Analysis method for 3D power spectrum of projected tensor field with fast estimator and window convolution modelling: an application to intrinsic alignments

In this paper we consider a 3D tensor field estimated from large-scale structure observables, whose components are denoted as $s_{ij}({\mathbf{x}})$ satisfying $s_{ij}=s_{ji}$. The Fourier transform is given by $s_{ij}({\mathbf{k}})\equiv\int_{{\mathbf{x}}}~{}s_{ij}({\mathbf{x}})~{}e^{-i{\mathbf{k}}\cdot{\mathbf{x}}}$. An example is the tensor field that can be inferred from shapes of galaxies or dark matter halos. The galaxy or halo shape is characterized by the second moments of their light or mass density profile, which form the tensor field sampled at galaxy/halo positions. The rank-2 tensor field $s_{ij}$ carries 6 components at each position, which correspond to scalar, vector and tensor modes, respectively, as discussed in the following. The tensor field is, without loss of generality, can be decomposed into 6 orthogonal polarization states, which we label $p=\{0,z,x,y,+,\times\}$ following the notations in Ref. [65]: $s_{ij}({\mathbf{k}})\equiv\sum_{p}\bar{s}^{(p)}({\mathbf{k}})\bar{\varepsilon}^{(p)}_{ij}(\hat{\mathbf{k}})$ where $\hat{\mathbf{k}}$ is the unit vector of ${\mathbf{k}}$ and $\bar{\varepsilon}^{(p)}_{ij}(\hat{\mathbf{k}})$ is the polarization basis. The 6 polarization bases are; $\bar{\varepsilon}^{(0)}_{ij}=\sqrt{1/3}\delta^{K}_{ij}$ is the scalar mode of the trace component where $\delta^{K}_{ij}$ is the Kronecker delta function; $\bar{\varepsilon}^{(z)}_{ij}=\sqrt{3/2}(\hat{k}_{i}\hat{k}_{j}-\delta^{K}_{ij}/3)$ is the longitudinal scalar mode ($\bar{\varepsilon}^{(z)}_{ii}=0$); $\bar{\varepsilon}^{(x,y)}_{ij}=\sqrt{1/2}(\hat{k}_{i}\hat{w}^{(x,y)}_{j}+\hat{k}_{j}\hat{w}^{(x,y)}_{i})$ are the vector modes, where $\hat{\mathbf{w}}^{(x,y)}(\hat{\mathbf{k}})$ are two orthonormal and transverse vectors satisfying $\hat{\mathbf{w}}^{(p)}\cdot\hat{\mathbf{w}}^{(p^{\prime})}=\delta^{K}_{pp^{\prime}}$ and $\hat{\mathbf{k}}\cdot\hat{\mathbf{w}}^{(p)}=0$; $\bar{\varepsilon}^{(+,\times)}_{ij}=\sqrt{1/2}(\hat{w}^{(x)}_{i}\hat{w}^{(x)}_{j}-\hat{w}^{(y)}_{i}\hat{w}^{(y)}_{j},\hat{w}^{(x)}_{i}\hat{w}^{(y)}_{j}+\hat{w}^{(x)}_{j}\hat{w}^{(y)}_{i})$ are two traceless and transverse tensor modes which satisfy $\bar{\varepsilon}^{(+,\times)}_{ii}=0$ and $\bar{\varepsilon}^{(+,\times)}_{ij}\hat{k}_{j}=0$. Note that the results we will show below are independent of the specific choice of $\hat{\mathbf{w}}^{(x,y)}$. We define the normalization factor of each basis so that $\bar{\varepsilon}^{(p)}_{ij}\bar{\varepsilon}^{(p^{\prime})}_{ij}=\delta^{K}_{pp^{\prime}}$. For a galaxy shape case the trace component $\bar{s}^{(0)}({\mathbf{k}})$ characterizes a size of galaxy. The trace component would be generally an independent observable from the traceless scalar component [see e.g. 66, 67, 68]. However the analysis method of the trace component is the same as that for the density fluctuation field, which is well studied for a galaxy clustering analysis. Hence from now on, we denote the trace component simply as $\delta({\mathbf{x}})$ and focus on other 5 traceless components.

As more convenient expressions, we alternatively expand the tensor field by their transformation properties under a rotation around $\hat{\mathbf{k}}$ as

$\displaystyle s_{ij}({\mathbf{k}})=\sum_{m=-2}^{2}s^{(m)}({\mathbf{k}})~{}\varepsilon^{(m)}_{ij}(\hat{\mathbf{k}}),$

(1)

where the new basis $\left\{\varepsilon^{(m)}_{ij}\right\}$ is the so-called helicity basis related to the polarization basis as

$\displaystyle\varepsilon^{(0)}_{ij}=\bar{\varepsilon}^{(z)}_{ij},~{}\varepsilon^{(\pm 1)}_{ij}=\frac{1}{\sqrt{2}}\left(\bar{\varepsilon}^{(x)}_{ij}\pm i\bar{\varepsilon}^{(y)}_{ij}\right),~{}\varepsilon^{(\pm 2)}_{ij}=\frac{1}{\sqrt{2}}\left(\bar{\varepsilon}^{(+)}_{ij}\pm i\bar{\varepsilon}^{(\times)}_{ij}\right).$

(2)

Note that the helicity tensors transform as $\varepsilon^{(m)}_{ij}\to e^{im\psi}\varepsilon^{(m)}_{ij}$ under a rotation around $\hat{\mathbf{k}}$ by angle $\psi$. Requiring statistical homogeneity and isotropy and parity invariance, we can write the auto-power spectra of the tensor field as a sum over power spectra of each helicity field [22]:

$\displaystyle\langle s_{ij}({\mathbf{k}})~{}s_{kl}({\mathbf{k}}^{\prime})\rangle\equiv(2\pi)^{3}\delta^{3}_{D}({\mathbf{k}}+{\mathbf{k}}^{\prime})\left\{\Lambda^{(0)}_{ij,kl}(\hat{\mathbf{k}})~{}P_{ss}^{(0)}({\mathbf{k}})+\sum_{\lambda=1}^{2}\Lambda^{(\lambda)}_{ij,kl}(\hat{\mathbf{k}})~{}\frac{P_{ss}^{(\lambda)}({\mathbf{k}})}{2}\right\},$

(3)

or equivalently,

$\displaystyle\langle s_{ij}({\mathbf{k}})~{}s_{ij}({\mathbf{k}}^{\prime})\rangle\equiv(2\pi)^{3}\delta^{3}_{D}({\mathbf{k}}+{\mathbf{k}}^{\prime})\sum_{\lambda=0}^{2}P_{ss}^{(\lambda)}({\mathbf{k}}),$

(4)

where $\delta^{3}_{D}({\mathbf{k}})$ is Dirac delta function and $\Lambda^{(\lambda)}_{ij,kl}\equiv\sum_{m=\pm\lambda}\varepsilon^{(m)}_{ij}\varepsilon^{(m)*}_{kl}$ are the rotationally invariant tensors in the plane normal to $\hat{\mathbf{k}}$ with respect to the helicity $\lambda\equiv|m|$, and the explicit forms are:

	$\displaystyle\Lambda^{(0)}_{ij,kl}(\hat{\mathbf{k}})$	$\displaystyle=\frac{3}{2}\left(\hat{k}_{i}\hat{k}_{j}-\frac{1}{3}\delta^{K}_{ij}\right)\left(\hat{k}_{k}\hat{k}_{l}-\frac{1}{3}\delta^{K}_{kl}\right),$		(5)
	$\displaystyle\Lambda^{(1)}_{ij,kl}(\hat{\mathbf{k}})$	$\displaystyle=\frac{1}{2}\left(\mathcal{P}_{ik}(\hat{\mathbf{k}})\hat{k}_{j}\hat{k}_{l}+\mathcal{P}_{il}(\hat{\mathbf{k}})\hat{k}_{j}\hat{k}_{k}+\mathcal{P}_{jk}(\hat{\mathbf{k}})\hat{k}_{i}\hat{k}_{l}+\mathcal{P}_{jl}(\hat{\mathbf{k}})\hat{k}_{i}\hat{k}_{k}\right),$		(6)
	$\displaystyle\Lambda^{(2)}_{ij,kl}(\hat{\mathbf{k}})$	$\displaystyle=\frac{1}{2}\left(\mathcal{P}_{ik}(\hat{\mathbf{k}})\mathcal{P}_{jl}(\hat{\mathbf{k}})+\mathcal{P}_{il}(\hat{\mathbf{k}})\mathcal{P}_{jk}(\hat{\mathbf{k}})-\mathcal{P}_{ij}(\hat{\mathbf{k}})\mathcal{P}_{kl}(\hat{\mathbf{k}})\right),$		(7)

where $\mathcal{P}_{ij}(\hat{\mathbf{a}})\equiv\delta^{K}_{ij}-\hat{a}_{i}\hat{a}_{j}$ is the projection tensor with respect to $\hat{\mathbf{a}}$.

In addition to the auto-power spectra, we define the cross-power spectra with the density field as

$\displaystyle\langle s_{ij}({\mathbf{k}})~{}\delta({\mathbf{k}}^{\prime})\rangle\equiv(2\pi)^{3}\delta^{3}_{D}({\mathbf{k}}+{\mathbf{k}}^{\prime})\varepsilon^{(0)}_{ij}(\hat{\mathbf{k}})~{}P_{s\delta}^{(0)}({\mathbf{k}}).$

(8)

It is usually difficult to estimate the 3D tensor field from large-scale structure observables. What is easier to measure instead is a projected tensor field, where the projection is onto plane perpendicular to the LOS direction at each position. For a galaxy shape case, we can estimate the projected shape of a galaxy from its observed surface brightness distribution. Correlations of the projected shape field, i.e. IA correlations [11, 18], have been observed from actual data [e.g., 46, 69]. In this paper we do not discuss another source of correlations of observed galaxy shapes, which arise from cosmological weak lensing effects due to the foreground large-scale structure (the so-called cosmic shear), so please do not confuse IA with cosmic shear. We hereafter denote the projected tensor field as $\gamma_{ij}({\mathbf{x}})$ satisfying $\hat{x}_{i}\gamma_{ij}({\mathbf{x}})=0$, where we set an observer’s position to the coordinate origin and $\hat{\mathbf{x}}$ becomes the unit vector along the line-of-sight direction to ${\mathbf{x}}$ (in this case $\hat{x}^{i}$ is equivalent to the angular position of ${\mathbf{x}}$ on the celestial sphere of an observer in our choice of the coordinate). The projected tensor field can be defined by

$\displaystyle\gamma_{ij}({\mathbf{x}})\equiv\Lambda^{(2)}_{ij,kl}(\hat{\mathbf{x}})s_{kl}({\mathbf{x}}).$

(9)

Note that the last term of $\Lambda^{(2)}_{ij,kl}$ in Eq. (7) ensures that $\gamma_{ij}$ is traceless; $\gamma_{ii}=0$. Hence $\gamma_{ij}$ carries two modes. The coordinated-independent two modes, often used in the literature, are $E$- and $B$-modes, as discussed below.

For later convenience, we introduce the complex representation of the projected tensor field using the helicity basis in real space:

${}_{\pm 2}\gamma({\mathbf{x}})\equiv\gamma_{1}({\mathbf{x}})\pm i\gamma_{2}({\mathbf{x}})\equiv e^{(\pm 2)}_{ij}(\hat{\mathbf{x}})\gamma_{ij}({\mathbf{x}})=e^{(\pm 2)}_{ij}(\hat{\mathbf{x}})s_{ij}({\mathbf{x}}),$

(10)

where $e^{(\pm 2)}_{ij}$ is defined in terms of the orthonormal basis; $\{\hat{\mathbf{x}},\hat{\mathbf{e}}_{\theta},\hat{\mathbf{e}}_{\phi}\}$ as

$\displaystyle e^{(\pm 2)}_{ij}(\hat{\mathbf{x}})\equiv\hat{e}^{(\pm 1)}_{i}(\hat{\mathbf{x}})\hat{e}^{(\pm 1)}_{j}(\hat{\mathbf{x}}),$

(11)

with

$\displaystyle\hat{\mathbf{e}}^{(\pm 1)}(\hat{\mathbf{x}})\equiv\frac{1}{\sqrt{2}}\left(\hat{\mathbf{e}}_{\theta}\pm i\hat{\mathbf{e}}_{\phi}\right),$

(12)

and $\hat{\mathbf{e}}_{\theta}$ and $\hat{\mathbf{e}}_{\phi}$ are the unit vectors in the 2D plane perpendicular to the line-of-sight direction $\hat{\mathbf{x}}$, e.g. the RA and Dec directions in the celestial coordinate. We have used the relation $e^{(\pm 2)}_{ij}\Lambda^{(2)}_{ij,kl}=e^{(\pm 2)}_{kl}$ in the last equality of Eq. (10). The helicity basis $e^{(\pm 2)}_{ij}$ satisfies identities, $e^{(\pm 2)}_{ii}=e^{(\pm 2)}_{ij}e^{(\pm 2)}_{ij}=0$ and $e^{(\pm 2)}_{ij}e^{(\mp 2)}_{ij}=1$. The projected tensor components $\gamma_{1}$ and $\gamma_{2}$ correspond to the distortions along the directions of coordinate axes ($\hat{\mathbf{e}}_{\theta}$ or $\hat{\mathbf{e}}_{\phi}$) and the directions rotated by $45^{\circ}$ from coordinate axes, respectively [also see Ref. 56, for the definition]. Hereafter we will use only the “$+2$” component because the results in the following sections are identical to those in the case of using the “$-2$” component; we will omit the label “$+2$” for notational simplicity, e.g. $\gamma({\mathbf{x}})\equiv{}_{+2}\gamma({\mathbf{x}})$, $e_{ij}(\hat{\mathbf{x}})\equiv e^{(+2)}_{ij}(\hat{\mathbf{x}})$ from here on.

In this section we briefly review derivation of power spectra of the projected tensor field assuming the distant observer approximation or equivalently the global plane-parallel (GPP) approximation.

As we described above (Eq. 9), the projected tensor field, $\gamma_{ij}({\mathbf{x}})$, is obtained from projection of $s_{ij}({\mathbf{x}})$ onto plane perpendicular to the LOS direction ($\hat{\mathbf{x}}$). Hence the projected components of tensor field varies with the LOS direction, and its Fourier transform is generally given by a convolution form as $\gamma_{ij}({\mathbf{k}})=\int_{{\mathbf{k}}^{\prime}}\Lambda^{(2)}_{ij,kl}({\mathbf{k}}-{\mathbf{k}}^{\prime})\,s_{kl}({\mathbf{k}}^{\prime})$. Assuming the GPP approximation, the tensor field shares the same, global LOS direction, which we denote as a constant unit vector $\hat{\mathbf{n}}$ in the following. In this case the projection tensor becomes a constant tensor, and the helicity basis in real space, $e_{ij}$, and the orthonormal vectors, $\hat{\mathbf{e}}_{\theta}$ and $\hat{\mathbf{e}}_{\phi}$, become independent of $\hat{\mathbf{x}}$. The projected tensor field reduces to $\gamma_{ij}({\mathbf{x}};\hat{\mathbf{n}})\simeq\Lambda^{(2)}_{ij,kl}(\hat{\mathbf{n}})s_{kl}({\mathbf{x}})$ and also its Fourier transform is simply expressed by a multiplication form as $\gamma_{ij}({\mathbf{k}};\hat{\mathbf{n}})\simeq\Lambda^{(2)}_{ij,kl}(\hat{\mathbf{n}})s_{kl}({\mathbf{k}})$.

Since the complex projected tensor field in Fourier space is given as $\gamma({\mathbf{k}};\hat{\mathbf{n}})\equiv e_{ij}(\hat{\mathbf{n}})s_{ij}({\mathbf{k}})$, we can construct the so-called $E$/$B$-mode fields as

$\displaystyle\Gamma({\mathbf{k}};\hat{\mathbf{n}})$

$\displaystyle\equiv E({\mathbf{k}};\hat{\mathbf{n}})+iB({\mathbf{k}};\hat{\mathbf{n}})\equiv\gamma({\mathbf{k}};\hat{\mathbf{n}})e^{-2i\phi_{\hat{\mathbf{k}},\hat{\mathbf{n}}}},$

(13)

where the phase factor is given as

$\displaystyle e^{2i\phi_{\hat{\mathbf{k}},\hat{\mathbf{n}}}}\equiv\frac{2e_{ij}(\hat{\mathbf{n}})\hat{k}_{i}\hat{k}_{j}}{\mathcal{P}_{ij}(\hat{\mathbf{n}})\hat{k}_{i}\hat{k}_{j}},$

(14)

which determines a rotation of the shear components on the plane perpendicular to the LOS direction by angle between $\hat{\mathbf{e}}_{\theta}(\hat{\mathbf{n}})$ and the projected wavevector; $k_{\perp i}\equiv\mathcal{P}_{ij}(\hat{\mathbf{n}})k_{j}$. Using these modes, we can define the coordinate-independent power spectra from the observed fields $\delta$ and $\Gamma$ as

	$\displaystyle\left\langle\Gamma({\mathbf{k}})\delta^{*}({\mathbf{k}}^{\prime})\right\rangle$	$\displaystyle\equiv(2\pi)^{3}\delta^{3}_{D}({\mathbf{k}}-{\mathbf{k}}^{\prime})P_{\gamma\delta}({\mathbf{k}})\equiv(2\pi)^{3}\delta^{3}_{D}({\mathbf{k}}-{\mathbf{k}}^{\prime})P_{E\delta}({\mathbf{k}}),$		(15)
	$\displaystyle\left\langle\Gamma({\mathbf{k}})\Gamma^{*}({\mathbf{k}}^{\prime})\right\rangle$	$\displaystyle\equiv(2\pi)^{3}\delta^{3}_{D}({\mathbf{k}}-{\mathbf{k}}^{\prime})P_{+}({\mathbf{k}})\equiv(2\pi)^{3}\delta^{3}_{D}({\mathbf{k}}-{\mathbf{k}}^{\prime})[P_{EE}({\mathbf{k}})+P_{BB}({\mathbf{k}})],$		(16)
	$\displaystyle\left\langle\Gamma({\mathbf{k}})\Gamma({\mathbf{k}}^{\prime})\right\rangle$	$\displaystyle\equiv(2\pi)^{3}\delta^{3}_{D}({\mathbf{k}}-{\mathbf{k}}^{\prime})P_{-}({\mathbf{k}})\equiv(2\pi)^{3}\delta^{3}_{D}({\mathbf{k}}-{\mathbf{k}}^{\prime})[P_{EE}({\mathbf{k}})-P_{BB}({\mathbf{k}})],$		(17)

where we have introduced the $E$- and $B$-mode power spectra, $P_{EE}$ and $P_{BB}$, and we have assumed that the imaginary part corresponding to parity-odd power spectrum vanishes due to parity invariance: $\left\langle B\delta\right\rangle=\left\langle EB\right\rangle=0$. The power spectra generally depend on the norm of Fourier modes $k\equiv|{\mathbf{k}}|$ and the angle, $\mu_{k}\equiv\hat{\mathbf{k}}\cdot\hat{\mathbf{n}}$; $P({\mathbf{k}})=P(k,\mu_{k})$.

We can relate the projected tensor power spectra to the full 3D power spectra of the tensor field (see Appendix A for the derivation):

	$\displaystyle P_{\gamma\delta}({\mathbf{k}})$	$\displaystyle=\sqrt{\frac{3}{8}}(1-\mu_{k}^{2})P_{s\delta}^{(0)}({\mathbf{k}}),$		(18)
	$\displaystyle P_{+}({\mathbf{k}})$	$\displaystyle=\frac{3}{8}(1-\mu_{k}^{2})^{2}P_{ss}^{(0)}({\mathbf{k}})+\frac{1}{8}(1-\mu_{k}^{2})\{(1-\mu_{k})^{2}+(1+\mu_{k})^{2}\}P_{ss}^{(1)}({\mathbf{k}})+\frac{1}{32}\{(1-\mu_{k})^{4}+(1+\mu_{k})^{4}\}P_{ss}^{(2)}({\mathbf{k}}),$		(19)
	$\displaystyle P_{-}({\mathbf{k}})$	$\displaystyle=(1-\mu_{k}^{2})^{2}\left[\frac{3}{8}P_{ss}^{(0)}({\mathbf{k}})-\frac{1}{4}P_{ss}^{(1)}({\mathbf{k}})+\frac{1}{16}P_{ss}^{(2)}({\mathbf{k}})\right].$		(20)

The spectra, $P^{(0)}_{s\delta}$ and $P^{(\lambda)}_{ss}$ ($\lambda=0,1,2$), generally depend on $|{\mathbf{k}}|$ and $\mu_{k}$, e.g. due to the RSD effect [8], the primordial fossil effects [65], and also the super-survey tidal effect [70, 71]. Hence we use vector notation ${\mathbf{k}}$ in the argument of power spectra to keep generality of our discussion. We should stress that the $\mu_{k}$-dependent prefactors in front of the underlying tensor power spectra on the r.h.s., such as $(1-\mu_{k}^{2})$, are purely from geometrical effects due to the projection from $s_{ij}$ to $\gamma_{ij}$. In particular, the factorized form of the cross- and “minus”-power spectra become important when we construct FFT-based estimators of the power spectra of the projected tensor field and derive the associated Hankel transforms that give the 1D integral relations to the correlation functions.

The $E$- and $B$-mode auto-power spectra of the projected tensor field are explicitly given as

	$\displaystyle P_{EE}$	$\displaystyle=\frac{1}{2}\left(P_{+}+P_{-}\right)=\frac{3}{8}(1-\mu_{k}^{2})^{2}P_{ss}^{(0)}+\frac{1}{4}\mu_{k}^{2}(1-\mu_{k}^{2})P_{ss}^{(1)}+\frac{1}{16}(1+\mu_{k}^{2})^{2}P_{ss}^{(2)},$
	$\displaystyle P_{BB}$	$\displaystyle=\frac{1}{2}\left(P_{+}-P_{-}\right)=\frac{1}{4}(1-\mu_{k}^{2})P_{ss}^{(1)}+\frac{1}{2}\mu_{k}^{2}P_{ss}^{(2)}.$		(21)

Thus the $E$-mode power spectrum arises from the scalar (helicity-0), vector (helicity-1) and tensor (helicity-2) modes, while the $B$-mode spectrum is from the vector and tensor modes, as in the CMB polarization power spectra [72, 73, 74], the cosmic shear spectra [75, 76, 77], and the IA spectra [78, 31]. Note that Ref. [79] employed the Limber approximation to derive the $E/B$-mode angular power spectra of the IA shear, where only the Fourier modes with $\mu_{k}=0$ are considered, and arrived at conclusion that the $B$-mode angular power spectrum is only from the vector mode.

In this section we derive the Hankel transforms to relate the power spectra of the projected tensor field to the two-point correlation functions. We will later use the Hankel transform expressions to derive the formula of power spectrum including the effect of survey window convolution.

Let us begin with defining coordinate-independent correlation functions of the projected tensor field. For this purpose we introduce the projected tensor field at the position ${\mathbf{x}}$, defined with respect to line connecting ${\mathbf{x}}$ and ${\mathbf{x}}^{\prime}$, denoted as $\gamma_{r}({\mathbf{x}};{\mathbf{x}}^{\prime})$:

$\displaystyle\gamma_{r}({\mathbf{x}};{\mathbf{x}}^{\prime})\equiv\gamma_{+}({\mathbf{x}};{\mathbf{x}}^{\prime})+i\gamma_{\times}({\mathbf{x}};{\mathbf{x}}^{\prime})\equiv\gamma({\mathbf{x}})e^{-2i\phi_{\hat{\mathbf{r}},\hat{\mathbf{n}}}},$

(22)

with

$\displaystyle e^{2i\phi_{\hat{\mathbf{r}},\hat{\mathbf{n}}}}\equiv\frac{e_{ij}(\hat{\mathbf{n}})\hat{r}_{i}\hat{r}_{j}}{\mathcal{P}_{ij}(\hat{\mathbf{n}})\hat{r}_{i}\hat{r}_{j}},$

(23)

where ${\mathbf{r}}\equiv{\mathbf{x}}-{\mathbf{x}}^{\prime}$, the vector connecting ${\mathbf{x}}$ and ${\mathbf{x}}^{\prime}$. $\phi_{\hat{\mathbf{r}},\hat{\mathbf{n}}}$ in Eq. (23) is the phase factor rotating the tensor components on the plane perpendicular to the LOS direction and thus $\gamma_{+}(\gamma_{\times})$ is the tangential (cross) component with respect to the projected relative vector; ${\mathbf{r}}_{\perp i}\equiv\mathcal{P}_{ij}(\hat{\mathbf{n}}){\mathbf{r}}_{j}$³³3In the case of $(\hat{\mathbf{n}},\hat{\mathbf{e}}_{\theta},\hat{\mathbf{e}}_{\phi})=(\hat{\mathbf{x}}_{3},\hat{\mathbf{x}}_{1},\hat{\mathbf{x}}_{2})$ for instance, we reproduce the standard matrix representation as $\displaystyle\begin{pmatrix}\gamma_{+}\\ \gamma_{\times}\\ \end{pmatrix}=\begin{pmatrix}\cos{2\phi}&\sin{2\phi}\\ -\sin{2\phi}&\cos{2\phi}\\ \end{pmatrix}\begin{pmatrix}\gamma_{1}\\ \gamma_{2}\\ \end{pmatrix},$ (24) where $\phi\equiv\phi_{\hat{\mathbf{r}},\hat{\mathbf{x}}_{3}}$ is the angle between ${\mathbf{r}}_{\perp}$ and $\hat{\mathbf{x}}_{1}$.

Using the above fields, we can now define the coordinate-independent correlation functions as

	$\displaystyle\xi_{\gamma\delta}({\mathbf{r}})$	$\displaystyle\equiv\left\langle\gamma_{r}({\mathbf{x}};{\mathbf{x}}^{\prime})\delta({\mathbf{x}}^{\prime})\right\rangle=\left\langle\gamma({\mathbf{x}})\delta({\mathbf{x}}^{\prime})\right\rangle e^{-2i\phi_{\hat{\mathbf{r}},\hat{\mathbf{n}}}},$		(25)
	$\displaystyle\xi_{+}({\mathbf{r}})$	$\displaystyle\equiv\left\langle\gamma_{r}({\mathbf{x}};{\mathbf{x}}^{\prime})\gamma_{r}^{*}({\mathbf{x}}^{\prime};{\mathbf{x}})\right\rangle=\left\langle\gamma({\mathbf{x}})\gamma^{*}({\mathbf{x}}^{\prime})\right\rangle,$		(26)
	$\displaystyle\xi_{-}({\mathbf{r}})$	$\displaystyle\equiv\left\langle\gamma_{r}({\mathbf{x}};{\mathbf{x}}^{\prime})\gamma_{r}({\mathbf{x}}^{\prime};{\mathbf{x}})\right\rangle=\left\langle\gamma({\mathbf{x}})\gamma({\mathbf{x}}^{\prime})\right\rangle e^{-4i\phi_{\hat{\mathbf{r}},\hat{\mathbf{n}}}}.$		(27)

These correlation functions are free of choices of the coordinates $\hat{\mathbf{e}}_{\theta}$ and $\hat{\mathbf{e}}_{\phi}$; e.g. a rotation of the $\hat{\mathbf{e}}_{\theta}$ and $\hat{\mathbf{e}}_{\phi}$ coordinates does not change these correlation functions. The correlation functions generally depend on the distance between two points, $r\equiv|{\mathbf{r}}|$, and angle, $\mu_{r}\equiv\hat{\mathbf{r}}\cdot\hat{\mathbf{n}}$; $\xi({\mathbf{r}})=\xi(r,\mu_{r})$, e.g. due to the RSD effect.

Using the definitions of the power spectra in Eqs. (15)–(17) and those of the correlation functions in Eqs. (25)–(26), we can find relations to connect these as⁴⁴4The expressions of Eqs. (28)–(30) are in analogy with the 2D (angular) statistics. If we replace the 3D vectors, $({\mathbf{r}},{\mathbf{k}})$ with the 2D vectors, $(\bm{\theta},\bm{\ell})$ in Eqs. (28)–(30), the expressions just correspond to the well-known relation between the angular power spectrum and the angular 2D correlation function under the global plane-parallel approximation: e.g. $\displaystyle C_{\gamma\delta}(\ell)=\int_{\bm{\theta}}\xi^{2D}_{\gamma\delta}(\bm{\theta})e^{2i(\phi_{\bm{\theta}}-\phi_{\bm{\ell}})}e^{-i\bm{\ell}\cdot\bm{\theta}}=2\pi\int\theta\mathrm{d}\theta\xi^{2D}_{\gamma\delta}(\theta)J_{2}(\ell\theta),$ where $J_{m}$ is the Bessel function of order $m$.

	$\displaystyle P_{\gamma\delta}({\mathbf{k}})$	$\displaystyle=\int_{\mathbf{r}}\xi_{\gamma\delta}({\mathbf{r}})e^{2i(\phi_{\hat{\mathbf{r}},\hat{\mathbf{n}}}-\phi_{\hat{\mathbf{k}},\hat{\mathbf{n}}})}e^{-i{\mathbf{k}}\cdot{\mathbf{r}}},$		(28)
	$\displaystyle P_{+}({\mathbf{k}})$	$\displaystyle=\int_{\mathbf{r}}\xi_{+}({\mathbf{r}})e^{-i{\mathbf{k}}\cdot{\mathbf{r}}},$		(29)
	$\displaystyle P_{-}({\mathbf{k}})$	$\displaystyle=\int_{\mathbf{r}}\xi_{-}({\mathbf{r}})e^{4i(\phi_{\hat{\mathbf{r}},\hat{\mathbf{n}}}-\phi_{\hat{\mathbf{k}},\hat{\mathbf{n}}})}e^{-i{\mathbf{k}}\cdot{\mathbf{r}}}.$		(30)

We consider multipole moments of the power spectrum, which is defined by the angle average with respect to $\hat{\mathbf{k}}$, and derive the Hankel transform. The easiest case is the Hankel transform for the “plus” auto-power spectrum, $P_{+}({\mathbf{k}})$ because it is similar to that for the redshift-space power spectrum of the density field:

	$\displaystyle P^{(\ell)}_{+}(k)$	$\displaystyle\equiv(2\ell+1)\int_{\hat{\mathbf{k}}}~{}P_{+}({\mathbf{k}}){\cal L}_{\ell}(\mu_{k})$
		$\displaystyle=4\pi(-i)^{\ell}\int r^{2}\mathrm{d}r~{}\xi^{(\ell)}_{+}(r)j_{\ell}(kr)\left(=\mathcal{H}_{\ell}\left[\xi^{(\ell)}_{+}(r)\right](k)\right),$		(31)

where ${\cal L}_{\ell}(x)$ is the $\ell$-th order Legendre polynomial, and we have defined multipole moments of the power spectrum and correlation function as $P_{+}({\mathbf{k}})=\sum_{\ell}P^{(\ell)}_{+}(k)\mathcal{L}_{\ell}(\mu_{k})$ and $\xi_{+}({\mathbf{r}})=\sum_{\ell}\xi^{(\ell)}_{+}(r)\mathcal{L}_{\ell}(\mu_{r})$.

On the other hand, in the case of the cross-power spectrum $P_{\gamma\delta}$ and the “minus” auto-power spectrum $P_{-}$, there are additional phase factors and thus the angle integration is not trivial unlike the density auto or “plus” auto power spectra. To overcome this complexity, we use the associated Legendre polynomials, instead of ${\cal L}_{\ell}$, which are defined as

$\displaystyle\mathcal{L}^{m}_{L}(\mu)\equiv(-1)^{m}(1-\mu^{2})^{m/2}\frac{{\rm d}^{m}}{{\rm d}\mu^{m}}\mathcal{L}_{L}(\mu).$

(32)

The associated Legendre polynomials satisfy the following orthogonal relation and the integration identity (see Appendix E for the proof):

	$\displaystyle\int_{-1}^{1}\frac{\mathrm{d}\mu}{2}~{}{\cal L}_{L}^{m}(\mu){\cal L}_{L^{\prime}}^{m}(\mu)$	$\displaystyle=\frac{(L+m)!}{(2L+1)(L-m)!}\delta^{K}_{LL^{\prime}}$		(33)
	$\displaystyle\int_{\hat{\mathbf{k}}}e^{-im\phi_{\hat{\mathbf{k}},\hat{\mathbf{n}}}}\mathcal{L}^{m}_{L}(\hat{\mathbf{k}}\cdot\hat{\mathbf{n}})\mathcal{L}_{\ell^{\prime}}(\hat{\mathbf{k}}\cdot\hat{\mathbf{r}})$	$\displaystyle=\frac{1}{2L+1}\delta^{K}_{L\ell^{\prime}}\mathcal{L}^{m}_{L}(\hat{\mathbf{r}}\cdot\hat{\mathbf{n}})e^{-im\phi_{\hat{\mathbf{r}},\hat{\mathbf{n}}}},$		(34)

Throughout this paper we use the capital letter “$L$” to denote the order of associated Legendre polynomials (${\cal L}^{m}_{L})$, while we use the lower letter “$\ell$” to denote the order of Legendre polynomials (${\cal L}_{\ell}$). Let us first consider the cross power spectrum (Eq. 28). Introducing the multipole moments of the cross spectrum and the cross correlation, expanded in terms of the associated Legendre polynomials with $m=2$, as

	$\displaystyle P^{(L)}_{\gamma\delta}(k)$	$\displaystyle\equiv(2L+1)\frac{(L-2)!}{(L+2)!}\int_{\hat{\mathbf{k}}}P_{\gamma\delta}({\mathbf{k}}){\cal L}^{m=2}_{L}(\hat{\mathbf{k}}\cdot\hat{\mathbf{n}}),$		(35)
	$\displaystyle\xi_{\gamma\delta}({\mathbf{r}})$	$\displaystyle\equiv\sum_{L=2}\xi^{(L)}_{\gamma\delta}(r){\cal L}^{m=2}_{L}(\hat{\mathbf{r}}\cdot\hat{\mathbf{n}}),$		(36)

we can find the following the Hankel transform (see Appendix C):

$\displaystyle P^{(L)}_{\gamma\delta}(k)=4\pi(-i)^{L}\int r^{2}\mathrm{d}r~{}\xi^{(L)}_{\gamma\delta}(r)j_{L}(kr)\left(=\mathcal{H}_{L}\left[\xi^{(L)}_{\gamma\delta}(r)\right](k)\right),$

(37)

for integer $L$ with $L\geq 2$, and we introduced the factor $(L-2)!/(L+2)!$ in Eq. (35) accounting of the orthogonality relation of the associated Legendre polynomials (Eq. 33). Also note $P_{\gamma\delta}({\mathbf{k}})=\sum_{L=2}P^{(L)}_{\gamma\delta}(k){\cal L}^{m=2}_{L}(\hat{\mathbf{k}}\cdot\hat{\mathbf{n}})$. Thus the associated Legendre polynomials with order $m=2$ can nicely deal with the phase factor in Eq. (28)⁵⁵5We can expand the cross statistics in terms of the usual Legendre polynomials instead as in the case of the galaxy clustering; $P_{\gamma\delta}({\mathbf{k}})=\sum_{\ell=0}P^{(\ell)}_{\gamma\delta}(k){\cal L}_{\ell}(\hat{\mathbf{k}}\cdot\hat{\mathbf{n}})$ and $\xi_{\gamma\delta}({\mathbf{r}})=\sum_{\ell=0}\xi^{(\ell)}_{\gamma\delta}(r){\cal L}_{\ell}(\hat{\mathbf{r}}\cdot\hat{\mathbf{n}})$, however, the moments of the same order $\left(P^{(\ell)}_{\gamma\delta},\xi^{(\ell)}_{\gamma\delta}\right)$ are ${\it not}$ directly associated with each other by the Hankel and inverse Hankel transformations.. The cross power spectrum of projected tensor field is particularly important, e.g. for IA measurements, so this is one of the main results of this paper.

Similarly, using the associated Legendre polynomials with order $m=4$, we can find the Hankel transform for the minus auto-power spectrum of shear as

	$\displaystyle P^{(L)}_{-}(k)$	$\displaystyle\equiv(2L+1)\frac{(L-4)!}{(L+4)!}\int_{\hat{\mathbf{k}}}~{}P_{-}({\mathbf{k}}){\cal L}^{m=4}_{L}(\hat{\mathbf{k}}\cdot\hat{\mathbf{n}})$
		$\displaystyle=4\pi(-i)^{L}\int r^{2}\mathrm{d}r~{}\xi^{(L)}_{-}(r)j_{L}(kr)\left(=\mathcal{H}_{L}\left[\xi^{(L)}_{-}(r)\right](k)\right),$		(38)

for integer $L$ with $L\geq 4$, and note $P_{-}({\mathbf{k}})=\sum_{L=4}~{}P^{(L)}_{-}(k){\cal L}^{m=4}_{L}(\hat{\mathbf{k}}\cdot\hat{\mathbf{n}})$.

Before proceeding we would like to notice the following things. One might think that Eqs. (37) and (38) are not a complete expansion of the underlying power spectra, $P_{\gamma\delta}$ and $P_{-}$, at first glace, because a set of the associated Legendre polynomials $\left\{\mathcal{L}^{m}_{L}(\mu)\right\}$ with fixed $m(\neq 0)$ is not a complete basis. An arbitrary function $f(\mu)$ cannot be generally expanded by a series of $\mathcal{L}^{m}_{L}(\mu)$; e.g., $f(\mu)=1$ cannot be expanded by $\{{\cal L}^{m=2}_{L}\}$, because the lowest order ${\cal L}^{m=2}_{L}$ is ${\cal L}^{m=2}_{L=2}(\mu)=3(1-\mu^{2})$. However, this is not true. As can be found from Eqs. (18) and (20), the power spectra $P_{\gamma\delta}$ and $P_{-}$ are given by the underlying 3D power spectra of the tensor fields, $P^{(0)}_{s\delta}$ and $P_{ss}^{(\lambda)}$, with geometrical prefactors $(1-\mu_{k}^{2})$ and $(1-\mu_{k}^{2})^{2}$, respectively: $P_{\gamma\delta}\propto(1-\mu_{k}^{2})P^{(0)}_{\delta s}$ and $P_{-}\propto(1-\mu_{k}^{2})^{2}\left[P_{ss}^{(0)}+\cdots\right]$. Recalling the definitions $P_{\gamma\delta}({\mathbf{k}})=\sum_{L=2}P^{(L)}_{\gamma\delta}(k){\cal L}^{m=2}_{L}(\hat{\mathbf{k}}\cdot\hat{\mathbf{n}})$ and $P_{-}({\mathbf{k}})=\sum_{L=4}P^{(L)}_{-}(k){\cal L}^{m=4}_{L}(\hat{\mathbf{k}}\cdot\hat{\mathbf{n}})$, we can find that a set of $\left\{{\cal L}^{m=2}_{L}(\mu_{k})/(1-\mu_{k}^{2})\right\}$ and $\left\{{\cal L}^{m=4}_{L}(\mu_{k})/(1-\mu_{k}^{2})^{2}\right\}$ includes a complete set of $\mu$-polynomials, $\left\{\mu_{k}^{0},\mu_{k}^{1},\mu_{k}^{2},\cdots\right\}$. Hence a set of the multipole moments, $P_{\gamma\delta}^{(L)}(k)$ or $P_{-}^{(L)}(k)$, carries the full information on the underlying power spectra, $P_{s\delta}^{(0)}({\mathbf{k}})$ and $P_{ss}^{(\lambda)}({\mathbf{k}})$.