Gamma^{(\alpha)}$ vanish if they are measured in the only reference frame that makes sense for two points - namely, the line connecting them; and so $\Gamma^{(0)}=\xi_-$, $\Gamma^{(1)}=\xi_+$. Note that if different projection directions are taken, the imaginary parts of the $\Gamma^{(\alpha)}$ are not zero.

4.4 Generalization

The discussion on obtaining the natural components of the shear 3PCF immediately suggests how to generalize it to the natural components of higher-order correlation functions. Here, we shall give the results for the four-point function (also see the recent work by Takada & Jain 2002a for a detailed consideration of the kurtosis of the cosmic shear field); further generalizations are straightforward to obtain:

If $\gamma_{\mu\nu\lambda\kappa}$ is the four-point correlation function of the shear (for notational simplicity, we skip the arguments of this function), and we consider a general rotation of the directions relative to which tangential and cross components are defined, the transformation of the shear four-point correlation function reads

 

$$% \gamma'_{\mu\nu\lambda\kappa} \equiv \left\langle \gamma'_\mu(\vec{X}_1)\gamma'_\nu(\vec{X}_2)\gamma'_\lambda(\vec{X}_3)\gamma'_\kappa(\vec{X}_4) \right\rangle$$

$$R_{\mu\alpha}(2\zeta_1)~R_{\nu\beta}(2\zeta_2)~R_{\lambda\gamma}(2\zeta_3)~ R_{\kappa\delta}(2\zeta_4) \gamma_{\alpha\beta\gamma\delta}.$$ = (29)

In analogy to the treatment for the 3PCF, there are now eight complex conjugate pairs of eigenvalues, which are

 

$$\lambda^{(0)}_{1,2} = \exp\left( \pm 2{\rm i}[\zeta_1+\zeta_2+\zeta_3+\zeta_4] \right);$$

$$\lambda^{(1)}_{1,2} = \exp\left( \pm 2{\rm i}[\zeta_1+\zeta_2-\zeta_3-\zeta_4] \right);$$

$$\lambda^{(2)}_{1,2} = \exp\left( \pm 2{\rm i} [\zeta_1-\zeta_2+\zeta_3-\zeta_4] \right);$$

$$\lambda^{(3)}_{1,2} = \exp\left( \pm 2{\rm i} [-\zeta_1+\zeta_2+\zeta_3-\zeta_4] \right);$$

$$\lambda^{(4)}_{1,2} = \exp\left( \pm 2{\rm i}[\zeta_1+\zeta_2+\zeta_3-\zeta_4] \right);$$

$$\lambda^{(5)}_{1,2} = \exp\left( \pm 2{\rm i} [\zeta_1+\zeta_2-\zeta_3+\zeta_4] \right);$$

$$\lambda^{(6)}_{1,2} = \exp\left( \pm 2{\rm i} [\zeta_1-\zeta_2+\zeta_3+\zeta_4] \right);$$

$$\lambda^{(7)}_{1,2} = \exp\left( \pm 2{\rm i} [-\zeta_1+\zeta_2+\zeta_3+\zeta_4] \right).$$ (30)

The corresponding natural components are then

$$\Gamma^{(0)} = \left\langle \gamma(\vec{X}_1)\gamma(\vec{X}_2)\gamma(\vec{X}_3)\gamma(\vec{X}_4) \right\rangle;$$

$$\Gamma^{(1)} = \left\langle \gamma(\vec{X}_1)\gamma(\vec{X}_2)\gamma^*(\vec{X}_3)\gamma^*(\vec{X}_4) \right\rangle;$$

$$\Gamma^{(2)} = \left\langle \gamma(\vec{X}_1)\gamma^*(\vec{X}_2)\gamma(\vec{X}_3)\gamma^*(\vec{X}_4) \right\rangle;$$

$$\Gamma^{(3)} = \left\langle \gamma^*(\vec{X}_1)\gamma(\vec{X}_2)\gamma(\vec{X}_3)\gamma^*(\vec{X}_4) \right\rangle;$$

$$\Gamma^{(4)} = \left\langle \gamma(\vec{X}_1)\gamma(\vec{X}_2)\gamma(\vec{X}_3)\gamma^*(\vec{X}_4) \right\rangle;$$

$$\Gamma^{(5)} = \left\langle \gamma(\vec{X}_1)\gamma(\vec{X}_2)\gamma^*(\vec{X}_3)\gamma(\vec{X}_4) \right\rangle;$$

$$\Gamma^{(6)} = \left\langle \gamma(\vec{X}_1)\gamma^*(\vec{X}_2)\gamma(\vec{X}_3)\gamma(\vec{X}_4) \right\rangle;$$

$$\Gamma^{(7)} = \left\langle \gamma^*(\vec{X}_1)\gamma(\vec{X}_2)\gamma(\vec{X}_3)\gamma(\vec{X}_4) \right\rangle\cdot$$ (31)

For reference, we shall write down the first of these explicitly,

$$% \Gamma^{(0)} \gamma_{\rm tttt}+\gamma_{\times\times\times\times} -\gamma_{\rm ... ...times t t \times} -\gamma_{\rm\times t \times t} -\gamma_{\rm\times \times t t}$$ =

$$+ {\rm i}\left\lbrack \gamma_{\rm ttt\times}+\gamma_{\rm tt\times... ..._{\rm\times \times t \times} -\gamma_{\rm\times \times \times t} \right\rbrack.$$ (32)

These natural components of the shear four-point correlation function will transform under the rotation (29) like

$$% \left( \Gamma^{(\alpha)} \right)'=\lambda_2^{(\alpha)}~\Gamma^{(\alpha)}.$$ (33)

These relations make it obvious how generalizations to higher-order correlations can be obtained.

4.5 The estimator of Bernardeau et al. (2002)

In their paper, Bernardeau et al. (2002b) considered, for the first time, a specific shear 3PCF, which they then applied successfully to observational cosmic shear data (Bernardeau et al. 2002a). Here, we shall write the Bernardeau et al. (2002b) estimator in our notation. They considered one side of a triangle (say, $\vec{x}_3$) as the reference direction, and project the shear at all three points along this direction. Hence, their projected components (here written as $\gamma^{(3)}_{\rm t,\times}$) read

$$% \gamma^{(3)}_\mu(\vec{X}_l)=-R_{\mu\nu}(2\varphi_3)\gamma_\... ...\mu\nu}(2\varphi_3-2\varphi_l)\gamma_\nu^{(\rm s)}(\vec{X}_l),$$ (34)

and, of course, $\gamma^{(3)}(\vec{X}_3)=\gamma^{(\rm s)}(\vec{X}_3)$. The 3PCF defined by Bernardeau et al. (2002b) then becomes

$$% \Gamma_\mu^{(\rm B)}=\left\langle \left\lbrack \gamma_{\rm ... ...) \right\rbrack \gamma_\mu^{(3)}(\vec{X}_3) \right\rangle\cdot$$ (35)

Since

$$\begin{eqnarray*}&& \gamma_{\rm t}^{(3)}(\vec{X}_1) \gamma_{\rm t}^{(3)}(\vec{X}... ...(\rm s)}(\vec{X}_1)\gamma_\lambda^{(\rm s)}(\vec{X}_2),\nonumber \end{eqnarray*}$$

one obtains

$$% \Gamma_\mu^{(\rm B)}=R_{\nu\lambda}(2\phi_3) \gamma^{(\rm s)}_{\nu\lambda\mu},$$ (36)

provided the triangle formed by the points $\vec{X}_l$ has the orientation defined at the beginning of Sect. 3.