# The integrated perturbation theory for cosmological tensor fields IV: Full-
sky formulation

Takahiko Matsubara<EMAIL_ADDRESS>Institute of Particle and Nuclear
Studies, High Energy Accelerator Research Organization (KEK), Oho 1-1, Tsukuba
305-0801, Japan  The Graduate Institute for Advanced Studies, SOKENDAI,
Tsukuba 305-0801, Japan

###### Abstract

In Papers I–III [1, 2, 3], we use the flat-sky and distant-observer
approximations to develop a formalism with which the correlation statistics of
cosmological tensor fields are calculated by the nonlinear perturbation
theory, generalizing the integrated perturbation theory for scalar fields. In
this work, the formalism is extended to include the full-sky and wide-angle
effects in evaluating the power spectra and correlation functions of
cosmological tensor fields of any ranks. With the newly developed formalism,
one can evaluate the nonlinear power spectra and correlation functions to
arbitrary higher orders in principle. After describing the general formalism,
we explicitly derive and give analytic results of the lowest-order linear
theory for an illustrative purpose in this paper. The derived linear formulas
with full-sky and wide-angle effects are numerically compared with the
previous formulas with flat-sky and distant-observer limits in a simple model
of tensor bias.

##  I Introduction

The large-scale structure of the Universe is one of the most invaluable
sources of information about the nature of our Universe itself. The spatial
distributions of galaxies have been playing important roles to unveil the
structure of the Universe, where three-dimensional positions of galaxies are
measured by various kinds of modern galaxy surveys (e.g., Refs. [4, 5, 6, 7,
8, 9, 10]). In order that maximal information can be extracted from such
modern surveys of large-scale structure, we can use not only the positions of
objects, but also various properties of objects, such as angular momentum [11,
12, 13, 14] and shape moments [15, 16, 17, 18], etc. The correlation
statistics of intrinsic alignment of galaxies, characterized by second moments
of shapes, have been extensively studied in recent years, and they provide
complementary information of cosmology (for review, see, e.g., Ref. [19] and
references therein).

Theoretical modeling of the large-scale structure of the Universe has a long
history. Among others, the perturbation theory of density field plays an
important role in analyzing the correlation statistics on large scales, where
the nonlinearity in gravitational evolutions of density field is considered to
be weak enough [20, 21]. As galaxy surveys become larger and larger, methods
using perturbation theory are expected to become increasingly useful.
Recently, methods of perturbation theory have been actively applied to the
correlation statistics of intrinsic alignment by many authors. The intrinsic
alignment is usually characterized by the second moments of galaxy shapes
[22], and the rank-2 tensor is attached to each galaxy. One can also consider
higher moments of galaxy shapes as well [18]. When angular moments of galaxies
can be measured, a rank-1 tensor (i.e., a vector) is attached to each galaxy.

With the above examples in mind, we develop a generalized formalism to predict
correlation statistics of tensor fields in previous Papers I–III [1, 2, 3] in
the series. The formalism is a generalization of the integrated perturbation
theory (iPT) [23, 24, 25, 26], which is originally formulated to describe
gravitational evolutions of scalar-type biased fields. We generalize the
theory so that tensor-type biased fields can also be described by nonlinear
perturbation theory, using irreducible representations of tensor fields
throughout. In Paper I, the basic formulations of the iPT for tensor fields
are described and some analytic results are derived in lowest-order
approximations. In Paper II, methods of calculating higher-order corrections
of nonlinear evolutions are developed in the formalism. In Paper III, the
previous formalism are extended to predict statistics of projected tensors,
because observed tensors usually correspond to two-dimensional tensors
projected onto the sky from originally three-dimensionally tensors.

In the developments of previous Papers I–III, we derive most of the results
consistently assuming flat-sky and distant-observer approximations. In this
Paper IV, we consider the corresponding problem without assuming the last
approximations, including full-sky and wide-angle effects in power spectra and
correlation functions for tensor fields in the large-scale structure of the
Universe. Generalizing the method of rotationally invariant functions
developed in previous Papers I–III, we derive various formulas which are
generally useful for full-sky analyses of cosmological tensor fields, which is
independent of the dynamics using the perturbation theory.

The full-sky and wide-angle effects in power spectra and correlation functions
of density fields with redshift-space distortions have been extensively
investigated mostly using the linear perturbation theory (for pioneering work,
see Refs. [27, 28, 29]). In this paper, we develop a systematic way of
evaluating the full-sky and wide-angle effects in correlation statistics of
tensor fields. Applying the iPT, we explicitly derive analytic expressions in
linear theory, and extensions of the results to include nonlinear corrections
using the higher-order perturbation theory are straightforward by using the
present formalism. The expressions of the correlation functions are natural
extensions of the geometric method of evaluating the linear correlation
function in redshift space [30, 31, 32, 33], which is recently generalized
also to derive linear correlation functions of galaxy ellipticity [34]. The
formalism of this paper generalizes these previous studies for the correlation
functions, and provides a way to evaluate nonlinear correlation functions of
tensor fields with arbitrary ranks in redshift space by the iPT, and also to
evaluate full-sky effects in the power spectrum.

This paper is organized as follows. In Sec. II, basic ingredients and
notations are introduced to describe the full-sky and wide-angle effects of
the power spectrum and correlation function of tensor fields, and rotationally
invariant functions which play central roles in this paper are introduced.
Relations between the full-sky and wide-angle formalism to the flat-sky and
distant-observer formalism are also derived in this section. In Sec. III, we
explain how one can apply the iPT to calculate the invariant functions in the
general formalism, and explicit results of applying the linear theory of
gravitational evolution are analytically derived. In Sec. IV, we consider the
tensor fields projected onto the sky, and corresponding formulas for E/B power
spectra and $+/\times$ correlation functions with full-sky and wide-angle
effects are derived. Analytic formulas of the linear theory are explicitly
derived, and they are numerically compared with the corresponding formulas of
the previous results in the flat-sky and distant-observer limits. The
conclusions are summarized in Sec. V. In Appendix A, the formulas of large-
scale limits of the power spectra and correlation functions are explicitly
given, where the scale-dependencies of the bias functions of tensor fields are
neglected,

##  II Irreducible tensor fields in full-sky

###  II.1 Spherical bases in Cartesian and spherical coordinates

In this section, we first review the concept and conventions of spherical
basis in Cartesian coordinates introduced in Paper I, and then introduce the
corresponding basis in spherical coordinates to describe the tensors in the
full-sky formulation of this paper. As explained in Paper I, any symmetric
tensor of rank-$l$, $T_{i_{1}i_{2}\cdots i_{l}}$, can be decomposed into
traceless parts, $T^{(s)}_{i_{i}\cdots i_{s}}$, where
$s=l,l-2,\ldots,(0\mathrm{\ or\ }1)$. A traceless part of the tensor
components forms an irreducible representation of the rotation group SO(3). In
Paper I, a traceless tensor of rank-$s$ is decomposed into the spherical basis
as

$T^{(s)}_{i_{1}i_{2}\cdots
i_{s}}=A_{s}\sum_{\sigma=-s}^{s}T_{s\sigma}\mathsf{Y}^{(\sigma)}_{i_{1}i_{2}\cdots
i_{s}},$ (1)

where

$A_{s}\equiv\sqrt{\frac{s!}{(2s-1)!!}}$ (2)

is a normalization factor of our convention, and $T_{s\sigma}$ is the
irreducible representation of the tensor, and
$\mathsf{Y}^{(\sigma)}_{i_{1}i_{2}\cdots i_{s}}$ with $\sigma=0,\pm 1,\pm
2,\cdots,\pm s$ is the spherical basis of the Cartesian coordinates.111In
Paper I, the indices $s$ and $\sigma$ are denoted by $l$ and $m$ for the
irreducible tensors. In this paper, the letters like $s$ and $\sigma$ are used
instead for indices of irreducible tensors. The letters like $l$, $m$ are
saved to indices of multipoles of the angular positions $(\theta,\phi)$ of the
field. The spherical basis of the rank-$s$ tensor is constructed from a set of
vector basis in Cartesian coordinates,

$\mathbf{e}^{0}=\hat{\mathbf{e}}_{3},\quad\mathbf{e}^{\pm}=\mp\frac{\hat{\mathbf{e}}_{1}\mp
i\hat{\mathbf{e}}_{2}}{\sqrt{2}},$ (3)

where $\hat{\mathbf{e}}_{i}$ with $i=1,2,3$ are the unit base vectors of the
Cartesian coordinates $(x_{1},x_{2},x_{3})$. For example, the spherical bases
of the rank-$0,1$ tensors are respectively given by

$\mathsf{Y}^{(0)}=1,\quad\mathsf{Y}^{(\sigma)}_{i}={\mathrm{e}^{\sigma}}_{i},$
(4)

and those of the rank-$2$ tensors are given by

$\displaystyle\mathsf{Y}^{(0)}_{ij}=\sqrt{\frac{3}{2}}\left({\mathrm{e}^{0}}_{i}\,{\mathrm{e}^{0}}_{j}-\frac{1}{3}\delta_{ij}\right),$
(5) $\displaystyle\mathsf{Y}^{(\pm
1)}_{ij}=\sqrt{2}\,{\mathrm{e}^{0}}_{(i}\,{\mathrm{e}^{\pm}}_{j)},$ (6)
$\displaystyle\mathsf{Y}^{(\pm
2)}_{ij}={\mathrm{e}^{\pm}}_{i}\,{\mathrm{e}^{\pm}}_{j},$ (7)

where
${\mathrm{e}^{\sigma}}_{i}=[\mathbf{e}^{\sigma}]_{i}=\mathbf{e}^{\sigma}\cdot\hat{\mathbf{e}}_{i}$
with $\sigma=0,\pm$ are Cartesian components of the spherical basis, and round
brackets in the indices of the right-hand side (RHS) of Eq. (6) indicate
symmetrization with respect to the indices inside the brackets, e.g.,
$a_{(i}b_{j)}=(a_{i}b_{j}+a_{j}b_{i})/2$.

The spherical bases for rank-3 and 4 tensors are explicitly given in Appendix
A of Paper I, and how to systematically construct spherical bases for
arbitrary higher-order tensors is also explained there. A relation of the
spherical basis to the spherical harmonics is quite useful. For an arbitrary
unit vector $\bm{n}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$, the
contraction of the indices in the spherical basis with those in unit vectors
is given by

$\mathsf{Y}^{(\sigma)}_{i_{1}i_{2}\cdots i_{s}}n_{i_{1}}n_{i_{2}}\cdots
n_{i_{s}}=A_{s}\sqrt{\frac{4\pi}{2s+1}}\,Y_{s\sigma}^{*}(\theta,\phi),$ (8)

where $Y_{lm}^{*}(\theta,\phi)$ (with $l=s$ and $m=\sigma$) is the complex
conjugate of the spherical harmonics222The Condon-Shortley phase is included
in our definition of the spherical harmonics $Y_{lm}(\theta,\phi)$ (see also
Paper I)., $Y_{lm}(\theta,\phi)$. Using the above equation, explicit forms of
higher-rank spherical bases $\mathsf{Y}^{(\sigma)}_{i_{1}i_{2}\cdots i_{s}}$
can be uniquely determined (Appendix A of Paper I).

The spherical basis $\mathsf{Y}^{(\sigma)}_{i_{1}\cdots i_{s}}$ satisfies an
orthonormality relation,

$\mathsf{Y}^{(\sigma)*}_{i_{1}\cdots
i_{s}}\mathsf{Y}^{(\sigma^{\prime})}_{i_{1}\cdots
i_{s}}=\delta_{\sigma}^{\sigma^{\prime}},$ (9)

where $\delta_{\sigma}^{\sigma^{\prime}}$ is the Kronecker’s delta symbol, and
the indices $i_{1},\ldots,i_{l}$ are summed over with omitting the summation
symbol $\sum_{i_{1},\ldots,i_{l}}$, as we adopt the Einstein summation
convention for repeated indices of Cartesian coordinates. From the
construction of the spherical basis in Paper I, the complex conjugate of the
basis is given by

$\mathsf{Y}^{(\sigma)*}_{i_{1}\cdots
i_{s}}=(-1)^{\sigma}\mathsf{Y}^{(-\sigma)}_{i_{1}\cdots
i_{s}}=\sum_{\sigma^{\prime}}g^{(s)}_{\sigma\sigma^{\prime}}\mathsf{Y}^{(\sigma^{\prime})}_{i_{1}\cdots
i_{s}},$ (10)

where, in the last equation, the spherical metric is defined by

$g^{(s)}_{\sigma\sigma^{\prime}}\equiv(-1)^{\sigma}\delta_{\sigma,-\sigma^{\prime}},$
(11)

for $\sigma,\sigma^{\prime}=0,\pm 1,\ldots,\pm s$. The spherical metric of Eq.
(11) is considered to lower the spherical index $\sigma$ of the spherical
basis. The inverse metric $g_{(s)}^{\sigma\sigma^{\prime}}$ which satisfy
$\sum_{\sigma^{\prime\prime}}g_{(s)}^{\sigma\sigma^{\prime\prime}}g^{(s)}_{\sigma^{\prime\prime}\sigma^{\prime}}=\delta_{\sigma^{\prime}}^{\sigma}$
has the same matrix elements as the RHS of Eq. (11), i.e.,
$g_{(s)}^{\sigma\sigma^{\prime}}=g^{(s)}_{\sigma\sigma^{\prime}}$. With the
above notations, The orthonormality relation of Eq. (9) is equivalently
represented by

$\mathsf{Y}^{(\sigma)}_{i_{1}\cdots
i_{s}}\mathsf{Y}^{(\sigma^{\prime})}_{i_{1}\cdots
i_{s}}=g_{(s)}^{\sigma\sigma^{\prime}},\quad\mathsf{Y}^{(\sigma)*}_{i_{1}\cdots
i_{s}}\mathsf{Y}^{(\sigma^{\prime})*}_{i_{1}\cdots
i_{s}}=g^{(s)}_{\sigma\sigma^{\prime}}.$ (12)

The decomposition of Eq. (1) for the traceless tensor is unique, because the
relation is inverted by the orthonormality relation of Eq. (9), and we have

$T_{s\sigma}=\frac{1}{A_{s}}T^{(s)}_{i_{1}i_{2}\cdots
i_{s}}\mathsf{Y}^{(\sigma)*}_{i_{1}i_{2}\cdots i_{s}}.$ (13)

On one hand, the spherical basis defined above is suitable for working in
Cartesian coordinates system. In the context of observational cosmology, the
statistics of cosmological field in the distant-observer approximation are
described in Cartesian coordinates systems, where the lines of sight are
approximately considered to have a fixed direction throughout the space. On
the other hand, when the distant-observer approximation is not sufficiently
valid and wide-angle effects are not negligible, it is natural to work in a
spherical coordinates system in which the coordinates origin is located at the
position of the observer. The line-of-sight direction matches with the radial
direction of the spherical coordinates.

As explained in Paper III, the spherical basis of a tensor in a spherical
coordinates system can be constructed by a set of vector basis,

$\tilde{\mathbf{e}}^{0}=\hat{\mathbf{e}}_{r},\quad\tilde{\mathbf{e}}^{\pm}=\mp\frac{\hat{\mathbf{e}}_{\theta}\mp
i\hat{\mathbf{e}}_{\phi}}{\sqrt{2}},$ (14)

where $\hat{\mathbf{e}}_{r},\hat{\mathbf{e}}_{\theta},\hat{\mathbf{e}}_{\phi}$
are the unit base vectors of the spherical coordinates $(r,\theta,\phi)$.
Contrary to the Cartesian case where the base vectors have fixed directions
throughout the space, the above base vectors do not have fixed directions and
depend on the angular coordinates $(\theta,\phi)$ of each position. The
spherical basis $\tilde{\mathsf{Y}}^{(\sigma)}_{i_{1}i_{2}\cdots
i_{s}}(\theta,\phi)$ of rank-$s$ tensors in spherical coordinates is defined
in the same way for that in the Cartesian coordinates as given in Eqs.
(4)–(7), replacing the basis vectors $\mathbf{e}^{\sigma}$ with
$\tilde{\mathbf{e}}^{\sigma}$.

Defining a $3\times 3$ orthogonal matrix

$S(\theta,\phi)=\begin{pmatrix}\cos\theta\,\cos\phi&\cos\theta\,\sin\phi&-\sin\theta\\\
-\sin\phi&\cos\phi&0\\\
\sin\theta\,\cos\phi&\sin\theta\,\sin\phi&\cos\theta\end{pmatrix},$ (15)

the orthonormal bases in the Cartesian and spherical coordinates systems are
related by

$\begin{pmatrix}\hat{\mathbf{e}}_{\theta}(\theta,\phi)\\\
\hat{\mathbf{e}}_{\phi}(\theta,\phi)\\\
\hat{\mathbf{e}}_{r}(\theta,\phi)\end{pmatrix}=S(\theta,\phi)\begin{pmatrix}\hat{\mathbf{e}}_{1}\\\
\hat{\mathbf{e}}_{2}\\\ \hat{\mathbf{e}}_{3}\end{pmatrix}.$ (16)

The components of the spherical vectors, Eqs. (3) and (14) are related by

${\tilde{\mathrm{e}}^{\sigma}\phantom{}}_{j}(\theta,\phi)={\mathrm{e}^{\sigma}}_{i}\,S_{ij}(\theta,\phi),$
(17)

where
$\tilde{\mathrm{e}}^{\sigma}_{\phantom{\sigma}i}=[\tilde{\mathbf{e}}^{\sigma}]_{i}=\tilde{\mathbf{e}}^{\sigma}\cdot\hat{\mathbf{e}}_{i}$
with $\sigma=0,\pm$ are Cartesian components of Eq. (14), and
$S_{ij}(\theta,\phi)$ are matrix elements of Eq. (15). Accordingly, the
spherical bases of traceless tensors in Cartesian and spherical coordinates
are related by

$\tilde{\mathsf{Y}}^{(\sigma)}_{j_{1}\cdots
j_{s}}(\theta,\phi)={\mathsf{Y}}^{(\sigma)}_{i_{1}\cdots
i_{s}}S_{i_{1}j_{1}}(\theta,\phi)\cdots S_{i_{s}j_{s}}(\theta,\phi).$ (18)

As the matrix $S$ of Eq. (15) is an orthogonal matrix,
$S^{-1}=S^{\mathrm{T}}$, the orthonormality relations of spherical basis in
Cartesian coordinates, Eqs. (9) and (12) are translated to those in spherical
coordinates,

$\displaystyle\tilde{\mathsf{Y}}^{(\sigma)*}_{i_{1}\cdots
i_{s}}(\theta,\phi)\tilde{\mathsf{Y}}^{(\sigma^{\prime})}_{i_{1}\cdots
i_{s}}(\theta,\phi)$ $\displaystyle=\delta_{\sigma}^{\sigma^{\prime}},$ (19)
$\displaystyle\tilde{\mathsf{Y}}^{(\sigma)}_{i_{1}\cdots
i_{s}}(\theta,\phi)\tilde{\mathsf{Y}}^{(\sigma^{\prime})}_{i_{1}\cdots
i_{s}}(\theta,\phi)$ $\displaystyle=g_{(s)}^{\sigma\sigma^{\prime}},$ (20)
$\displaystyle\tilde{\mathsf{Y}}^{(\sigma)*}_{i_{1}\cdots
i_{s}}(\theta,\phi)\tilde{\mathsf{Y}}^{(\sigma^{\prime})*}_{i_{1}\cdots
i_{s}}(\theta,\phi)$ $\displaystyle=g^{(s)}_{\sigma\sigma^{\prime}}.$ (21)

Similarly to Eqs. (1) and (13), traceless tensors $T^{(s)}_{i_{1}i_{2}\cdots
i_{s}}$ of rank $s$ are uniquely decomposed into irreducible tensors
$\tilde{T}_{s\sigma}$ in spherical coordinates, depending on angular positions
$\theta,\phi$.

The relation of the basis vectors, Eq. (16), is considered as a local rotation
of the coordinate basis at a position of $\bm{r}$. Applying the rotational
transformation of the spherical basis derived in a previous paper,333To derive
the following equation, we substitute
$R(\phi,\theta,0)=S^{\mathrm{T}}(\phi,\theta)=S^{-1}(\phi,\theta)$ in Eqs.
(21), (24) and (25) of Paper I. the following identity is shown to hold:

$\tilde{\mathsf{Y}}^{(\sigma)}_{i_{1}i_{2}\cdots
i_{s}}(\theta,\phi)=\sum_{\sigma^{\prime}}\mathsf{Y}^{(\sigma^{\prime})}_{i_{1}i_{2}\cdots
i_{s}}D^{\sigma^{\prime}*}_{(s)\sigma}(\phi,\theta,0),$ (22)

where $D^{m^{\prime}}_{(l)m}(\alpha,\beta,\gamma)$ is the Wigner’s rotation
matrix [35, 36] for the rotation with Euler’s angles $(\alpha,\beta,\gamma)$.
The Wigner’s matrix is unitary, and the Wigner’s matrix of the inverse
rotation is given by

$\displaystyle D^{m^{\prime}}_{(l)m}(-\gamma,-\beta,-\alpha)$
$\displaystyle=D^{m*}_{(l)m^{\prime}}(\alpha,\beta,\gamma)$ (23)
$\displaystyle=(-1)^{m-m^{\prime}}D^{-m}_{(l)-m^{\prime}}(\alpha,\beta,\gamma).$
(24)

###  II.2 Irreducible tensor fields in Cartesian coordinates

We generally consider a tensor field $F_{Xi_{1}\cdots i_{s}}(\bm{r})$ as in
previous Papers I–III, where the label $X$ denotes the class of objects in a
given sample, such as a certain type of galaxies etc. The traceless part of
the tensor, $F^{(s)}_{Xi_{1}\cdots i_{s}}(\bm{r})$, can be decomposed into
spherical basis both in Cartesian and spherical coordinates as

$\displaystyle F^{(s)}_{Xi_{1}i_{2}\cdots i_{s}}(\bm{r})$
$\displaystyle=A_{s}\sum_{\sigma}F_{Xs\sigma}(\bm{r})\mathsf{Y}^{(\sigma)}_{i_{1}i_{2}\cdots
i_{s}}$ (25)
$\displaystyle=A_{s}\sum_{\sigma}\tilde{F}_{Xs\sigma}(\bm{r})\tilde{\mathsf{Y}}^{(\sigma)}_{i_{1}i_{2}\cdots
i_{s}}(\theta,\phi),$ (26)

where $F_{Xs\sigma}(\bm{r})$ and $\tilde{F}_{Xs\sigma}(\bm{r})$ are components
of the irreducible tensors in Cartesian and spherical coordinates,
respectively, and $(\theta,\phi)$ are angular coordinates of the position
$\bm{r}$. Using the orthonormality of the spherical bases, Eqs. (9) and (19),
the irreducible tensors are given by

$\displaystyle F_{Xs\sigma}(\bm{r})$
$\displaystyle=\frac{1}{A_{s}}F^{(s)}_{Xi_{1}i_{2}\cdots
i_{s}}(\bm{r})\mathsf{Y}^{(\sigma)*}_{i_{1}i_{2}\cdots i_{s}},$ (27)
$\displaystyle\tilde{F}_{Xs\sigma}(\bm{r})$
$\displaystyle=\frac{1}{A_{s}}F^{(s)}_{Xi_{1}i_{2}\cdots
i_{s}}(\bm{r})\tilde{\mathsf{Y}}^{(\sigma)*}_{i_{1}i_{2}\cdots
i_{s}}(\theta,\phi).$ (28)

Combining Eqs. (22)–(28), the two irreducible tensors are related by

$\tilde{F}_{Xs\sigma}(\bm{r})=\sum_{\sigma^{\prime}}{F}_{Xs\sigma^{\prime}}(\bm{r})D^{\sigma^{\prime}}_{(s)\sigma}(\phi,\theta,0).$
(29)

We consider a passive rotation of the Cartesian coordinates system around the
origin,

$\hat{\mathbf{e}}_{i}\xrightarrow{\mathbb{R}}\hat{\mathbf{e}}_{i}^{\prime}=\hat{\mathbf{e}}_{j}R_{ji},$
(30)

where $R_{ji}=\hat{\mathbf{e}}_{j}\cdot\hat{\mathbf{e}}_{i}^{\prime}$ are
components of a real orthogonal matrix which satisfies $R^{\mathrm{T}}R=I$ and
$I$ is the $3\times 3$ unit matrix. Under the above rotation, the Cartesian
tensor field transforms as

$F^{(s)}_{Xi_{1}i_{2}\cdots
i_{s}}(\bm{r})\xrightarrow{\mathbb{R}}F^{\prime(s)}_{Xi_{1}i_{2}\cdots
i_{s}}(\bm{r}^{\prime})\\\ =F^{(s)}_{Xj_{1}j_{2}\cdots
j_{s}}(\bm{r})R_{j_{1}i_{1}}R_{j_{2}i_{2}}\cdots R_{j_{s}i_{s}},$ (31)

and the spherical tensor in Cartesian coordinates transforms as (Paper I),

$F_{Xs\sigma}(\bm{r})\xrightarrow{\mathbb{R}}F^{\prime}_{Xs\sigma}(\bm{r}^{\prime})=\sum_{\sigma^{\prime}}F_{Xs\sigma^{\prime}}(\bm{r})D^{\sigma^{\prime}}_{(s)\sigma}(R),$
(32)

where $D^{\sigma^{\prime}}_{(s)\sigma}(R)$ is the Wigner’s rotation matrix
corresponding the the rotation $R$.

Due to the property of the rotational transformation above, it is natural to
expand the angular dependence of $F_{Xs\sigma}(\bm{r})$ by the spherical
harmonics as

$F_{Xs\sigma}(\bm{r})=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\sqrt{\frac{4\pi}{2l+1}}\,F_{Xs\sigma}^{lm}(r)Y_{lm}(\theta,\phi),$
(33)

where the functions $\sqrt{4\pi/(2l+1)}\,Y_{lm}(\theta,\phi)$ are spherical
harmonics of Racah’s normalization, which are extensively used in Papers I–III
and are denoted as $C_{lm}(\theta,\phi)$ there. At a distance $r=|\bm{r}|$
from the observer, the function $F^{lm}_{Xs\sigma}(r)$ in the above definition
corresponds to the expansion coefficients with the latter normalization of the
spherical harmonics. From the orthonormality of the spherical harmonics, the
coefficients of the expansion is inversely given by

$F_{Xs\sigma}^{lm}(r)=\sqrt{\frac{2l+1}{4\pi}}\int\sin\theta\,d\theta\,d\phi\,F_{Xs\sigma}(\bm{r})Y_{lm}^{*}(\theta,\phi).$
(34)

Under the rotation of the coordinates system around the origin, the spherical
harmonics transform as

$Y_{lm}(\theta,\phi)\xrightarrow{\mathbb{R}}Y_{lm}(\theta^{\prime},\phi^{\prime})=\sum_{m^{\prime}}Y_{lm^{\prime}}(\theta,\phi)D_{(l)m}^{m^{\prime}}(R),$
(35)

and thus the expansion coefficients of Eq. (33) transform as

$F_{Xs\sigma}^{lm}(r)\xrightarrow{\mathbb{R}}F_{Xs\sigma}^{\prime
lm}(r)=\sum_{\sigma^{\prime},m^{\prime}}F_{Xs\sigma^{\prime}}^{lm^{\prime}}(r)D^{\sigma^{\prime}}_{(s)\sigma}(R)D^{m^{\prime}*}_{(l)m}(R).$
(36)

The complex conjugate of the Wigner rotation matrix satisfies

$D^{m^{\prime}*}_{(l)m}(R)=D^{m}_{(l)m^{\prime}}(R^{-1})=\sum_{m_{1},m_{1}^{\prime}}g^{(l)}_{m^{\prime}m_{1}^{\prime}}g_{(l)}^{mm_{1}}D^{m_{1}^{\prime}}_{(l)m_{1}}(R).$
(37)

Below, we consider other symmetries than that of rotation. Assuming the
original tensor field $F^{(s)}_{Xi_{1}i_{2}\cdots i_{s}}(\bm{r})$ is real, the
complex conjugates of Eq. (27) is given by

$F_{Xs\sigma}^{*}(\bm{r})=(-1)^{\sigma}F_{Xs,-\sigma}(\bm{r})=\sum_{\sigma^{\prime}}g_{(s)}^{\sigma\sigma^{\prime}}F_{Xs\sigma^{\prime}}(\bm{r}),$
(38)

and the complex conjugate of the multipole coefficients, Eq. (34), is given by

$F^{lm\,*}_{Xs\sigma}(r)=(-1)^{m+\sigma}F^{l,-m}_{Xs,-\sigma}(r)=\sum_{m^{\prime},\sigma^{\prime}}g^{(l)}_{mm^{\prime}}g_{(s)}^{\sigma\sigma^{\prime}}F^{lm^{\prime}}_{Xs\sigma^{\prime}}(r).$
(39)

The parity transformation of the original tensor field is given by

$F^{(s)}_{Xi_{1}\cdots
i_{s}}(\bm{r})\xrightarrow{\mathbb{P}}F^{\prime(s)}_{Xi_{1}\cdots
i_{s}}(\bm{r})=(-1)^{p_{X}+s}F^{(s)}_{Xi_{1}\cdots i_{s}}(-\bm{r}),$ (40)

where $p_{X}=0$ for ordinary tensors and $p_{X}=1$ for pseudotensors.
Correspondingly, the parity transformation of irreducible tensors, Eq. (27),
is given by

$F_{Xs\sigma}(\bm{r})\xrightarrow{\mathbb{P}}F_{Xs\sigma}^{\prime}(\bm{r})=(-1)^{p_{X}+s}F_{Xs\sigma}(-\bm{r}),$
(41)

and that of multiple coefficients, Eq. (34), is given by

$F^{lm}_{Xs\sigma}(r)\xrightarrow{\mathbb{P}}F^{\prime
lm}_{Xs\sigma}(r)=(-1)^{p_{X}+s+l}F^{lm}_{Xs\sigma}(r).$ (42)

###  II.3 Irreducible tensor fields in spherical coordinates

The radial base vector $\hat{\mathbf{e}}_{r}$ in spherical coordinates at a
physical position does not change under the rotation, while the angular base
vectors $(\hat{\mathbf{e}}_{\theta},\hat{\mathbf{e}}_{\phi})$ rotates
depending on the rotation $R$. We denote the last rotation angle by $\gamma$,
which is somehow a complicated function of $R_{ij}$ and $(\theta,\phi)$. Thus
the base vectors in spherical coordinates at a given direction transform as

$\begin{pmatrix}\hat{\mathbf{e}}_{\theta}(\theta,\phi)\\\
\hat{\mathbf{e}}_{\phi}(\theta,\phi)\\\
\hat{\mathbf{e}}_{r}(\theta,\phi)\end{pmatrix}\xrightarrow{\mathbb{R}}\begin{pmatrix}\hat{\mathbf{e}}_{\theta}^{\prime}(\theta^{\prime},\phi^{\prime})\\\
\hat{\mathbf{e}}_{\phi}^{\prime}(\theta^{\prime},\phi^{\prime})\\\
\hat{\mathbf{e}}_{r}^{\prime}(\theta^{\prime},\phi^{\prime})\end{pmatrix}=\begin{pmatrix}\cos\gamma&\sin\gamma&0\\\
-\sin\gamma&\cos\gamma&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}\hat{\mathbf{e}}_{\theta}(\theta,\phi)\\\
\hat{\mathbf{e}}_{\phi}(\theta,\phi)\\\
\hat{\mathbf{e}}_{r}(\theta,\phi)\end{pmatrix},$ (43)

where $(\theta^{\prime},\phi^{\prime})$ are transformed angular coordinates in
rotated coordinates system. In terms of the spherical basis of Eq. (14), the
above transformation is equivalent to

$\begin{cases}\tilde{\mathbf{e}}^{0}(\theta,\phi)\xrightarrow{\mathbb{R}}\tilde{\mathbf{e}}^{\prime
0}(\theta^{\prime},\phi^{\prime})=\tilde{\mathbf{e}}^{0}(\theta,\phi),\\\
\tilde{\mathbf{e}}^{\pm}(\theta,\phi)\xrightarrow{\mathbb{R}}\tilde{\mathbf{e}}^{\prime\pm}(\theta^{\prime},\phi^{\prime})=e^{\pm
i\gamma}\tilde{\mathbf{e}}^{\pm}(\theta,\phi).\end{cases}$ (44)

In terms of the Cartesian components of the base vectors,

$\displaystyle{\tilde{\mathrm{e}}^{\sigma}}_{\phantom{\sigma}i}(\theta,\phi)$
$\displaystyle=[\tilde{\mathbf{e}}^{\sigma}(\theta,\phi)]_{i}=\tilde{\mathbf{e}}^{\sigma}(\theta,\phi)\cdot\hat{\mathbf{e}}_{i},$
(45)
$\displaystyle{\tilde{\mathrm{e}}^{\prime\sigma}}_{\phantom{\sigma}i}(\theta^{\prime},\phi^{\prime})$
$\displaystyle=[\tilde{\mathbf{e}}^{\prime\sigma}(\theta^{\prime},\phi^{\prime})]_{i}=\tilde{\mathbf{e}}^{\prime\sigma}(\theta^{\prime},\phi^{\prime})\cdot\hat{\mathbf{e}}^{\prime}_{i}$
(46)

with $\sigma=0,\pm$, the transformation of Eq. (44) is given by

${\tilde{\mathrm{e}}^{\sigma}}_{\phantom{\sigma}i}(\theta,\phi)\xrightarrow{\mathbb{R}}{\tilde{\mathrm{e}}^{\prime\sigma}}_{\phantom{\sigma}i}(\theta^{\prime},\phi^{\prime})=e^{i\sigma\gamma}{\tilde{\mathrm{e}}^{\sigma}}_{\phantom{\sigma}j}(\theta,\phi)R_{ji}$
(47)

According to the transformation rule of Eq. (47), the spherical tensor basis
in spherical coordinates transforms as

$\tilde{\mathsf{Y}}^{(\sigma)}_{i_{1}i_{2}\cdots
i_{s}}(\theta,\phi)\xrightarrow{\mathbb{R}}\tilde{\mathsf{Y}}^{\prime(\sigma)}_{i_{1}i_{2}\cdots
i_{s}}(\theta^{\prime},\phi^{\prime})\\\
=e^{i\sigma\gamma}\tilde{\mathsf{Y}}^{(\sigma)}_{j_{1}j_{2}\cdots
j_{s}}(\theta,\phi)R_{j_{1}i_{1}}R_{j_{2}i_{2}}\cdots R_{j_{s}i_{s}}.$ (48)

Combining Eqs. (28), (31) and (48), the spherical tensor field in spherical
coordinates simply transforms as

$\tilde{F}_{Xs\sigma}(\bm{r})\xrightarrow{\mathbb{R}}\tilde{F}^{\prime}_{Xs\sigma}(\bm{r}^{\prime})=e^{-i\sigma\gamma}\tilde{F}_{Xs\sigma}(\bm{r}).$
(49)

The spherical tensor field is naturally expanded by the spin-weighted
spherical harmonics ${}_{\sigma}Y_{lm}(\theta,\phi)$ [37, 38] as

$\tilde{F}_{Xs\sigma}(\bm{r})=\sum_{l=|\sigma|}^{\infty}\sum_{m=-l}^{l}\sqrt{\frac{4\pi}{2l+1}}\,\tilde{F}^{lm}_{Xs\sigma}(r)\,{}_{\sigma}Y_{lm}(\theta,\phi),$
(50)

where $\tilde{F}^{lm}_{Xs\sigma}(r)$ are the expansion coefficients of the
multipoles of angular dependence with a fixed distance $r$ from the
coordinates origin. The spin-weighted spherical harmonics are represented by
the Wigner’s rotation matrix as

${}_{\sigma}Y_{lm}(\theta,\phi)=(-1)^{m}\sqrt{\frac{2l+1}{4\pi}}D^{-m}_{(l)\sigma}(\phi,\theta,0).$
(51)

From the orthonormality of spin-weighted spherical harmonics,

$\int\sin\theta\,d\theta\,d\phi\,{}_{\sigma}Y_{lm}^{*}(\theta,\phi){}_{\sigma}Y_{l^{\prime}m^{\prime}}(\theta,\phi)=\delta_{ll^{\prime}}\delta^{m}_{m^{\prime}},$
(52)

the above expansion is inverted as

$\tilde{F}^{lm}_{Xs\sigma}(r)=\sqrt{\frac{2l+1}{4\pi}}\int\sin\theta\,d\theta\,d\phi\,\tilde{F}_{Xs\sigma}(\bm{r})\,{}_{\sigma}Y_{lm}^{*}(\theta,\phi).$
(53)

Under the passive rotation of Eq. (30), the spin-weighted spherical harmonics
transform as [39]

${}_{\sigma}Y_{lm}(\theta,\phi)\xrightarrow{\mathbb{R}}{}_{\sigma}Y_{lm}(\theta^{\prime},\phi^{\prime})=e^{-i\sigma\gamma}\sum_{m^{\prime}}{}_{\sigma}Y_{lm^{\prime}}(\theta,\phi)D_{(l)m}^{m^{\prime}}(R),$
(54)

where the angle $\gamma$ is exactly the same as the one in Eq. (43).
Therefore, the phase factor $e^{-i\sigma\gamma}$ cancels under the rotational
transformations of Eqs. (49) and (54), and the multipole coefficients simply
transform as

$\tilde{F}^{lm}_{Xs\sigma}(r)\xrightarrow{\mathbb{R}}\tilde{F}^{\prime
lm}_{Xs\sigma}(r)=\sum_{m^{\prime}}\tilde{F}^{lm^{\prime}}_{Xs\sigma}(r)D^{m^{\prime}*}_{(l)m}(R),$
(55)

Comparing the above transformation with the one in Cartesian coordinates, Eq.
(36), one notices the absence of the rotation matrix regarding the index
$\sigma$ of irreducible tensor in spherical coordinates, as the tensor basis
rotates together with the coordinates axis.

Substituting Eq. (29) into Eq. (53), the relation between the multipoles
$F^{lm}_{Xs\sigma}(r)$ and the irreducible tensor $F_{Xs\sigma}(\bm{r})$ in
Cartesian coordinates is obtained. The spin-weighted spherical harmonics in
the integrand of Eq. (53) is substituted by Eq. (51), and we use a formula for
a product of Wigner’s rotation matrices [35, 1],

$D_{(l_{1})m_{1}}^{m_{1}^{\prime}}(R)D_{(l_{2})m_{2}}^{m_{2}^{\prime}}(R)=(-1)^{l_{1}+l_{2}}\sum_{l,m,m^{\prime}}(-1)^{l}(2l+1)\\\
\times\left(l_{1}\,l_{2}\,l\right)_{m_{1}m_{2}}^{\phantom{m_{1}m_{2}}m}\left(l_{1}\,l_{2}\,l\right)^{m_{1}^{\prime}m_{2}^{\prime}}_{\phantom{m_{1}^{\prime}m_{2}^{\prime}}m^{\prime}}D_{(l)m}^{m^{\prime}}(R),$
(56)

where the notation introduced in Paper I for Wigner $3j$-symbols is employed,
i.e., $(l_{1}\,l_{2}\,l_{3})_{m_{1}m_{2}m_{3}}$ is the usual $3j$-symbol, and
azimuthal indices $m_{1},m_{2},\ldots$ can be raised by the spherical metric
$g_{(l)}^{mm^{\prime}}$ (see Paper I). As a result, we eventually derive the
relation between multipole coefficients in spherical and Cartesian
coordinates:

$\tilde{F}^{lm}_{Xs\sigma}(r)=(-1)^{\sigma}(2l+1)\sum_{l^{\prime},m^{\prime},\sigma^{\prime}}\begin{pmatrix}s&l^{\prime}&l\\\
\sigma&0&-\sigma\end{pmatrix}\\\
\times\left(s\,l\,l^{\prime}\right)^{\sigma^{\prime}m}_{\phantom{\sigma^{\prime}m}m^{\prime}}F_{Xs\sigma^{\prime}}^{l^{\prime}m^{\prime}}(r).$
(57)

Below, we consider other symmetries than that of rotation. Assuming the
original tensor field $F^{(s)}_{Xi_{1}i_{2}\cdots i_{s}}(\bm{r})$ is real, the
complex conjugates of Eq. (28) is given by

$\tilde{F}_{Xs\sigma}^{*}(\bm{r})=(-1)^{\sigma}\tilde{F}_{Xs,-\sigma}(\bm{r})=\sum_{\sigma^{\prime}}g_{(s)}^{\sigma\sigma^{\prime}}\tilde{F}_{Xs\sigma^{\prime}}(\bm{r}),$
(58)

and the complex conjugate of the multipole coefficients, Eq. (53), is given by

$\tilde{F}^{lm\,*}_{Xs\sigma}(r)=(-1)^{m}\tilde{F}^{l,-m}_{Xs,-\sigma}(r)=\sum_{m^{\prime}}g^{(l)}_{mm^{\prime}}\tilde{F}^{lm^{\prime}}_{Xs,-\sigma}(r).$
(59)

The parity transformation of the original tensor field is given by Eq. (40).
Correspondingly, the parity transformation of irreducible tensors in spherical
coordinates, Eq. (28), is given by

$\tilde{F}_{Xs\sigma}(\bm{r})\xrightarrow{\mathbb{P}}\tilde{F}_{Xs\sigma}^{\prime}(\bm{r})=(-1)^{p_{X}}\tilde{F}_{Xs,-\sigma}(-\bm{r}),$
(60)

where we use the parity property of the spherical basis,

$\tilde{\mathsf{Y}}^{(\sigma)}_{i_{1}\cdots
i_{s}}(\pi-\theta,\phi+\pi)=(-1)^{s}\,\tilde{\mathsf{Y}}^{(-\sigma)}_{i_{1}\cdots
i_{s}}(\theta,\phi),$ (61)

which can be shown from the definition. Accordingly, the parity of multiple
coefficients, Eq. (53), is given by

$\tilde{F}^{lm}_{Xs\sigma}(r)\xrightarrow{\mathbb{P}}\tilde{F}^{\prime
lm}_{Xs\sigma}(r)=(-1)^{p_{X}+l}\tilde{F}^{lm}_{Xs,-\sigma}(r),$ (62)

where we use the parity of spin-weighted spherical harmonics,

${}_{\sigma}Y_{lm}(\pi-\theta,\phi+\pi)=(-1)^{l}{}_{-\sigma}Y_{lm}(\theta,\phi).$
(63)

###  II.4 The power spectrum

The statistics of irreducible tensors can be theoretically predicted by
applying the iPT, as we have shown in previous Papers I–III in the limit of
flat-sky and distant-observer approximations. In this paper, we are
generalizing the previous results, without assuming the last approximations.
First we consider statistics of irreducible tensors in Cartesian coordinates,
$F_{Xs\sigma}(\bm{r})$ defined by Eq. (25), which are naturally treated by the
perturbation theory as we show below. The irreducible tensors in spherical
coordinates, $\tilde{F}_{Xs\sigma}(\bm{r})$ and $\tilde{F}^{lm}_{Xs\sigma}(r)$
defined by Eqs. (26) and (50), are linearly related to those in Cartesian
coordinates by Eqs. (29) and (57), and thus statistics of the latter variables
in spherical coordinates are derived from the former variables in Cartesian
coordinates.

We first define the power spectrum of the multipoles in Cartesian coordinates
by

$C^{(s_{1}s_{2})\,l_{1}l_{2};m_{1}m_{2}}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})\equiv\left\langle
F^{l_{1}m_{1}}_{X_{1}s_{1}\sigma_{1}}(r_{1})F^{l_{2}m_{2}}_{X_{2}s_{2}\sigma_{2}}(r_{2})\right\rangle.$
(64)

Under the passive rotation of Eq. (30), the multipoles transform as Eq. (36).
Accordingly, the power spectrum above transforms as

$C^{(s_{1}s_{2})\,l_{1}l_{2};m_{1}m_{2}}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})\xrightarrow{\mathbb{R}}\sum_{\sigma^{\prime}_{1},\sigma^{\prime}_{2},m^{\prime}_{1},m^{\prime}_{2}}C^{(s_{1}s_{2})\,l_{1}l_{2};m_{1}^{\prime}m_{2}^{\prime}}_{X_{1}X_{2}\sigma^{\prime}_{1}\sigma^{\prime}_{2}}(r_{1},r_{2})\\\
\times
D^{\sigma^{\prime}_{1}}_{(s_{1})\sigma_{1}}(R)D^{\sigma^{\prime}_{2}}_{(s_{2})\sigma_{2}}(R)D^{m^{\prime}_{1}*}_{(l_{1})m_{1}}(R)D^{m^{\prime}_{2}*}_{(l_{2})m_{2}}(R).$
(65)

In the statistically isotropic Universe, the correlation function of the
multipoles, Eq. (64), should be rotationally invariant as the function is
given by an ensemble average of the field variables, and should not depend on
any choice of the Cartesian basis $\hat{\mathbf{e}}_{i}$. Therefore, the
functions are invariant under the transformation of Eq. (65) for any rotation
$R$. Averaging over the rotation in particular, the functions satisfy an
identity,

$C^{(s_{1}s_{2})\,l_{1}l_{2};m_{1}m_{2}}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})=\sum_{\sigma^{\prime}_{1},\sigma^{\prime}_{2},m^{\prime}_{1},m^{\prime}_{2}}C^{(s_{1}s_{2})\,l_{1}l_{2};m_{1}^{\prime}m_{2}^{\prime}}_{X_{1}X_{2}\sigma^{\prime}_{1}\sigma^{\prime}_{2}}(r_{1},r_{2})\\\
\times\int\frac{[dR]}{8\pi^{2}}D^{\sigma^{\prime}_{1}}_{(s_{1})\sigma_{1}}(R)D^{\sigma^{\prime}_{2}}_{(s_{2})\sigma_{2}}(R)D^{m^{\prime}_{1}*}_{(l_{1})m_{1}}(R)D^{m^{\prime}_{2}*}_{(l_{2})m_{2}}(R),$
(66)

where the last integral represents the average over conceivable rotations $R$,
or integrals over the Euler angles of the rotation $R$ divided by the total
volume $8\pi^{2}$ of possible values of the Euler angles. The products of
Wigner’s rotation matrices are evaluated by the method described in Paper I,
and the integral in Eq. (66) reduces to a summation of the products of
$3j$-symbols,

$\int\frac{[dR]}{8\pi^{2}}D^{\sigma^{\prime}_{1}}_{(s_{1})\sigma_{1}}(R)D^{\sigma^{\prime}_{2}}_{(s_{2})\sigma_{2}}(R)D^{m^{\prime}_{1}*}_{(l_{1})m_{1}}(R)D^{m^{\prime}_{2}*}_{(l_{2})m_{2}}(R)\\\
=(-1)^{s_{1}+s_{2}+l_{1}+l_{2}}\sum_{l,m,m^{\prime}}(2l+1)\left(s_{1}\,l_{1}\,l\right)_{\sigma_{1}}^{\phantom{\sigma_{1}}m_{1}m}\\\
\times\left(s_{2}\,l_{2}\,l\right)_{\sigma_{2}\phantom{m_{2}}m}^{\phantom{\sigma_{2}}m_{2}}\left(s_{1}\,l_{1}\,l\right)^{\sigma_{1}^{\prime}\phantom{m_{1}^{\prime}}m^{\prime}}_{\phantom{\sigma_{1}^{\prime}}m_{1}^{\prime}}\left(s_{2}\,l_{2}\,l\right)^{\sigma_{2}^{\prime}}_{\phantom{\sigma_{2}^{\prime}}m_{2}^{\prime}m^{\prime}}.$
(67)

Substituting the above into Eq. (66), we find the correlation function is
given by

$C^{(s_{1}s_{2})\,l_{1}l_{2};m_{1}m_{2}}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})=i^{s_{1}+s_{2}+l_{1}+l_{2}}\sum_{l,m}\left(s_{1}\,l_{1}\,l\right)_{\sigma_{1}}^{\phantom{\sigma_{1}}m_{1}m}\\\
\times\left(s_{2}\,l_{2}\,l\right)_{\sigma_{2}\phantom{m_{2}}m}^{\phantom{\sigma_{2}}m_{2}}C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}\,l}(r_{1},r_{2}),$
(68)

where

$C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}\,l}(r_{1},r_{2})\equiv
i^{s_{1}+s_{2}+l_{1}+l_{2}}(2l+1)\sum_{\begin{subarray}{c}\sigma_{1},\sigma_{2}\\\
m_{1},m_{2},m\end{subarray}}\left(s_{1}\,l_{1}\,l\right)^{\sigma_{1}}_{\phantom{\sigma_{1}}m_{1}m}\\\
\times\left(s_{2}\,l_{2}\,l\right)^{\sigma_{2}\phantom{m_{2}}m}_{\phantom{\sigma_{2}}m_{2}}C^{(s_{1}s_{2})\,l_{1}l_{2};m_{1}m_{2}}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2}).$
(69)

The last functions $C^{l_{1}l_{2};l}_{X_{1}X_{2};s_{1}s_{2}}(r_{1},r_{2})$ are
invariant under rotations of coordinates system, and thus represent physical
degrees of freedom of the spectrum, which do not depend on the choice of
coordinates system. We call the function as the invariant spectrum of the
tensor fields in the present paper of the full-sky analysis. As we show in
later sections below, one can naturally predict the invariant spectrum by the
iPT.

The property of complex conjugate, Eq. (39), shows that

$C^{(s_{1}s_{2})\,l_{1}l_{2};m_{1}m_{2}\,*}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})=\sum_{m_{1}^{\prime},m_{2}^{\prime},\sigma_{1}^{\prime},\sigma_{2}^{\prime}}g_{(s_{1})}^{\sigma_{1}\sigma_{1}^{\prime}}g_{(s_{2})}^{\sigma_{2}\sigma_{2}^{\prime}}g^{(l_{1})}_{m_{1}m_{1}^{\prime}}g^{(l_{2})}_{m_{2}m_{2}^{\prime}}\\\
\times
C^{(s_{1}s_{2})\,l_{1}l_{2};m_{1}^{\prime}m_{2}^{\prime}}_{X_{1}X_{2}\sigma_{1}^{\prime}\sigma_{2}^{\prime}}(r_{1},r_{2}).$
(70)

The parity property of Eq. (42) shows that

$C^{(s_{1}s_{2})\,l_{1}l_{2};m_{1}m_{2}}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})\xrightarrow{\mathbb{P}}(-1)^{p_{X_{1}}+p_{X_{2}}+s_{1}+s_{2}+l_{1}+l_{2}}\\\
\times
C^{(s_{1}s_{2})\,l_{1}l_{2};m_{1}m_{2}}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2}).$
(71)

Substituting the above equations into Eq. (69), the invariant spectrum
satisfies the property of complex conjugate,

$C^{s_{1}s_{2};l_{1}l_{2}\,*}_{X_{1}X_{2}\,l}(r_{1},r_{2})=C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}\,l}(r_{1},r_{2}),$
(72)

i.e., they are real functions, and the parity transformation is given by

$C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}\,l}(r_{1},r_{2})\xrightarrow{\mathbb{P}}(-1)^{p_{X_{1}}+p_{X_{2}}+s_{1}+s_{2}+l_{1}+l_{2}}C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}\,l}(r_{1},r_{2}).$
(73)

Next we consider the power spectrum of the irreducible tensor field
$\tilde{F}^{lm}_{Xs\sigma}(r)$ on the basis of spherical coordinates. Due to
the property of rotational transformation of Eq. (55), and following the
similar (and simpler) considerations as in the above, one finds that the power
spectrum of the fields necessarily have a form,

$\left\langle\tilde{F}^{l_{1}m_{1}}_{X_{1}s_{1}\sigma_{1}}(r_{1})\tilde{F}^{l_{2}m_{2}}_{X_{2}s_{2}\sigma_{2}}(r_{1})\right\rangle=\delta_{l_{1}l_{2}}\,g_{(l_{1})}^{m_{1}m_{2}}\,\tilde{C}^{(s_{1}s_{2})\,l_{1}}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2}),$
(74)

and the last factor corresponds to the reduced power spectrum of the
irreducible tensors in the spherical coordinates. Substituting Eq. (57) into
the left-hand side (LHS) of the above equation, one really confirms the power
spectrum have the form of the RHS. Using the orthogonality relations of the
$3j$-symbol and formulas of the $6j$-symbol quoted in Appendix C of Paper I,
we find the reduced power spectrum introduced above is given by

$\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})=i^{s_{1}+s_{2}}(-1)^{\sigma_{1}+\sigma_{2}}\sum_{l_{1},l_{2}}i^{l_{1}+l_{2}}\begin{pmatrix}s_{1}&l_{1}&l\\\
\sigma_{1}&0&-\sigma_{1}\end{pmatrix}\\\ \times\begin{pmatrix}s_{2}&l_{2}&l\\\
\sigma_{2}&0&-\sigma_{2}\end{pmatrix}C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}\,l}(r_{1},r_{2}).$
(75)

Using the orthogonality relations of $3j$-symbol [36], the above equation can
be inverted as

$C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}\,l}(r_{1},r_{2})=(-i)^{s_{1}+s_{2}+l_{1}+l_{2}}(2l_{1}+1)(2l_{2}+1)\\\
\times\sum_{\sigma_{1},\sigma_{2}}(-1)^{\sigma_{1}+\sigma_{2}}\begin{pmatrix}s_{1}&l_{1}&l\\\
\sigma_{1}&0&-\sigma_{1}\end{pmatrix}\begin{pmatrix}s_{2}&l_{2}&l\\\
\sigma_{2}&0&-\sigma_{2}\end{pmatrix}\\\
\times\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2}).$
(76)

The property of complex conjugate, Eq. (59), shows that

$\tilde{C}^{(s_{1}s_{2})\,l\,*}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})=\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2},-\sigma_{1},-\sigma_{2}}(r_{1},r_{2}),$
(77)

Similarly, the parity property of Eq. (62) shows that

$\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})\xrightarrow{\mathbb{P}}(-1)^{p_{X_{1}}+p_{X_{2}}}\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2},-\sigma_{1},-\sigma_{2}}(r_{1},r_{2}).$
(78)

The above Eqs. (77) and (78) are consistent with Eq. (75) with Eqs. (72) and
(73). When the parity symmetry holds and the power spectrum is invariant under
the transformation of Eq. (78), the power spectrum is real when
$s_{1}+s_{2}+p_{X_{1}}+p_{X_{2}}=\mathrm{even}$, and pure imaginary when
$s_{1}+s_{2}+p_{X_{1}}+p_{X_{2}}=\mathrm{odd}$.

###  II.5 The correlation function

####  II.5.1 Two-point correlation function in Cartesian coordinates

The two-point correlation functions of the tensor fields are also given in
terms of the invariant spectrum defined by Eq. (69). First we consider the
correlation function of the tensor field in Cartesian coordinates, defined by

$\left\langle
F_{X_{1}s_{1}\sigma_{1}}(\bm{r}_{1})F_{X_{2}s_{2}\sigma_{2}}(\bm{r}_{2})\right\rangle=\xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r}_{1},\bm{r}_{2}).$
(79)

Combining Eqs. (33), (64) and (68), the above correlation function is given by
the invariant spectrum. A straightforward calculation gives

$\xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r}_{1},\bm{r}_{2})=4\pi\sum_{\begin{subarray}{c}l_{1},l_{2},l\\\
m_{1},m_{2},m\end{subarray}}\frac{i^{s_{1}+s_{2}+l_{1}+l_{2}}}{\sqrt{(2l_{1}+1)(2l_{2}+1)}}\\\
\times\left(s_{1}\,l_{1}\,l\right)_{\sigma_{1}}^{\phantom{\sigma_{1}}m_{1}m}\left(s_{2}\,l_{2}\,l\right)_{\sigma_{2}\phantom{m_{2}}m}^{\phantom{\sigma_{2}}m_{2}}Y_{l_{1}m_{1}}(\Omega_{1})Y_{l_{2}m_{2}}(\Omega_{2})\\\
\times C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}\,l}(r_{1},r_{2}),$ (80)

where $\Omega_{i}=(\theta_{i},\phi_{i})$ represents the angular coordinates of
$\hat{\bm{r}}_{i}$ with $i=1,2$. Applying a recoupling formula of $3j$-symbols
using the $6j$-symbol (see Appendix C of Paper I in the present notations),
the above equation is equivalent to

$\xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r}_{1},\bm{r}_{2})=i^{s_{1}-s_{2}}\sum_{l_{1},l_{2}}(-i)^{l_{1}-l_{2}}\sum_{l,m}(2l+1)\\\
\times\left(s_{1}\,s_{2}\,l\right)_{\sigma_{1}\sigma_{2}}^{\phantom{\sigma_{1}\sigma_{2}}m}X^{l_{1}l_{2}}_{lm}(\hat{\bm{r}}_{1},\hat{\bm{r}}_{2})\sum_{l^{\prime}}\begin{Bmatrix}s_{1}&s_{2}&l\\\
l_{2}&l_{1}&l^{\prime}\end{Bmatrix}\\\ \times
C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}\,l^{\prime}}(r_{1},r_{2}),$ (81)

where

$X^{l_{1}l_{2}}_{lm}(\hat{\bm{r}}_{1},\hat{\bm{r}}_{2})=\frac{4\pi}{\sqrt{(2l_{1}+1)(2l_{2}+1)}}\\\
\times\sum_{m_{1},m_{2}}\left(l\,l_{1}\,l_{2}\right)_{m}^{\phantom{m}m_{1}m_{2}}Y_{l_{1}m_{1}}(\Omega_{1})Y_{l_{2}m_{2}}(\Omega_{2}).$
(82)

are the bipolar spherical harmonics with a normalization introduced in Paper
I. Using orthogonality relations of the spherical harmonics, $3j$\- and
$6j$-symbols, the Eqs. (80) and (81) can be inverted as

$C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}\,l}(r_{1},r_{2})=\frac{i^{s_{1}+s_{2}+l_{1}+l_{2}}}{4\pi}(2l+1)\sqrt{(2l_{1}+1)(2l_{2}+1)}\\\
\times\sum_{\sigma_{1},\sigma_{2},m_{1},m_{2},m}\left(s_{1}\,l_{1}\,l\right)^{\sigma_{1}m_{1}}_{\phantom{\sigma_{1}m_{1}}m}\left(s_{2}\,l_{2}\,l\right)^{\sigma_{2}m_{2}m}\\\
\times\int
d\Omega_{1}d\Omega_{2}Y_{l_{1}m_{1}}(\Omega_{1})Y_{l_{2}m_{2}}(\Omega_{2})\xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r}_{1},\bm{r}_{2}),$
(83)

and

$C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}\,l}(r_{1},r_{2})=i^{s_{1}-s_{2}}\,(-i)^{l_{1}-l_{2}}(2l+1)(2l_{1}+1)(2l_{2}+1)\\\
\times\sum_{l^{\prime},m^{\prime},\sigma_{1},\sigma_{2}}(2l^{\prime}+1)\left(s_{1}\,s_{2}\,l^{\prime}\right)^{\sigma_{1}\sigma_{2}m^{\prime}}\begin{Bmatrix}s_{1}&s_{2}&l^{\prime}\\\
l_{2}&l_{1}&l\end{Bmatrix}\\\
\times\int\frac{d\Omega_{1}}{4\pi}\frac{d\Omega_{2}}{4\pi}X^{l_{1}l_{2}}_{l^{\prime}m^{\prime}}(\hat{\bm{r}}_{1},\hat{\bm{r}}_{2})\xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r}_{1},\bm{r}_{2}).$
(84)

Substituting Eq. (38) into Eq. (79), the complex conjugate of the correlation
function is given by

$\xi^{(s_{1}s_{2})\,*}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r}_{1},\bm{r}_{2})=(-1)^{\sigma_{1}+\sigma_{2}}\xi^{(s_{1}s_{2})}_{X_{1}X_{2},-\sigma_{1},-\sigma_{2}}(\bm{r}_{1},\bm{r}_{2}),$
(85)

which is consistent with Eqs. (72) and (80). Similarly, the parity
transformation of Eq. (41) shows that

$\xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r}_{1},\bm{r}_{2})\xrightarrow{\mathbb{P}}(-1)^{p_{X_{1}}+p_{X_{2}}+s_{1}+s_{2}}\xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(-\bm{r}_{1},-\bm{r}_{2}),$
(86)

which is consistent with Eqs. (73) and (80).

In the definition of the correlation function, Eq. (79), the statistical
average corresponds to the ensemble average, which cannot be evaluated for an
observer whose position is fixed near the Earth. In practice, one should
average over the sky positions of the correlation function to obtain
statistically meaningful results. In this regard, one can set a coordinates
system for each pair of astronomical objects to probe the correlation function
of tensor fields. For each pair of objects, 1 and 2, it is always possible to
set the spherical coordinates system $(r,\theta,\phi)$ in which the first
object is located at $\bm{r}_{1}:(r_{1},\theta,0)$ and the second object is
located at $\bm{r}_{2}:(r_{2},0,0)$. Averaging over all the pairs of objects,
one can observationally estimate the correlation function for this particular
values of argument. We denote the correlation function with this particular
setup by
$\xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2},\theta)$.
Just substituting these particular values, $\Omega_{1}=(r_{1},\theta,0)$ and
$\Omega_{2}=(r_{2},0,0)$, Eq. (80) reduces to

$\xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2},\theta)=i^{s_{1}+s_{2}}(-1)^{\sigma_{1}}\sum_{l_{1},l_{2},l}i^{l_{1}+l_{2}}\begin{pmatrix}s_{1}&l_{1}&l\\\
\sigma_{1}&-\sigma_{12}&\sigma_{2}\end{pmatrix}\\\
\times\begin{pmatrix}s_{2}&l_{2}&l\\\
\sigma_{2}&0&-\sigma_{2}\end{pmatrix}\tilde{P}_{l_{1}}^{\sigma_{12}}(\cos\theta)C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}\,l}(r_{1},r_{2}),$
(87)

where we define normalized Legendre polynomials,

$\tilde{P}_{l}^{m}(\cos\theta)\equiv\sqrt{\frac{(l-m)!}{(l+m)!}}\,P_{l}^{m}(\cos\theta)=\sqrt{\frac{4\pi}{2l+1}}\,Y_{l}^{m}(\theta,0).$
(88)

####  II.5.2 Two-point correlation function in spherical coordinates

Next we consider the correlation function of the irreducible tensor field
$\tilde{F}_{Xs\sigma}(\bm{r})$ on the basis of spherical coordinates, defined
by

$\left\langle\tilde{F}_{X_{1}s_{1}\sigma_{1}}(\bm{r}_{1})\tilde{F}_{X_{2}s_{2}\sigma_{2}}(\bm{r}_{2})\right\rangle=\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r}_{1},\bm{r}_{2}).$
(89)

Due to the relation of Eq. (29), the correlation function in spherical
coordinates above is related to that in Cartesian coordinates of Eq. (79) by

$\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r}_{1},\bm{r}_{2})=\sum_{\sigma_{1}^{\prime},\sigma_{2}^{\prime}}\xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}^{\prime}\sigma_{2}^{\prime}}(\bm{r}_{1},\bm{r}_{2})\\\
\times
D^{\sigma_{1}^{\prime}}_{(s_{1})\sigma_{1}}(\phi_{1},\theta_{1},0)D^{\sigma_{2}^{\prime}}_{(s_{2})\sigma_{2}}(\phi_{2},\theta_{2},0).$
(90)

Substituting Eq. (80) into the above equation, using a formula of Eq. (56) and
applying an orthogonality relation of $3j$-symbols, we derive

$\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r}_{1},\bm{r}_{2})=i^{s_{1}+s_{2}}(-1)^{\sigma_{1}+\sigma_{2}}\sum_{l_{1},l_{2},l}i^{l_{1}+l_{2}}\begin{pmatrix}s_{1}&l_{1}&l\\\
\sigma_{1}&0&-\sigma_{1}\end{pmatrix}\\\ \times\begin{pmatrix}s_{2}&l_{2}&l\\\
\sigma_{2}&0&-\sigma_{2}\end{pmatrix}A^{(l)}_{\sigma_{1}\sigma_{2}}(\hat{\bm{r}}_{1},\hat{\bm{r}}_{2})C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}\,l}(r_{1},r_{2}),$
(91)

where

$A^{(l)}_{\sigma_{1}\sigma_{2}}(\hat{\bm{r}}_{1},\hat{\bm{r}}_{2})\equiv\sum_{m_{1},m_{2}}g^{(l)}_{m_{1}m_{2}}D^{m_{1}}_{(l)\sigma_{1}}(\phi_{1},\theta_{1},0)D^{m_{2}}_{(l)\sigma_{2}}(\phi_{2},\theta_{2},0)$
(92)

are functions of the two directions, $\hat{\bm{r}}_{1}\equiv\bm{r}_{1}/r_{1}$
and $\hat{\bm{r}}_{2}\equiv\bm{r}_{2}/r_{2}$ with spherical coordinates
$(\theta_{1},\phi_{1})$ and $(\theta_{2},\phi_{2})$, respectively. The same
equation can be derived as well by substituting Eqs. (50), (74) and (75) into
the above Eq. (89). Due to the properties of Wigner’s rotation matrix, the
complex conjugate of the last function is given by

$A^{(l)\,*}_{\sigma_{1}\sigma_{2}}(\hat{\bm{r}}_{1},\hat{\bm{r}}_{2})=(-1)^{\sigma_{1}+\sigma_{2}}A^{(l)}_{-\sigma_{1},-\sigma_{2}}(\hat{\bm{r}}_{1},\hat{\bm{r}}_{2}),$
(93)

and the parity property is given by

$A^{(l)}_{\sigma_{1}\sigma_{2}}(-\hat{\bm{r}}_{1},-\hat{\bm{r}}_{2})=A^{(l)}_{-\sigma_{1},-\sigma_{2}}(\hat{\bm{r}}_{1},\hat{\bm{r}}_{2}).$
(94)

Applying the addition theorem of the Wigner rotation matrix [36], the last
functions reduces to an expression with a single Wigner matrix as

$A^{(l)}_{\sigma_{1}\sigma_{2}}(\hat{\bm{r}}_{1},\hat{\bm{r}}_{2})=(-1)^{\sigma_{2}}D^{-\sigma_{2}}_{(l)\;\sigma_{1}}(\alpha,\beta,\gamma),$
(95)

where $(\alpha,\beta,\gamma)$ are Euler angles of the consecutive rotations by
$(\phi_{2},\theta_{2},0)$ and $(0,-\theta_{1},-\phi_{1})$ in this order.
Explicitly, the Euler angles are related to the angular coordinates of
$\Omega_{1}$ and $\Omega_{2}$ by

$\displaystyle\cot\alpha$
$\displaystyle=\cos\theta_{2}\cot(\phi_{1}-\phi_{2})-\frac{\cot\theta_{1}\sin\theta_{2}}{\sin(\phi_{1}-\phi_{2})},$
(96) $\displaystyle\cos\beta$
$\displaystyle=\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}\cos(\phi_{1}-\phi_{2}),$
(97) $\displaystyle\cot\gamma$
$\displaystyle=\cos\theta_{1}\cot(\phi_{1}-\phi_{2})-\frac{\sin\theta_{1}\cot\theta_{2}}{\sin(\phi_{1}-\phi_{2})}.$
(98)

Substituting Eq. (58) into Eq. (89), the complex conjugate of the correlation
function is given by

$\tilde{\xi}^{(s_{1}s_{2})\,*}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r}_{1},\bm{r}_{2})=(-1)^{\sigma_{1}+\sigma_{2}}\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2},-\sigma_{1},-\sigma_{2}}(\bm{r}_{1},\bm{r}_{2}),$
(99)

which is consistent with Eqs. (72) and (91). Similarly, the parity
transformation of Eq. (60) shows that

$\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r}_{1},\bm{r}_{2})\xrightarrow{\mathbb{P}}(-1)^{p_{X_{1}}+p_{X_{2}}}\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2},-\sigma_{1},-\sigma_{2}}(-\bm{r}_{1},-\bm{r}_{2}),$
(100)

which is consistent with Eqs. (73) and (80).

As in the previous case, the correlation function of Eq. (91) corresponds to
the ensemble average, and we consider the coordinates systems are set for each
pair of objects. Just in the previous case, we consider they are chosen as
$(\theta_{1},\phi_{1})=(\theta,0)$ and $(\theta_{2},\phi_{2})=(0,0)$.
Substituting these coordinates values into Eq. (91), we derive

$\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2},\theta)=i^{s_{1}+s_{2}}(-1)^{\sigma_{1}+\sigma_{2}}\sum_{l,l_{1},l_{2}}i^{l_{1}+l_{2}}u^{(l)}_{\sigma_{1}\sigma_{2}}(\theta)\\\
\times\begin{pmatrix}s_{1}&l_{1}&l\\\
\sigma_{1}&0&-\sigma_{1}\end{pmatrix}\begin{pmatrix}s_{2}&l_{2}&l\\\
\sigma_{2}&0&-\sigma_{2}\end{pmatrix}C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}\,l}(r_{1},r_{2}),$
(101)

where

$u^{(l)}_{m^{\prime}m}(\theta)\equiv(-1)^{m}D^{-m}_{(l)\,m^{\prime}}(0,\theta,0)=(-1)^{m}d^{(l)}_{-m,m^{\prime}}(\theta).$
(102)

The function $d^{(l)}_{m^{\prime}m}(\theta)\equiv
D^{m^{\prime}}_{(l)m}(0,\theta,0)$ is known to be the Wigner’s $d$-function.
An explicit formula of Wigner’s rotation matrix [36] is translated into a
formula,

$u^{(l)}_{m^{\prime}m}(\theta)=(-1)^{m}\sqrt{(l+m)!\,(l-m)!\,(l+m^{\prime})!\,(l-m^{\prime})!}\\\
\times\sum_{r}\frac{(-1)^{r}\left[\cos(\theta/2)\right]^{2l-m-m^{\prime}-2r}\left[\sin(\theta/2)\right]^{m+m^{\prime}+2r}}{r!\,(l-m-r)!\,(l-m^{\prime}-r)!\,(m+m^{\prime}+r)!},$
(103)

where $r$ runs over all integers for which the factorial arguments are non-
negative. We have

$u^{(l)\,*}_{m^{\prime}m}(\theta)=u^{(l)}_{m^{\prime}m}(\theta),$ (104)

i.e., they are real functions, and we also have symmetries,

$u^{(l)}_{m^{\prime}m}(\theta)=u^{(l)}_{-m,-m^{\prime}}(\theta)=(-1)^{m^{\prime}+m}u^{(l)}_{mm^{\prime}}(\theta),$
(105)

and

$u^{(l)}_{m^{\prime}m}(-\theta)=u^{(l)}_{mm^{\prime}}(\theta).$ (106)

From the last symmetry, the correlation function of Eq. (101) satisfies the
interchange symmetry:

$\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2},\theta)=\tilde{\xi}^{(s_{2}s_{1})}_{X_{2}X_{1}\sigma_{2}\sigma_{1}}(r_{2},r_{1},-\theta).$
(107)

Comparing Eq. (101) with Eq. (75), we have a simple relation between the
correlation function and the power spectrum in spherical coordinates,

$\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2},\theta)=\sum_{l}u^{(l)}_{\sigma_{1}\sigma_{2}}(\theta)\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2}).$
(108)

From the orthonormality relation of Wigner’s $d$-function [36], we have the
orthonormality relation,

$\int_{0}^{\pi}\sin\theta\,d\theta\,u^{(l^{\prime})}_{m^{\prime}m}(\theta)\,u^{(l)}_{m^{\prime}m}(\theta)=\frac{2}{2l+1}\delta_{l^{\prime}l}.$
(109)

Thus the inverse relation of Eq. (108) is given by

$\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})=\frac{2l+1}{2}\int_{0}^{\pi}\sin\theta\,d\theta\,u^{(l)}_{\sigma_{1}\sigma_{2}}(\theta)\,\\\
\times\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2},\theta).$
(110)

The parity transformation of Eq. (101) is given by

$\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2},\theta)\xrightarrow{\mathbb{P}}(-1)^{p_{X_{1}}+p_{X_{2}}}\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2},-\sigma_{1},-\sigma_{2}}(r_{1},r_{2},\theta).$
(111)

###  II.6 Alternative expressions of the correlation functions

The evaluations of the correlation functions from the invariant power spectrum
by the formulas of Eqs. (87), (101) and (108) are numerically difficult,
because we need to evaluate the power spectrum
$C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}\,l}(r_{1},r_{2})$ theoretically up to
sufficiently large $l$ to ensure the convergence of the series expansion. In
order to see the problem, we consider a small angle limit $\theta\rightarrow
0$, for example. In this limit, we have
$u^{(l)}_{\sigma_{1}\sigma_{2}}(\theta)\rightarrow
g^{(l)}_{\sigma_{1}\sigma_{2}}$, where the RHS is defined by Eq. (11).
Therefore, we have

$\lim_{\theta\rightarrow
0}\,\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2},\theta)=g^{(l)}_{\sigma_{1}\sigma_{2}}\sum_{l}\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2}\sigma_{1},-\sigma_{1}}(r_{1},r_{2}).$
(112)

The RHS of the above equation vanishes when $\sigma_{1}+\sigma_{2}\neq 0$,
while whole multipoles $l$ of the power spectrum contribute when
$\sigma_{1}+\sigma_{2}=0$. Unless the absolute values of the power spectrum
for large $l$ are small enough and the summation converges fast, the above
summation cannot be evaluated by truncating the infinite sum over $l$.

A useful expression of the correlation function of scalar fields without
assuming the distant-observer approximation is proposed in Ref. [30], in which
the correlation function is parameterized by the separation vector,

$\bm{x}\equiv\bm{r}_{1}-\bm{r}_{2},$ (113)

and the opening angle $\theta$. This method is recently generalized in order
to describe the correlation function of two-dimensionally projected shapes of
galaxies [34, 40]. Below we consider essentially the same way of description
with our formalism, generalizing the linear analysis of the previous work to
those applicable in more general situations. For that purpose, we consider a
redundant function

$\Xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{x};\hat{\bm{z}}_{1},\hat{\bm{z}}_{2}),$
(114)

in which the arguments $\bm{x},\hat{\bm{z}}_{1},\hat{\bm{z}}_{2}$ are
independent vectors, and the last two vectors
$\hat{\bm{z}}_{1},\hat{\bm{z}}_{2}$ are unit vectors. When the arguments are
given by $\bm{x}=\bm{r}_{1}-\bm{r}_{2}$,
$\hat{\bm{z}}_{1}=\hat{\bm{r}}_{1}=\bm{r}_{1}/r_{1}$,
$\hat{\bm{z}}_{2}=\hat{\bm{r}}_{2}=\bm{r}_{2}/r_{2}$, the values of the above
function are defined to be the same as the values of the correlation function
of Eq. (79):

$\xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r}_{1},\bm{r}_{2})=\Xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r}_{1}-\bm{r}_{2};\hat{\bm{r}}_{1},\hat{\bm{r}}_{2}).$
(115)

The redundant function of Eq. (114) is defined so that the homogeneity and
isotropy are satisfied, i.e., the function is independent under the
translation and rotation of the coordinates system. As long as the last
condition is satisfied, the function by definition can take arbitrary values
when the above relation of Eq. (115) is not physically satisfied, e.g., when
the three vectors $\bm{x}$, $\bm{z}_{1}$, $\bm{z}_{2}$ are not coplanar, etc.
As we will see in the next section, the redundant function with unphysical
arguments are conveniently defined in the iPT so that the lines of sight
$\hat{\bm{z}}_{1}$, $\hat{\bm{z}}_{2}$ can be oriented to any directions even
though they do not correspond to physical lines of sight.

Following the similar derivation with the rotational symmetry given in Paper
I, the redundant function can be represented by

$\Xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{x};\hat{\bm{z}}_{1},\hat{\bm{z}}_{2})=i^{s_{1}+s_{2}}\sum_{\begin{subarray}{c}l_{1},l_{2},l,l_{z1},l_{z2}\\\
m_{1},m_{2},m,m_{z1},m_{z2}\end{subarray}}i^{l}C_{lm}(\hat{\bm{x}})\left(l_{1}\,l_{2}\,l\right)_{m_{1}m_{2}}^{\phantom{m_{1}m_{2}}m}\\\
\times\left(s_{1}\,l_{z1}\,l_{1}\right)_{\sigma_{1}}^{\phantom{\sigma_{1}}m_{z1}m_{1}}\left(s_{2}\,l_{z2}\,l_{2}\right)_{\sigma_{2}}^{\phantom{\sigma_{2}}m_{z2}m_{2}}C_{l_{z1}m_{z1}}(\hat{\bm{z}}_{1})C_{l_{z2}m_{z2}}(\hat{\bm{z}}_{2})\\\
\times(-1)^{l_{2}}\Xi^{s_{1}s_{2};l_{1}l_{2}l}_{X_{1}X_{2}l_{z1}l_{z2}}(x),$
(116)

where $\Xi^{s_{1}s_{2};l_{1}l_{2}l}_{X_{1}X_{2}l_{z1}l_{z2}}(x)$ is a
rotationally invariant function, $x=|\bm{x}|$, and we use a notation
$C_{lm}(\hat{\bm{r}})=\sqrt{4\pi/(2l+1)}\,Y_{lm}(\theta,\phi)$ for spherical
harmonics with Racah’s normalization with spherical coordinates
$(\theta,\phi)$ of the direction $\hat{\bm{r}}$. The invariant function is
evaluated by the iPT as explicitly shown in the next section.

The flip symmetry in redshift space, which is discussed in Paper I, indicates
that the correlation function in redshift space is invariant under the
flipping of the direction to the lines of sight. That requires the function in
Eq. (116) is invariant under the flipping,
$\hat{\bm{z}}_{1}\rightarrow-\hat{\bm{z}}_{1}$ and
$\hat{\bm{z}}_{2}\rightarrow-\hat{\bm{z}}_{2}$. In order to satisfy the flip
symmetry, the indices $l_{z1}$ and $l_{z2}$ should be non-negative even
integers. This property is really satisfied in predictions of the iPT, as
explicitly shown in the next section. We assume this property in the
following.

Applying a recoupling formula of $3j$-symbols using the $9j$-symbol, (see
Appendix C of Paper I in the present notations), the above equation is
equivalent to

$\Xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{x};\hat{\bm{z}}_{1},\hat{\bm{z}}_{2})=i^{s_{1}+s_{2}}\sum_{s,\sigma}(2s+1)\left(s_{1}\,s_{2}\,s\right)_{\sigma_{1}\sigma_{2}}^{\phantom{\sigma_{1}\sigma_{2}}\sigma}\\\
\times\sum_{l,l_{z1},l_{z2},l_{z}}(-i)^{l}(-1)^{l_{z}}\sqrt{2l_{z}+1}\,X^{l\,l_{z1}l_{z2}}_{l_{z};s\sigma}(\hat{\bm{x}},\hat{\bm{z}}_{1},\hat{\bm{z}}_{2})\\\
\times\sum_{l_{1},l_{2}}(-1)^{l_{1}}\begin{Bmatrix}s_{1}&s_{2}&s\\\
l_{z1}&l_{z2}&l_{z}\\\
l_{1}&l_{2}&l\end{Bmatrix}\Xi^{s_{1}s_{2};l_{1}l_{2}l}_{X_{1}X_{2}l_{z1}l_{z2}}(x),$
(117)

With the above expressions, the correlation function in Cartesian coordinates
is given by Eq. (115) in terms of the invariant function.

From the correlation function in Cartesian coordinates given above, the
correlation function in spherical coordinates is given by Eq. (90). Using the
formulas for a product of Wigner’s rotation matrices and an orthogonality
relation of $3j$-symbols [35], one can show an identity

$\sum_{\sigma^{\prime},m_{z}}\left(s\,l_{z}\,l\right)_{\sigma^{\prime}}^{\phantom{\sigma^{\prime}}m_{z}m}C_{l_{z}m_{z}}(\theta,\phi)D^{\sigma^{\prime}}_{(s)\sigma}(\phi,\theta,0)\\\
=(-1)^{\sigma}\begin{pmatrix}s&l_{z}&l\\\
\sigma&0&-\sigma\end{pmatrix}D_{(l)\sigma}^{m}(\phi,\theta,0).$ (118)

Substituting the form of Eq. (116) into Eqs. (90) and (115), and applying the
above identity, we derive a relatively simple result:

$\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r}_{1},\bm{r}_{2})=i^{s_{1}+s_{2}}(-1)^{\sigma_{1}+\sigma_{2}}\sum_{l,m}i^{l}C_{lm}(\hat{\bm{x}})\\\
\times\sum_{l_{1},l_{2},m_{1},m_{2}}\left(l_{1}\,l_{2}\,l\right)_{m_{1}m_{2}}^{\phantom{m_{1}m_{2}}m}D_{(l_{1})\sigma_{1}}^{m_{1}}(\phi_{1},\theta_{1},0)D_{(l_{2})\sigma_{2}}^{m_{2}}(\phi_{2},\theta_{2},0)\\\
\times(-1)^{l_{2}}\tilde{\Xi}^{s_{1}s_{2};l_{1}l_{2}l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(x).$
(119)

where

$\tilde{\Xi}^{s_{1}s_{2};l_{1}l_{2}l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(x)\equiv\sum_{l_{z1},l_{z2}}\begin{pmatrix}s_{1}&l_{z1}&l_{1}\\\
\sigma_{1}&0&-\sigma_{1}\end{pmatrix}\begin{pmatrix}s_{2}&l_{z2}&l_{2}\\\
\sigma_{2}&0&-\sigma_{2}\end{pmatrix}\Xi^{s_{1}s_{2};l_{1}l_{2}l}_{X_{1}X_{2}l_{z1}l_{z2}}(x),$
(120)

and $x=|\bm{r}_{1}-\bm{r}_{2}|$.

As in the previous subsection, one can choose a coordinates system
$(\theta_{1},\phi_{1})=(\theta,0)$ and $(\theta_{2},\phi_{2})=(0,0)$ for each
pair of objects. In this case, the above equation reduces to

$\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2},\theta)=i^{s_{1}+s_{2}}(-1)^{\sigma_{1}}\sum_{l,m}i^{l}\tilde{P}_{l}^{m}(\mu_{2})\\\
\times\sum_{l_{1},l_{2}}u^{(l_{1})}_{\sigma_{1},\sigma_{2}-m}(\theta)(-1)^{l_{2}}\begin{pmatrix}l_{1}&l_{2}&l\\\
m-\sigma_{2}&\sigma_{2}&-m\end{pmatrix}\tilde{\Xi}^{s_{1}s_{2};l_{1}l_{2}l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(x),$
(121)

where

$x=\sqrt{{r_{1}}^{2}+{r_{2}}^{2}-2r_{1}r_{2}\cos\theta},\quad\mu_{2}=\frac{r_{1}\cos\theta-
r_{2}}{x}.$ (122)

Geometrically, the variable $\mu_{2}$ corresponds to the cosine of the angle
between $\bm{r}_{2}$ and $\bm{x}$. The expression of Eq. (121) is useful to
numerically evaluate the correlation function from the perturbation theory, as
demonstrated in the next section.

###  II.7 Correspondences to the distant-observer limit

In Papers I–III, the statistics of tensor fields are considered in the flat-
sky and distant-observer limits. These previous results should be recovered by
taking the corresponding limits of the full-sky formulation defined above. In
Paper III, the correspondence is already shown for projected tensor fields.
One can generalize the derivation to the general case without projection onto
the sky, as outlined below.

####  II.7.1 The power spectrum

First we consider correspondences between power spectra of distant-observer
limit and those of full-sky formulation. It is convenient to define a weighted
sum of the multipole expansion in spherical coordinates by

$\tilde{g}_{Xs\sigma}(\bar{\bm{l}},r)\equiv\frac{2\pi}{\bar{l}}\sum_{m}i^{m}e^{im\phi_{l}}\tilde{F}^{lm}_{Xs\sigma}(r),$
(123)

where $\phi_{l}$ is the azimuthal angle of the two-dimensional vector
$\bar{\bm{l}}$, and

$\bar{l}\equiv
l+\frac{1}{2},\quad\bar{\bm{l}}\equiv\left(\bar{l}\cos\phi_{l},\bar{l}\sin\phi_{l},0\right).$
(124)

The inverse relation of Eq. (123) is given by

$\tilde{F}^{lm}_{Xs\sigma}(r)=(-i)^{m}\frac{\bar{l}}{2\pi}\int\frac{d\phi_{l}}{2\pi}e^{-im\phi_{l}}\tilde{g}_{Xs\sigma}(\bar{\bm{l}},r).$
(125)

In the flat-sky and distant-observer limits, we consider the vicinity of
$\bm{r}\approx(0,0,r_{0})$, where $r_{0}$ is much larger than the clustering
scales we are interested in. Substituting the above equation into Eq. (50) in
the limits, and following a derivation similar to the one in Paper III, we
derive

$\tilde{F}_{Xs\sigma}(\bm{r})\approx(-i)^{\sigma}e^{-i\sigma\phi}\int\frac{d^{2}\bar{l}}{(2\pi)^{2}}e^{i\bar{\bm{l}}\cdot\bm{\theta}}e^{i\sigma\phi_{l}}\tilde{g}_{Xs\sigma}(\bar{\bm{l}},r),$
(126)

where

$\bm{r}=\left(r_{0}\theta\cos\phi,r_{0}\theta\sin\phi,r\right),\quad\bm{\theta}=\left(\theta\cos\phi,\theta\sin\phi,0\right).$
(127)

The variables $\theta$ and $\phi$ correspond to the spherical coordinates of
position vector $\bm{r}$ in the approximation, in which we have

$\theta\ll 1,\quad l\gg 1.$ (128)

In the same limit, the two-point correlation of
$\tilde{g}_{Xs\sigma}(\bar{\bm{l}},r)$ is derived by following exactly the
same manner with Paper III, resulting in

$\left\langle\tilde{g}_{X_{1}s_{1}\sigma_{1}}(\bar{\bm{l}},r)\tilde{g}_{X_{2}s_{2}\sigma_{2}}(\bar{\bm{l}}^{\prime},r^{\prime})\right\rangle\approx\frac{2\pi}{\bar{l}}(2\pi)^{2}\delta_{\mathrm{D}}^{2}(\bar{\bm{l}}+\bar{\bm{l}}^{\prime})\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r,r^{\prime}).$
(129)

In the limit of $\theta\ll 1$, the rotation matrix of Eq. (29) is given by
$D^{\sigma^{\prime}}_{(s)\sigma}(\phi,0,0)=\delta^{\sigma^{\prime}}_{\sigma}e^{-i\sigma\phi}$,
and thus we have

$\tilde{F}_{Xs\sigma}(\bm{r})\approx e^{-i\sigma\phi}F_{Xs\sigma}(\bm{r}).$
(130)

We define the tensor field with a displaced origin in the distant-observer
approximation,

$\bar{F}_{Xs\sigma}(\bm{x})\equiv
F_{Xs\sigma}(\bm{x}+r_{0}\hat{\mathbf{e}}_{3}).$ (131)

Combining Eqs. (126) and (130) the Fourier transform of Eq. (131) is given by
(note that we use the same symbols of functions for configuration space and
Fourier space)

$\displaystyle\bar{F}_{Xs\sigma}(\bm{k})$ $\displaystyle=\int
d^{3}x\,e^{-i\bm{k}\cdot\bm{x}}\bar{F}_{Xs\sigma}(\bm{x})$
$\displaystyle\approx(-i)^{\sigma}e^{i\sigma\phi_{k}}{r_{0}}^{2}\int
dx\,e^{-ik_{\parallel}x}\tilde{g}_{Xs\sigma}(\bar{\bm{l}},x+r_{0}),$ (132)

where the wavevector $\bm{k}$ in the last expression is parameterized by

$\bm{k}=\left(k_{\perp}\cos\phi_{k},k_{\perp}\sin\phi_{k},k_{\parallel}\right),$
(133)

with

$k_{\perp}\equiv\frac{\bar{l}}{r_{0}},\quad\phi_{k}\equiv\phi_{l}.$ (134)

From Eqs. (129) and (132), we derive

$\left\langle\bar{F}_{X_{1}s_{1}\sigma_{1}}(\bm{k})\bar{F}_{X_{2}s_{2}\sigma_{2}}(\bm{k}^{\prime})\right\rangle\approx(2\pi)^{3}\delta_{\mathrm{D}}^{3}(\bm{k}+\bm{k}^{\prime})P^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{k}),$
(135)

where

$P^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{k})=(-i)^{\sigma_{1}-\sigma_{2}}e^{i(\sigma_{1}+\sigma_{2})\phi_{k}}\\\
\times\frac{2\pi r_{0}}{k_{\perp}}\int
dx\,e^{-ik_{\parallel}x}\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(x+r_{0},r_{0}),$
(136)

and we have applied a relation
$\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(x+r_{0},x^{\prime}+r_{0})\approx\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(x-x^{\prime}+r_{0},r_{0})$
in the distant-observer approximation. The appearance of the delta function in
Eq. (135) shows that the translational invariance of the two-point statistics
is recovered in the Cartesian coordinates of the distant-observer limit. The
Eq. (136) is the correspondence of the full-sky power spectrum
$\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r,r^{\prime})$
with the flat-sky, distant-observer power spectrum
$P^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{k})$ in Cartesian
coordinates. Inverting Eq. (136), the full-sky power spectrum is approximately
given by an integral of flat-sky, distant-observer power spectrum as

$\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})\approx
i^{\sigma_{1}-\sigma_{2}}e^{-i(\sigma_{1}+\sigma_{2})\phi_{k}}\\\
\times\frac{k_{\perp}}{2\pi
r_{0}}\int\frac{dk_{\parallel}}{2\pi}\,e^{ik_{\parallel}(r_{1}-r_{2})}P^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{k}),$
(137)

where $|r_{1}-r_{2}|\ll r_{0}\equiv(r_{1}+r_{2})/2$ and
$l=k_{\perp}r_{0}-1/2\gg 1$.

Substituting Eq. (77) into the complex conjugate of Eq. (136), we derive

$P^{(s_{1}s_{2})\,*}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{k})=(-1)^{\sigma_{1}+\sigma_{2}}P^{(s_{1}s_{2})}_{X_{1}X_{2},-\sigma_{1},-\sigma_{2}}(-\bm{k}).$
(138)

This property of the complex conjugate is consistent with the power spectrum
in the distant-observer approximation described in Paper I. Similarly, the
parity transformation of Eq. (136) is given by

$P^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{k})\xrightarrow{\mathbb{P}}(-1)^{p_{X_{1}}+p_{X_{2}}+\sigma_{1}+\sigma_{2}}e^{2i(\sigma_{1}+\sigma_{2})\phi_{k}}P^{(s_{1}s_{2})}_{X_{1}X_{2},-\sigma_{1},-\sigma_{2}}(\bm{k}).$
(139)

On one hand, the last transformation corresponds to the parity around the
origin of the spherical coordinates. In Paper I, on the other hand, the parity
transformation in the distant-observer approximation is differently defined
and corresponds to the parity around the center of the sample, say,
$\bm{r}=r_{0}\hat{\mathbf{e}}_{3}$. As discussed in Paper III, the parity
transformation around the origin of spherical coordinates corresponds to the
combined transformation of the parity and flipping around the center of the
sample, instead of the origin of the spherical coordinates.

####  II.7.2 The correlation function

Next we consider correspondences between correlation functions of distant-
observer limit and those of full-sky formulation. In the distant-observer
approximation of Papers I–III, the correlation function of irreducible tensor
is defined by the field $F_{Xs\sigma}(\bm{r})$ in Cartesian coordinates. Thus
the correlation function in distant-observer limit corresponds to
$\xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})$ of Eq.
(79). The correlation function in spherical coordinates,
$\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})$ of
Eq. (89) is related to that in Cartesian coordinates by Eq. (90). One can
always choose the coordinates with $(\theta_{1},\phi_{1})=(\theta,0)$ and
$(\theta_{2},\phi_{2})=(0,0)$ as done in Eq. (101), and in this case we have

$\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2},\theta)=\sum_{m^{\prime}}d^{(s_{1})}_{m^{\prime}\sigma_{1}}(\theta)\xi^{(s_{1}s_{2})}_{X_{1}X_{2}m^{\prime}\sigma_{2}}(r_{1},r_{2},\theta).$
(140)

In the distant-observer approximation with $\theta\ll 1$, the Wigner’s
$d$-function is approximated by
$d^{(l)}_{m^{\prime}m}(\theta)\approx\delta_{m^{\prime}m}$, and we have

$\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2},\theta)\approx\xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2},\theta).$
(141)

The above property can also be understood by noting that the spherical basis
$\tilde{\mathbf{e}}^{m}$ of Eq. (14) in spherical coordinates and the basis
$\mathbf{e}^{m}$ of Eq. (3) in Cartesian coordinates are approximately the
same on the $x$-$z$ plane or the plane of $\phi=0$.

The correlation function in the distant-observer limit,
$\bar{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{x})$ is the
3-dimensional Fourier transform of the power spectrum in the same limit,

$\bar{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{x})=\int\frac{d^{3}k}{(2\pi)^{3}}P^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{k}).$
(142)

In Paper I, we have derived an equation to describe the correlation function
in distant-observer approximation when the line of sight is directed to the
$z$-axis, and the result is given by

$\bar{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{x})=i^{s_{1}+s_{2}}\sum_{l,l_{z},s}i^{l}(-1)^{s}\sqrt{2s+1}\begin{pmatrix}s_{1}&s_{2}&s\\\
\sigma_{1}&\sigma_{2}&-\sigma_{12}\end{pmatrix}\\\
\times\begin{pmatrix}s&l_{z}&l\\\
\sigma_{12}&0&-\sigma_{12}\end{pmatrix}C_{l\sigma_{12}}(\hat{\bm{x}})\xi^{s_{1}s_{2};l\,l_{z};s}_{X_{1}X_{2}}(x),$
(143)

where $\sigma_{12}=\sigma_{1}+\sigma_{2}$. Corresponding to the full-sky
correlation function of
$\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2},\theta)$
in the naturally chosen coordinates system, Eq. (101), one considers the
coordinates system with the azimuthal angle of $\bm{x}$ is zero, $\phi_{x}=0$,
and in this case we have

$\bar{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(x,\mu_{x})=i^{s_{1}+s_{2}}\sum_{l,l_{z},s}i^{l}(-1)^{s}\sqrt{2s+1}\begin{pmatrix}s_{1}&s_{2}&s\\\
\sigma_{1}&\sigma_{2}&-\sigma_{12}\end{pmatrix}\\\
\times\begin{pmatrix}s&l_{z}&l\\\
\sigma_{12}&0&-\sigma_{12}\end{pmatrix}\tilde{P}_{l}^{\sigma_{12}}(\mu_{x})\xi^{s_{1}s_{2};l\,l_{z};s}_{X_{1}X_{2}}(x),$
(144)

where $\mu_{x}=\cos\theta_{x}$ is the direction cosine of the vector $\bm{x}$
relative to the line of sight.

In the distant-observer limit of $\theta\rightarrow 0$, we have
$u^{(s_{1})}_{\sigma_{1},\sigma_{2}-m}(\theta)\rightarrow(-1)^{\sigma_{1}}\delta_{\sigma_{1}+\sigma_{2},m}$
in the expression of the correlation function with wide-angle effects, Eq.
(121). Comparing the result of Eq. (121) in this limit with the expression of
Eq. (144), the following correspondence between the invariant functions is
derived:

$\xi^{s_{1}s_{2};l\,l_{z};s}_{X_{1}X_{2}}(x)=(-1)^{l_{z}}(2l_{z}+1)\sqrt{2s+1}\sum_{l_{1},l_{2},l_{z1},l_{z2}}(-1)^{l_{1}}\\\
\times\begin{pmatrix}l_{z1}&l_{z2}&l_{z}\\\
0&0&0\end{pmatrix}\begin{Bmatrix}s_{1}&s_{2}&s\\\ l_{z1}&l_{z2}&l_{z}\\\
l_{1}&l_{2}&l\end{Bmatrix}\Xi^{s_{1}s_{2};l_{1}l_{2}l}_{X_{1}X_{2}l_{z1}l_{z2}}(x)$
(145)

in the distant-observer limit. Otherwise the expression of Eq. (121) is
inconsistent with the expression of Eq. (144) in the distant-observer limit.
In the next section, the above correspondence is explicitly shown to hold in
linear theory.

##  III Evaluation from the integrated perturbation theory

###  III.1 Applying the perturbation theory to full-sky statistics

A plausible property of the tensor fields in Cartesian coordinates is the
statistical homogeneity of the field in real space, because the directions of
the spherical basis in Cartesian coordinates, $\mathbf{e}^{m}$, are fixed
throughout the whole space, while those in spherical coordinates,
$\tilde{\mathbf{e}}^{m}$, vary and depend on the angular position
$(\theta,\phi)$ in space. However, in redshift space, the tensor fields are
not statistically homogeneous, because the statistical properties of spatial
clustering are distorted along the lines of sight, which directions depend on
angular positions of objects. In Papers I–III, where the distant-observer
approximation is applied, the direction to the line of sight is fixed in
Cartesian coordinates, and therefore the tensor fields are still homogeneous
in space, even though they are anisotropic due to the redshift-space
distortions.

To reconcile the redshift-space distortions even in the full-sky case, we
consider virtually homogeneous tensor fields
$F_{Xs\sigma}(\bm{r};\hat{\bm{z}})$ with a fictitious line-of-sight direction,
$\hat{\bm{z}}$, which is fixed throughout the space. In reality, the line-of-
sight direction $\hat{\bm{z}}$ must be the same as the radial direction
$\hat{\bm{r}}$ from the observer, and observable tensor fields correspond to a
special case of the virtual field with a constraint
$\hat{\bm{z}}=\hat{\bm{r}}$, i.e.,

$F_{Xs\sigma}(\bm{r})=F_{Xs\sigma}(\bm{r};\hat{\bm{r}}).$ (146)

As we explicitly show below, the introduction of the virtually homogeneous
field with a fixed line-of-sight direction turns out to be convenient in
applying the perturbation theory to derive the theoretical predictions of the
clustering in redshift space. In real space, such introduction of the
virtually homogeneous field is not necessary.

We consider the Fourier transform of the virtually homogeneous field in
redshift space,

$\displaystyle F_{Xs\sigma}(\bm{r};\hat{\bm{z}})$
$\displaystyle=\int\frac{d^{3}k}{(2\pi)^{3}}e^{i\bm{k}\cdot\bm{r}}F_{Xs\sigma}(\bm{k};\hat{\bm{z}}),$
(147) $\displaystyle F_{Xs\sigma}(\bm{k};\hat{\bm{z}})$ $\displaystyle=\int
d^{3}r\,e^{-i\bm{k}\cdot\bm{r}}F_{Xs\sigma}(\bm{r};\hat{\bm{z}}).$ (148)

The field $F_{Xs\sigma}(\bm{k};\hat{\bm{z}})$ is the Fourier transform and not
the same as the function $F_{Xs\sigma}(\bm{r};\hat{\bm{z}})$ in configuration
space, as we use the same symbols in both configuration and Fourier spaces for
simplicity. In the above Eqs. (147) and (148), the line-of-sight direction
$\hat{\bm{z}}$ is temporarily independent to the position $\bm{r}$.

The two-point correlation function of the virtual fields with different lines
of sight in configuration space is given by

$\left\langle
F_{X_{1}s_{1}\sigma_{1}}(\bm{r}_{1};\hat{\bm{z}}_{1})F_{X_{2}s_{2}\sigma_{2}}(\bm{r}_{2};\hat{\bm{z}}_{2})\right\rangle=\Xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r}_{1}-\bm{r}_{2};\hat{\bm{z}}_{1},\hat{\bm{z}}_{2}).$
(149)

That is, the correlation function of virtual fields with fixed lines of sight
only depends on the relative position $\bm{r}_{1}-\bm{r}_{2}$ due to the
homogeneity by construction, while the function explicitly depends on the two
different lines of sight, $\hat{\bm{z}}_{1}$ and $\hat{\bm{z}}_{2}$. The power
spectrum of the virtual fields in the Fourier space is given by

$\left\langle
F_{X_{1}s_{1}\sigma_{1}}(\bm{k}_{1};\hat{\bm{z}}_{1})F_{X_{2}s_{2}\sigma_{2}}(\bm{k}_{2};\hat{\bm{z}}_{2})\right\rangle\\\
=(2\pi)^{3}\delta_{\mathrm{D}}^{3}(\bm{k}_{1}+\bm{k}_{2})P^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{k}_{1};\hat{\bm{z}}_{1},\hat{\bm{z}}_{2}),$
(150)

where the Dirac’s delta function appears due to the homogeneity of the virtual
fields mentioned above. The power spectrum and the correlation function with
virtually fixed two lines of sight are related by the Fourier transform,

$\Xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r};\hat{\bm{z}}_{1},\hat{\bm{z}}_{2})=\int\frac{d^{3}k}{(2\pi)^{3}}e^{i\bm{k}\cdot\bm{r}}P^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{k};\hat{\bm{z}}_{1},\hat{\bm{z}}_{2}).$
(151)

As we show below, the power spectrum
$P^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{k};\hat{\bm{z}}_{1},\hat{\bm{z}}_{2})$
is a natural quantity to be be theoretically predicted by the perturbation
theory.

Once the last power spectrum is analytically evaluated by the perturbation
theory, the correlation function
$\Xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r};\hat{\bm{z}}_{1},\hat{\bm{z}}_{2})$
of Eq. (151) is obtained, and from that, the correlation function of Eq. (79)
is given by Eq. (115). Substituting the result obtained in the above way into
Eqs. (83) or (84), the expression of the invariant power spectrum
$C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}\,l}(r_{1},r_{2})$ is analytically
evaluated by the perturbation theory. The power spectrum
$\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})$
and the correlation function
$\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2},\theta)$
in spherical coordinates are finally evaluated by Eqs. (75) or (101),
respectively. The starting point of the above procedure is to evaluate the
power spectrum
$P^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{k};\hat{\bm{z}}_{1},\hat{\bm{z}}_{2})$,
to which we apply the iPT in the following.

###  III.2  Invariant power spectrum in Fourier space

In Paper I–III, the line of sight is fixed to a single direction
$\hat{\bm{z}}$ according to the distant-observer approximation. While we do
not assume the distant-observer approximation in this paper, the methods in
previous papers can be generalized to include the possibility of having
multiple numbers of line-of-sight directions. In terms of the iPT, we just use
the propagators in redshift space (Paper I),
$\Gamma^{(n)}_{Xs\sigma}(\bm{k}_{1},\ldots,\bm{k}_{n};\hat{\bm{z}}_{i})$, with
$i=1,2$ in evaluating the power spectrum of Eq. (150). In the course of
applying the iPT in this generalized situation, it is convenient to decompose
the direction dependencies of the arguments in the power spectrum of Eq. (150)
by polypolar spherical harmonics, which are extensively used in Paper I.

Following the same kind of calculations illustrated in Paper I, the
decomposition is given by

$P^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{k};\hat{\bm{z}}_{1},\hat{\bm{z}}_{2})=i^{s_{1}+s_{2}}\sum_{s,\sigma}(-1)^{s}\sqrt{2s+1}\left(s_{1}\,s_{2}\,s\right)_{\sigma_{1}\sigma_{2}}^{\phantom{\sigma_{1}\sigma_{2}}\sigma}\\\
\times\sum_{l,l_{z},l_{z1},l_{z2}}X^{l\,l_{z1}l_{z2}}_{l_{z};s\sigma}(\hat{\bm{k}},\hat{\bm{z}}_{1},\hat{\bm{z}}_{2})P^{s_{1}s_{2};s\,l_{z}}_{X_{1}X_{2};l\,l_{z1}l_{z2}}(k),$
(152)

where $X^{l_{1}l_{2}l_{3}}_{L;lm}(\bm{n}_{1},\bm{n}_{2},\bm{n}_{3})$ is the
tripolar spherical harmonics and the explicit definition is given in Paper I.
The last factor corresponds to the invariant spectrum. From the orthonormality
relation of the tripolar spherical harmonics and the orthogonality relation of
the $3j$-symbols (Paper I), the above expansion is unique, i.e., the invariant
spectrum is uniquely determined by the power spectrum on the LHS.

Substituting the expansion of Eq. (152) into Eq. (151), using the Rayleigh
expansion formula for the factor $e^{i\bm{k}\cdot\bm{r}}$, and various
formulas regarding $3j$-symbols and polypolar spherical harmonics (listed in
Appendices of Paper I), Eq. (115) eventually reduces to

$\xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{r}_{1},\bm{r}_{2})=i^{s_{1}+s_{2}}\sum_{s,\sigma}(-1)^{s}\sqrt{\\{s\\}}\left(s_{1}\,s_{2}\,s\right)_{\sigma_{1}\sigma_{2}}^{\phantom{\sigma_{1}\sigma_{2}}\sigma}\\\
\times\sum_{\begin{subarray}{c}l,l_{z},l_{z1},l_{z2}\\\
l_{1},l_{2},l_{1}^{\prime},l_{2}^{\prime}\end{subarray}}i^{l_{1}^{\prime}-l_{2}^{\prime}}(-1)^{l_{1}+l_{2}+l}\\{l_{1}\\}\\{l_{2}\\}\\{l_{1}^{\prime}\\}\\{l_{2}^{\prime}\\}\sqrt{\\{l_{z}\\}}\begin{pmatrix}l_{1}^{\prime}&l_{2}^{\prime}&l\\\
0&0&0\end{pmatrix}\\\ \times\begin{pmatrix}l_{1}&l_{z1}&l_{1}^{\prime}\\\
0&0&0\end{pmatrix}\begin{pmatrix}l_{2}&l_{z2}&l_{2}^{\prime}\\\
0&0&0\end{pmatrix}\begin{Bmatrix}l_{1}&l_{2}&s\\\
l_{1}^{\prime}&l_{2}^{\prime}&l\\\
l_{z1}&l_{z2}&l_{z}\end{Bmatrix}X^{l_{1}l_{2}}_{s\sigma}(\hat{\bm{r}}_{1},\hat{\bm{r}}_{2})\\\
\times\int\frac{k^{2}dk}{2\pi^{2}}j_{l_{1}^{\prime}}(kr_{1})j_{l_{2}^{\prime}}(kr_{2})P^{s_{1}s_{2};s\,l_{z}}_{X_{1}X_{2};l\,l_{z1}l_{z2}}(k),$
(153)

where the $3\times 3$ matrix with curly brackets is the $9j$-symbol, and we
use simple notations introduced in Paper I,

$\left\\{s\right\\}\equiv 2s+1,\quad\left\\{l_{1}\right\\}\equiv 2l_{1}+1,$
(154)

and so forth. Substituting the above result into Eq. (84), we have

$C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}\,l}(r_{1},r_{2})=(-1)^{s_{2}}(-i)^{l_{1}-l_{2}}\\{l\\}\\{l_{1}\\}\\{l_{2}\\}\sum_{s}(-1)^{s}\sqrt{\\{s\\}}\\\
\times\begin{Bmatrix}s_{1}&s_{2}&s\\\
l_{2}&l_{1}&l\end{Bmatrix}\sum_{\begin{subarray}{c}l^{\prime},l_{1}^{\prime},l_{2}^{\prime}\\\
l_{z},l_{z1},l_{z2}\end{subarray}}(-i)^{l_{1}^{\prime}-l_{2}^{\prime}}\\{l_{1}^{\prime}\\}\\{l_{2}^{\prime}\\}\sqrt{\\{l_{z}\\}}\begin{pmatrix}l_{1}^{\prime}&l_{2}^{\prime}&l^{\prime}\\\
0&0&0\end{pmatrix}\\\ \times\begin{pmatrix}l_{1}&l_{z1}&l_{1}^{\prime}\\\
0&0&0\end{pmatrix}\begin{pmatrix}l_{2}&l_{z2}&l_{2}^{\prime}\\\
0&0&0\end{pmatrix}\begin{Bmatrix}l_{1}&l_{2}&s\\\
l_{1}^{\prime}&l_{2}^{\prime}&l^{\prime}\\\
l_{z1}&l_{z2}&l_{z}\end{Bmatrix}\\\
\times\int\frac{k^{2}dk}{2\pi^{2}}j_{l_{1}^{\prime}}(kr_{1})j_{l_{2}^{\prime}}(kr_{2})P^{s_{1}s_{2};s\,l_{z}}_{X_{1}X_{2};l^{\prime}l_{z1}l_{z2}}(k).$
(155)

Once the LHS of Eq. (152) is evaluated by the perturbation theory, the
invariant function $P^{s_{1}s_{2};s\,l_{z}}_{X_{1}X_{2};l\,l_{z1}l_{z2}}(k)$
is uniquely determined, and finally the invariant spectrum
$C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(r_{1},r_{2})$ in the perturbation
theory is evaluated by the above equation.

###  III.3 The linear power spectra

In the lowest-order approximation, or in linear theory of gravitational
evolution, the power spectrum of the iPT is given by [23]

$P^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{k};\hat{\bm{z}}_{1},\hat{\bm{z}}_{2})=i^{s_{1}+s_{2}}\Gamma^{(1)}_{X_{1}s_{1}\sigma_{1}}(\bm{k};\hat{\bm{z}}_{1})\Gamma^{(1)}_{X_{2}s_{2}\sigma_{2}}(-\bm{k};\hat{\bm{z}}_{2})P_{\mathrm{L}}(k),$
(156)

where $P_{\mathrm{L}}(k)$ is the linear matter power spectrum, and
$\Gamma^{(1)}_{Xs\sigma}(\bm{k},\hat{\bm{z}})$ is the first-order propagator
of the irreducible tensor field in redshift space with a line of sight
$\hat{\bm{z}}$ [1]. Explicitly we have

$\Gamma^{(1)}_{Xs\sigma}(\bm{k},\hat{\bm{z}})=\sum_{l_{z},l}\Gamma^{(1)s\,l_{z}}_{Xl}(k)X^{l_{z}l}_{s\sigma}(\hat{\bm{z}},\hat{\bm{k}})$
(157)

in a form of expansion by bipolar spherical harmonics, where

$\Gamma^{(1)s\,l_{z}}_{Xl}(k)=\delta_{l_{z}0}\delta_{ls}\,c^{(1)}_{Xs}(k)\\\
+\delta_{s0}c^{(0)}_{X}\left[\left(1+\frac{f}{3}\right)\delta_{l_{z}0}\delta_{l0}+\frac{2\sqrt{5}f}{3}\delta_{l_{z}2}\delta_{l2}\right]$
(158)

is the invariant propagator in redshift space, $f=d\ln D(t)/d\ln a(t)$ is the
linear growth rate, $a(t)$, $D(t)$, respectively, are the scale factor and the
linear growth factor, and $c^{(0)}_{X}$, $c^{(1)}_{Xs}(k)$ are invariant
renormalized bias functions of zeroth and first orders, which are defined in
Paper I. In Paper I, we consider the general case that the propagators can
explicitly depend on the direction cosine $\mu=\hat{\bm{k}}\cdot\hat{\bm{z}}$.
In the above, we expand all the direction dependence in the propagator and the
propagator does not depend on $\mu$. We consider literally the lowest-order
perturbations, and do not include the resummation factor $\Pi(k,\mu)$
explained in Paper I.

The Kronecker’s delta symbols in Eq. (158) ensure that the indices always
satisfy $l=s+l_{z}$. Defining
$G_{Xs\,l_{z}}(k)\equiv\Gamma^{(1)s\,l_{z}}_{X\,s+l_{z}}(k)$, Eq. (158)
reduces to

$\Gamma^{(1)s\,l_{z}}_{Xl}(k)=\delta_{l,s+l_{z}}G_{Xs\,l_{z}}(k),$ (159)

where

$G_{Xs\,l_{z}}(k)=\delta_{l_{z}0}\,c^{(1)}_{Xs}(k)+\delta_{s0}c^{(0)}_{X}\left[\left(1+\frac{f}{3}\right)\delta_{l_{z}0}+\frac{2\sqrt{5}f}{3}\delta_{l_{z}2}\right].$
(160)

The index $l_{z}$ above takes only $0$ and $2$, and is an even number.

Substituting Eq. (157) into Eq. (156), and applying formulas of $3j$\- and
$9j$-symbols, products spherical harmonics, etc. (see, e.g., Appendices of
Paper I), the direction dependencies of the power spectrum of Eq. (156) are
expanded in a form of Eq. (152), from which the invariant power spectrum can
be read off. The result is given by

$P^{s_{1}s_{1};s\,l_{z}}_{X_{1}X_{2};l\,l_{z1}l_{z2}}(k)=(-1)^{s+l_{z}+s_{2}}\\{l\\}\sqrt{\\{s\\}\\{l_{z}\\}}\\\
\times\begin{pmatrix}s_{1}+l_{z1}&s_{2}+l_{z2}&l\\\
0&0&0\end{pmatrix}\begin{Bmatrix}s_{1}&s_{2}&s\\\ l_{z1}&l_{z2}&l_{z}\\\
s_{1}+l_{z1}&s_{2}+l_{z2}&l\end{Bmatrix}\\\ \times
G_{X_{1}s_{1}l_{z1}}(k)G_{X_{2}s_{2}l_{z2}}(k)P_{\mathrm{L}}(k).$ (161)

Substituting the above equation into Eq. (155), and using various kinds of sum
rules involving $3j$-, $6j$-, and $9j$-symbols [36], one can show that the
invariant spectrum of linear theory finally reduces to

$C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}\,l}(r_{1},r_{2})=(-i)^{l_{1}+l_{2}}(-1)^{s_{1}+s_{2}+l}\\{l\\}\\{l_{1}\\}\\{l_{2}\\}\\\
\times\sum_{l_{1}^{\prime},l_{2}^{\prime},l_{z1},l_{z2}}(-i)^{l_{1}^{\prime}+l_{2}^{\prime}}\\{l_{1}^{\prime}\\}\\{l_{2}^{\prime}\\}\begin{pmatrix}l_{1}&l_{z1}&l_{1}^{\prime}\\\
0&0&0\end{pmatrix}\begin{pmatrix}l_{2}&l_{z2}&l_{2}^{\prime}\\\
0&0&0\end{pmatrix}\\\ \times\begin{pmatrix}l&l_{1}^{\prime}&s_{1}+l_{z1}\\\
0&0&0\end{pmatrix}\begin{pmatrix}l&l_{2}^{\prime}&s_{2}+l_{z2}\\\
0&0&0\end{pmatrix}\\\ \times\begin{Bmatrix}s_{1}&l_{z1}&s_{1}+l_{z1}\\\
l_{1}^{\prime}&l&l_{1}\end{Bmatrix}\begin{Bmatrix}s_{2}&l_{z2}&s_{2}+l_{z2}\\\
l_{2}^{\prime}&l&l_{2}\end{Bmatrix}\\\
\times\int\frac{k^{2}dk}{2\pi^{2}}j_{l_{1}^{\prime}}(kr_{1})j_{l_{2}^{\prime}}(kr_{2})G_{X_{1}s_{1}l_{z1}}(k)G_{X_{2}s_{2}l_{z2}}(k)P_{\mathrm{L}}(k).$
(162)

With the last equation, the power spectra
$C^{(s_{1}s_{2})l_{1}l_{2};m_{1}m_{2}}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})$,
$\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})$,
and correlation functions
$\xi^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2},\theta)$,
$\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2},\theta)$
are predicted by Eqs. (68), (75), (87) and (101). Among them, the multipole
power spectrum in spherical coordinates,
$\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})$,
reduces to a form without $6j$-symbols. Applying the sum rules involving
$3j$\- and $6j$-symbols [36], the result is given by

$\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})=(-i)^{s_{1}+s_{2}}(-1)^{l}\\{l\\}\sum_{l_{1},l_{2},l_{z1},l_{z2}}(-i)^{l_{1}+l_{2}}\\{l_{1}\\}\\{l_{2}\\}\\\
\times\begin{bmatrix}l&l_{1}&s_{1}+l_{z1}\\\
\sigma_{1}&0&-\sigma_{1}\end{bmatrix}\begin{bmatrix}l&l_{2}&s_{2}+l_{z2}\\\
\sigma_{2}&0&-\sigma_{2}\end{bmatrix}\\\
\times\begin{pmatrix}s_{1}&l_{z1}&s_{1}+l_{z1}\\\
\sigma_{1}&0&-\sigma_{1}\end{pmatrix}\begin{pmatrix}s_{2}&l_{z2}&s_{2}+l_{z2}\\\
\sigma_{2}&0&-\sigma_{2}\end{pmatrix}\\\
\times\int\frac{k^{2}dk}{2\pi^{2}}j_{l_{1}}(kr_{1})j_{l_{2}}(kr_{2})G_{X_{1}s_{1}l_{z1}}(k)G_{X_{2}s_{2}l_{z2}}(k)P_{\mathrm{L}}(k).$
(163)

In the above equation, we employ a notation,

$\begin{bmatrix}l_{1}&l_{2}&l_{3}\\\
m_{1}&m_{2}&m_{3}\end{bmatrix}\equiv\begin{pmatrix}l_{1}&l_{2}&l_{3}\\\
0&0&0\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix},$ (164)

which is totally symmetric under permutations of the three columns, and non-
zero only when $l_{1}+l_{2}+l_{3}=\mathrm{even}$. Substituting Eq. (160), the
summations over $l_{z1}$ and $l_{z2}$ are trivially taken. The above results
are derived in redshift space from the beginning of the derivation. The
results in real space are trivially derived by substituting $l_{z}=0$ and
$f=0$ in Eq. (160).

For reader’s convenience, we note below more explicit forms of the results. In
the form of the linear propagator, Eq. (160), it is convenient to separately
consider the two cases, $s=0$ and $s\geq 1$, as the last term on the RHS of
Eq. (160) vanishes when $s\geq 1$. Accordingly it is also convenient to
separately write down the formula of the power spectrum in three cases,
$\tilde{C}^{(00)\,l}_{X_{1}X_{2}00}$, $\tilde{C}^{(s0)\,l}_{X_{1}X_{2}\sigma
0}$ and $\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}$ with
$s,s_{1},s_{2}\geq 1$.

When both $X_{1}$ and $X_{2}$ are scalar fields, we have $s_{1}=s_{2}=0$ in
Eq. (163). The result of substituting Eq. (160) into Eq. (163) in this case is
given by

$\frac{\tilde{C}^{(00)\,l}_{X_{1}X_{2}00}(r_{1},r_{2})}{2l+1}=I^{(00)}_{X_{1}X_{2}ll}(r_{1},r_{2})+\left(1+\frac{f}{3}\right)^{2}c^{(0)}_{X_{1}}c^{(0)}_{X_{2}}I_{ll}(r_{1},r_{2})\\\
+\left(1+\frac{f}{3}\right)c^{(0)}_{X_{1}}I^{(0)}_{X_{2}ll}(r_{1},r_{2})+(1\leftrightarrow
2)\\\
+\frac{2f}{3}c^{(0)}_{X_{1}}\sum_{l_{1}}(-1)^{(l_{1}-l)/2}(2l_{1}+1)\begin{pmatrix}l&l_{1}&2\\\
0&0&0\end{pmatrix}^{2}\\\
\times\left[I^{(0)}_{X_{2}l_{1}l}(r_{1},r_{2})+\left(1+\frac{f}{3}\right)c^{(0)}_{X_{2}}I_{l_{1}l}(r_{1},r_{2})\right]+(1\leftrightarrow
2)\\\
+\frac{4f^{2}}{9}c^{(0)}_{X_{1}}c^{(0)}_{X_{2}}\sum_{l_{1},l_{2}}(-1)^{(l_{1}+l_{2}-2l)/2}(2l_{1}+1)(2l_{2}+1)\\\
\times\begin{pmatrix}l&l_{1}&2\\\
0&0&0\end{pmatrix}^{2}\begin{pmatrix}l&l_{2}&2\\\
0&0&0\end{pmatrix}^{2}I_{l_{1}l_{2}}(r_{1},r_{2}),$ (165)

where $+(1\leftrightarrow 2)$ indicates adding a previous term with
replacements $X_{1}\leftrightarrow X_{2}$, $l_{1}\leftrightarrow l_{2}$ and
$r_{1}\leftrightarrow r_{2}$, and we define integrals,

$\displaystyle I_{l_{1}l_{2}}(r_{1},r_{2})$
$\displaystyle\equiv\int\frac{k^{2}dk}{2\pi^{2}}j_{l_{1}}(kr_{1})j_{l_{2}}(kr_{2})P_{\mathrm{L}}(k),$
(166) $\displaystyle I_{Xl_{1}l_{2}}^{(s)}(r_{1},r_{2})$
$\displaystyle\equiv\int\frac{k^{2}dk}{2\pi^{2}}j_{l_{1}}(kr_{1})j_{l_{2}}(kr_{2})c^{(1)}_{Xs}(k)P_{\mathrm{L}}(k),$
(167) $\displaystyle I_{X_{1}X_{2}l_{1}l_{2}}^{(s_{1}s_{2})}(r_{1},r_{2})$
$\displaystyle\equiv\int\frac{k^{2}dk}{2\pi^{2}}j_{l_{1}}(kr_{1})j_{l_{2}}(kr_{2})$
$\displaystyle\hskip 57.60004pt\times
c^{(1)}_{X_{1}s_{1}}(k)c^{(1)}_{X_{2}s_{2}}(k)P_{\mathrm{L}}(k).$ (168)

When $X_{1}$ is a scalar field and $X_{2}$ is a non-scalar field, we have
$s_{1}=0$ and $s_{2}=s$ where $s\geq 1$ in Eq. (163). The explicit form in
this case is given by

$\frac{\tilde{C}^{(s0)\,l}_{X_{1}X_{2}\sigma
0}(r_{1},r_{2})}{2l+1}=\frac{(-1)^{\sigma}i^{s}}{\sqrt{2s+1}}\sum_{l_{1}}(-i)^{l_{1}}(2l_{1}+1)\begin{bmatrix}s&l_{1}&l\\\
\sigma&0&-\sigma\end{bmatrix}\\\
\times\left[i^{l}I^{(s0)}_{X_{1}X_{2}l_{1}l}(r_{1},r_{2})+i^{l}c^{(0)}_{X_{2}}\left(1+\frac{f}{3}\right)I^{(s)}_{X_{1}l_{1}l}(r_{1},r_{2})\right.\\\
\left.+\frac{2f}{3}c^{(0)}_{X_{2}}\sum_{l_{2}}i^{l_{2}}(2l_{2}+1)\begin{pmatrix}l&l_{2}&2\\\
0&0&0\end{pmatrix}^{2}I^{(s)}_{X_{1}l_{1}l_{2}}(r_{1},r_{2})\right].$ (169)

When both $X_{1}$ and $X_{2}$ are non-scalar fields, we have $s_{1},s_{2}\geq
1$ in Eq. (163). The explicit form in this case is given by

$\frac{\tilde{C}^{(s_{1}s_{2})\,l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})}{2l+1}=\frac{(-1)^{l+\sigma_{1}+\sigma_{2}}i^{s_{1}+s_{2}}}{\sqrt{(2s_{1}+1)(2s_{2}+1)}}\sum_{l_{1},l_{2}}(-i)^{l_{1}+l_{2}}\\\
\times(2l_{1}+1)(2l_{2}+1)\begin{bmatrix}s_{1}&l_{1}&l\\\
\sigma_{1}&0&-\sigma_{1}\end{bmatrix}\begin{bmatrix}s_{2}&l_{2}&l\\\
\sigma_{2}&0&-\sigma_{2}\end{bmatrix}\\\ \times
I^{(s_{1}s_{2})l_{1}l_{2}}_{X_{1}X_{2}}(r_{1},r_{2}).$ (170)

###  III.4 The linear correlation function

The numerical evaluations of the linear correlation function is conveniently
evaluated by applying the expression of Eq. (121), rather than using a formal
relation of Eq. (108) between the correlation function and power spectrum in
spherical coordinates.

Identifying the virtual correlation function in Eq. (149) with that of
redundant function in Eq. (114), the decomposition of the power spectrum is
given by the inverse Fourier transform of Eq. (151) with Eq. (117). Applying
the Rayleigh expansion formula, the following result is derived:

$P^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{k};\hat{\bm{z}}_{1},\hat{\bm{z}}_{2})=i^{s_{1}+s_{2}}\sum_{s,\sigma}(2s+1)\left(s_{1}\,s_{2}\,s\right)_{\sigma_{1}\sigma_{2}}^{\phantom{\sigma_{1}\sigma_{2}}\sigma}\\\
\times\sum_{l,l_{z1},l_{z2},l_{z}}(-1)^{l+l_{z}}\sqrt{2l_{z}+1}\,X^{l\,l_{z1}l_{z2}}_{l_{z};s\sigma}(\hat{\bm{k}},\hat{\bm{z}}_{1},\hat{\bm{z}}_{2})\\\
\times\sum_{l_{1},l_{2}}(-1)^{l_{1}}\begin{Bmatrix}s_{1}&s_{2}&s\\\
l_{z1}&l_{z2}&l_{z}\\\
l_{1}&l_{2}&l\end{Bmatrix}\mathcal{P}^{s_{1}s_{2};l_{1}l_{2}l}_{X_{1}X_{2}l_{z1}l_{z2}}(k),$
(171)

where

$\mathcal{P}^{s_{1}s_{2};l_{1}l_{2}l}_{X_{1}X_{2}l_{z1}l_{z2}}(k)=4\pi\int
x^{2}dx\,j_{l}(kx)\,\Xi^{s_{1}s_{2};l_{1}l_{2}l}_{X_{1}X_{2}l_{z1}l_{z2}}(x).$
(172)

Substituting Eq. (161) into Eq. (152), the resulting expression for the power
spectrum really have the form of Eq. (171) with the invariant spectrum there
is given by

$\mathcal{P}^{s_{1}s_{2};l_{1}l_{2}l}_{X_{1}X_{2}l_{z1}l_{z2}}(k)=\delta_{l_{1},s_{1}+l_{z1}}\delta_{l_{2},s_{2}+l_{z2}}(2l+1)\begin{pmatrix}l_{1}&l_{2}&l\\\
0&0&0\end{pmatrix}\\\ \times
G_{X_{1}s_{1}l_{z1}}(k)G_{X_{2}s_{2}l_{z2}}(k)P_{\mathrm{L}}(k).$ (173)

Applying the inverse Hankel transform of Eq. (172), we have

$\Xi^{s_{1}s_{2};l_{1}l_{2}l}_{X_{1}X_{2}l_{z1}l_{z2}}(x)=\delta_{l_{1},s_{1}+l_{z1}}\delta_{l_{2},s_{2}+l_{z2}}(2l+1)\begin{pmatrix}l_{1}&l_{2}&l\\\
0&0&0\end{pmatrix}\\\
\times\int\frac{k^{2}dk}{2\pi}j_{l}(kx)G_{X_{1}s_{1}l_{z1}}(k)G_{X_{2}s_{2}l_{z2}}(k)P_{\mathrm{L}}(k),$
(174)

and Eq. (120) in this case reduces to

$\tilde{\Xi}^{s_{1}s_{2};l_{1}l_{2}l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(x)=(21+1)\begin{pmatrix}l_{1}&l_{2}&l\\\
0&0&0\end{pmatrix}\\\ \times\begin{pmatrix}s_{1}&l_{1}-s_{1}&l_{1}\\\
\sigma_{1}&0&-\sigma_{1}\end{pmatrix}\begin{pmatrix}s_{2}&l_{2}-s_{2}&l_{2}\\\
\sigma_{2}&0&-\sigma_{2}\end{pmatrix}\\\
\times\int\frac{k^{2}dk}{2\pi}j_{l}(kx)\tilde{G}_{X_{1}s_{1}l_{1}}(k)\tilde{G}_{X_{2}s_{2}l_{2}}(k)P_{\mathrm{L}}(k),$
(175)

where

$\displaystyle\tilde{G}_{Xsl}(k)$ $\displaystyle\equiv G_{Xs,l-s}(k)$
$\displaystyle=\delta_{ls}\,c^{(1)}_{Xs}(k)+\delta_{s0}c^{(0)}_{X}\left[\left(1+\frac{f}{3}\right)\delta_{l0}+\frac{2\sqrt{5}f}{3}\delta_{l2}\right],$
(176)

or we have

$\Gamma^{(1)s\,l_{z}}_{Xl}(k)=\delta_{l_{z},l-s}\tilde{G}_{Xsl}(k).$ (177)

Substituting the above equation into Eqs. (119) and (121), the predictions for
the correlation functions in linear theory are given.

##  IV Projected tensor fields

###  IV.1 The two-dimensionally projected irreducible tensors

In many practical observations, three-dimensional tensors are projected onto
the sky, and in these cases, we only observe projected two-dimensional tensors
of astronomical objects such as galaxy shapes, galaxy angular momenta, etc. In
Paper III, the relation between the irreducible decompositions of three-
dimensional tensors and those of projected two-dimensional tensors are derived
in Cartesian coordinates, assuming the distant-observer limit. The same
relation holds also in spherical coordinates, and we briefly summarize the
result below.

Any symmetric two-dimensional, rank-$s$ tensor $f_{Xi_{1}\cdots i_{s}}$ can be
decomposed into traceless parts $f^{(s)}_{Xi_{1}\cdots i_{s}}$, which is
decomposed by spherical basis on the tangential plane of the sphere as

$f^{(s)}_{Xi_{1}\cdots
i_{s}}=\tilde{f}^{(+s)}_{X}\tilde{\mathrm{m}}^{+}_{i_{1}}\cdots\tilde{\mathrm{m}}^{+}_{i_{s}}+\tilde{f}^{(-s)}_{X}\tilde{\mathrm{m}}^{-}_{i_{1}}\cdots\tilde{\mathrm{m}}^{-}_{i_{s}},$
(178)

where $\tilde{\mathrm{m}}^{\pm}_{i}$ are angular components of spherical basis
$\tilde{\mathbf{e}}^{\pm}$ of Eq. (14), i.e.,
$\tilde{\mathrm{m}}^{\pm}_{i}=\tilde{\mathrm{e}}^{\pm}_{i}$, and
$\tilde{f}^{(\pm s)}_{X}$ are the decomposed irreducible tensors. The
relations between irreducible components of the three-dimensional tensor
fields and those of projected two-dimensional tensor fields are given by
(Paper III)

$\tilde{f}^{(\pm s)}_{X}(\bm{r})=A_{s}\tilde{F}_{Xs,\pm s}(\bm{r}),$ (179)

where $\tilde{F}_{Xs,\pm s}$ on the RHS corresponds to $\tilde{F}_{Xs\sigma}$
of Eq. (28) with a substitution $\sigma=\pm s$. Due to Eq. (58), the complex
conjugate Eq. (179) is given by

$\tilde{f}^{(\pm s)*}_{X}(\bm{r})=(-1)^{s}\tilde{f}^{(\mp s)}_{X}(\bm{r}).$
(180)

Due to Eq. (60), the parity transformation of Eq. (179) is given by

$\tilde{f}^{(\pm s)}_{X}(\bm{r})\xrightarrow{\mathbb{P}}\tilde{f}^{(\pm
s)\prime}_{X}(\bm{r})=(-1)^{p_{X}}\tilde{f}^{(\mp s)}_{X}(-\bm{r}).$ (181)

###  IV.2 The power spectrum of projected tensors

####  IV.2.1 General formulas

The irreducible tensors are expanded by the spin-weighted spherical harmonics
by

$\tilde{f}^{(\pm
s)}_{X}(\bm{r})=\sum_{l,m}\sqrt{\frac{4\pi}{2l+1}}\,\tilde{f}^{(\pm
s)m}_{Xl}(r)\,{}_{\pm s}Y_{lm}(\hat{\bm{r}}).$ (182)

Comparing the expansion with Eq. (50), we have

$\tilde{f}^{(\pm s)m}_{Xl}(r)=A_{s}\tilde{F}^{lm}_{Xs,\pm s}(\bm{r}).$ (183)

It is convenient to define expansion coefficients with lowered azimuthal
indices as

$\tilde{f}^{(\pm
s)}_{Xlm}(r)\equiv\sum_{m^{\prime}}g^{(s)}_{mm^{\prime}}\tilde{f}^{(\pm
s)m^{\prime}}_{Xl}(r)=(-1)^{m}\tilde{f}^{(\pm s)\,-m}_{Xl}(r).$ (184)

The symmetric properties of Eq. (184), such as complex conjugate and parity,
are explicitly given in Paper III.

The two-dimensional irreducible tensors are decomposed into the so-called E
and B modes, which are defined by

$\tilde{f}^{(\pm s)}_{Xlm}(r)=\tilde{f}^{\mathrm{E}(s)}_{Xlm}(r)\pm
i\tilde{f}^{\mathrm{B}(s)}_{Xlm}(r).$ (185)

or equivalently

$\displaystyle\tilde{f}^{\mathrm{E}(s)}_{Xlm}(r)$
$\displaystyle=\frac{1}{2}\left[\tilde{f}^{(+s)}_{Xlm}(r)+\tilde{f}^{(-s)}_{Xlm}(r)\right],$
(186) $\displaystyle\tilde{f}^{\mathrm{B}(s)}_{Xlm}(r)$
$\displaystyle=\frac{1}{2i}\left[\tilde{f}^{(+s)}_{Xlm}(r)-\tilde{f}^{(-s)}_{Xlm}(r)\right].$
(187)

The two-point statistics of E/B modes are given by (Paper III)

$\displaystyle\left\langle\tilde{f}^{\mathrm{E}(s_{1})}_{X_{1}lm}(r_{1})\tilde{f}^{\mathrm{E}(s_{2})}_{X_{2}l^{\prime}m^{\prime}}(r_{2})\right\rangle$
$\displaystyle=\delta_{ll^{\prime}}g^{(l)}_{mm^{\prime}}C^{\mathrm{EE}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2}),$
(188)
$\displaystyle\left\langle\tilde{f}^{\mathrm{B}(s_{1})}_{X_{1}lm}(r_{1})\tilde{f}^{\mathrm{B}(s_{2})}_{X_{2}l^{\prime}m^{\prime}}(r_{2})\right\rangle$
$\displaystyle=\delta_{ll^{\prime}}g^{(l)}_{mm^{\prime}}C^{\mathrm{BB}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2}),$
(189)
$\displaystyle\left\langle\tilde{f}^{\mathrm{E}(s_{1})}_{X_{1}lm}(r_{1})\tilde{f}^{\mathrm{B}(s_{2})}_{X_{2}l^{\prime}m^{\prime}}(r_{2})\right\rangle$
$\displaystyle=\delta_{ll^{\prime}}g^{(l)}_{mm^{\prime}}C^{\mathrm{EB}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2}),$
(190)
$\displaystyle\left\langle\tilde{f}^{\mathrm{B}(s_{1})}_{X_{1}lm}(r_{1})\tilde{f}^{\mathrm{E}(s_{2})}_{X_{2}l^{\prime}m^{\prime}}(r_{2})\right\rangle$
$\displaystyle=\delta_{ll^{\prime}}g^{(l)}_{mm^{\prime}}C^{\mathrm{BE}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2}),$
(191)

where $C^{\mathrm{EE}(s_{1}s_{2})}_{X_{1}X_{2}l}$,
$C^{\mathrm{BB}(s_{1}s_{2})}_{X_{1}X_{2}l}$,
$C^{\mathrm{EB}(s_{1}s_{2})}_{X_{1}X_{2}l}$,
$C^{\mathrm{BE}(s_{1}s_{2})}_{X_{1}X_{2}l}$ are E/B power spectra. The last
spectra are rotationally invariant and real functions. Symmetric properties of
the E/B power spectra are explicitly given in Paper III.

Using the correspondences of Eqs. (183), (186) and (187), the power spectra of
E/B modes of Eqs. (188)–(191) are linearly related to the power spectra of
three-dimensional irreducible tensor fields
$\tilde{C}^{(s_{1}s_{2})l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})$,
defined by Eq. (74). The relations are explicitly given by

$\displaystyle C^{\mathrm{EE}}_{l}$
$\displaystyle=\frac{1}{4}\left[\left(C^{++}_{l}+C^{--}_{l}\right)+\left(C^{+-}_{l}+C^{-+}_{l}\right)\right],$
(192) $\displaystyle C^{\mathrm{BB}}_{l}$
$\displaystyle=-\frac{1}{4}\left[\left(C^{++}_{l}+C^{--}_{l}\right)-\left(C^{+-}_{l}+C^{-+}_{l}\right)\right],$
(193) $\displaystyle C^{\mathrm{EB}}_{l}$
$\displaystyle=\frac{1}{4i}\left[\left(C^{++}_{l}-C^{--}_{l}\right)-\left(C^{+-}_{l}-C^{-+}_{l}\right)\right],$
(194) $\displaystyle C^{\mathrm{BE}}_{l}$
$\displaystyle=\frac{1}{4i}\left[\left(C^{++}_{l}-C^{--}_{l}\right)+\left(C^{+-}_{l}-C^{-+}_{l}\right)\right],$
(195)

where abbreviations,

$\displaystyle
C^{\pm\pm}_{l}=A_{s_{1}}A_{s_{2}}\tilde{C}^{(s_{1}s_{2})l}_{X_{1}X_{2},\pm
s_{1},\pm s_{2}}(r_{1},r_{2}),$ (196) $\displaystyle
C^{\pm\mp}_{l}=A_{s_{1}}A_{s_{2}}\tilde{C}^{(s_{1},s_{2})l}_{X_{1}X_{2},\pm
s_{1},\mp s_{2}}(r_{1},r_{2}),$ (197) $\displaystyle
C^{\mathrm{EE}}_{l}=C^{\mathrm{EE}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2}),\quad
C^{\mathrm{BB}}_{l}=C^{\mathrm{BB}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2}),$
(198) $\displaystyle
C^{\mathrm{EB}}_{l}=C^{\mathrm{EB}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2}),\quad
C^{\mathrm{BE}}_{l}=C^{\mathrm{BE}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2}),$
(199)

are adopted just for simplicity of presentation. Combining Eqs. (77), (78) and
(192)–(195), the symmetries of complex conjugate and parity are given by

$C^{\mathrm{EE}*}_{l}=C^{\mathrm{EE}}_{l},\
C^{\mathrm{BB}*}_{l}=C^{\mathrm{BB}}_{l},\
C^{\mathrm{EB}*}_{l}=C^{\mathrm{EB}}_{l},\
C^{\mathrm{BE}*}_{l}=C^{\mathrm{BE}}_{l},$ (200)

i.e., they are all real functions, and

$\displaystyle C^{\mathrm{EE}}_{l}$
$\displaystyle\xrightarrow{\mathbb{P}}(-1)^{p_{X_{1}}+p_{X_{2}}}C^{\mathrm{EE}}_{l},$
(201) $\displaystyle C^{\mathrm{BB}}_{l}$
$\displaystyle\xrightarrow{\mathbb{P}}(-1)^{p_{X_{1}}+p_{X_{2}}}C^{\mathrm{BB}}_{l}.$
(202) $\displaystyle C^{\mathrm{EB}}_{l}$
$\displaystyle\xrightarrow{\mathbb{P}}(-1)^{p_{X_{1}}+p_{X_{2}}+1}C^{\mathrm{EB}}_{l},$
(203) $\displaystyle C^{\mathrm{BE}}_{l}$
$\displaystyle\xrightarrow{\mathbb{P}}(-1)^{p_{X_{1}}+p_{X_{2}}+1}C^{\mathrm{BE}}_{l}.$
(204)

Substituting Eq. (75) into Eqs. (192)–(195), we derive

$\displaystyle C^{\mathrm{EE}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2})$
$\displaystyle=\sum_{\begin{subarray}{c}l_{1},l_{2}\\\
l_{1}-s_{1}+l=\mathrm{even}\\\
l_{2}-s_{2}+l=\mathrm{even}\end{subarray}}(-1)^{(l_{1}+l_{2}-s_{1}-s_{2})/2}\hat{C}^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(r_{1},r_{2}),$
(205) $\displaystyle C^{\mathrm{BB}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2})$
$\displaystyle=\sum_{\begin{subarray}{c}l_{1},l_{2}\\\
l_{1}-s_{1}+l=\mathrm{odd}\\\
l_{2}-s_{2}+l=\mathrm{odd}\end{subarray}}(-1)^{(l_{1}+l_{2}-s_{1}-s_{2}-2)/2}\hat{C}^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(r_{1},r_{2}),$
(206) $\displaystyle C^{\mathrm{EB}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2})$
$\displaystyle=\sum_{\begin{subarray}{c}l_{1},l_{2}\\\
l_{1}-s_{1}+l=\mathrm{even}\\\
l_{2}-s_{2}+l=\mathrm{odd}\end{subarray}}(-1)^{(l_{1}+l_{2}-s_{1}-s_{2}-1)/2}\hat{C}^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(r_{1},r_{2}),$
(207) $\displaystyle C^{\mathrm{BE}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2})$
$\displaystyle=\sum_{\begin{subarray}{c}l_{1},l_{2}\\\
l_{1}-s_{1}+l=\mathrm{odd}\\\
l_{2}-s_{2}+l=\mathrm{even}\end{subarray}}(-1)^{(l_{1}+l_{2}-s_{1}-s_{2}-1)/2}\hat{C}^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(r_{1},r_{2}),$
(208)

where

$\hat{C}^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(r_{1},r_{2})\equiv
A_{s_{1}}A_{s_{2}}\begin{pmatrix}s_{1}&l_{1}&l\\\
s_{1}&0&-s_{1}\end{pmatrix}\begin{pmatrix}s_{2}&l_{2}&l\\\
s_{2}&0&-s_{2}\end{pmatrix}\\\ \times
C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(r_{1},r_{2}).$ (209)

The above results explicitly give the formula of power spectra of projected
E/B modes represented in terms of the invariant spectrum of three-dimensional
tensor fields.

####  IV.2.2 Relation to the power spectrum in the distant-observer limit

The power spectrum $P^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(\bm{k})$
of the tensor fields in the distant-observer limit, defined by Eq. (135), is
related to the full-sky power spectrum
$\tilde{C}^{(s_{1}s_{2})l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(r_{1},r_{2})$ by
Eqs. (136) and (137). These relations are transformed to the relations between
the power spectra of E/B modes, where the power spectra of E/B modes are
defined in Paper III. The calculations are straightforward, and the results
are naturally given by

$P^{\mathrm{EE/BB/EB/BE}(s_{1}s_{2})}_{X_{1}X_{2}}(\bm{k})\\\
\approx\frac{2\pi r_{0}}{k_{\perp}}\int
dx\,e^{-ik_{\parallel}x}\tilde{C}^{\mathrm{EE/BB/EB/BE}(s_{1}s_{2})}_{X_{1}X_{2}l}(x+r_{0},r_{0}),$
(210)

and

$C^{\mathrm{EE/BB/EB/BE}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2})\\\
\approx\frac{k_{\perp}}{2\pi
r_{0}}\int\frac{dk_{\parallel}}{2\pi}\,e^{ik_{\parallel}(r_{1}-r_{2})}P^{\mathrm{EE/BB/EB/BE}(s_{1}s_{2})}_{X_{1}X_{2}}(\bm{k}),$
(211)

where $|r_{1}-r_{2}|\ll r_{0}\equiv(r_{1}+r_{2})/2$ and
$l=k_{\perp}r_{0}-1/2\gg 1$. In a later section, we use the above
correspondences to check the validity of the distant-observer approximation in
the power spectrum. While the power spectra $C^{\mathrm{EE/BB/EB/BE}}$ in
spherical coordinates are all real functions as noted above,
$P^{\mathrm{EE/BB}}$ are real and $P^{\mathrm{EB/BE}}$ are pure imaginary as
shown in Paper III. Therefore, the exponential factors in the integrals of
Eqs. (210) and (211) can be replaced by
$e^{-ik_{\parallel}x}\rightarrow\cos(k_{\parallel}x)$ and
$e^{ik_{\parallel}(r_{1}-r_{2})}\rightarrow\cos[k_{\parallel}(r_{1}-r_{2})]$,
respectively for EE/BB power spectra, while they can be replaced by
$e^{-ik_{\parallel}x}\rightarrow-i\sin(k_{\parallel}x)$ and
$e^{ik_{\parallel}(r_{1}-r_{2})}\rightarrow i\sin[k_{\parallel}(r_{1}-r_{2})]$
for EB/BE power spectra.

####  IV.2.3 The linear power spectrum

In the lowest-order approximation with iPT, the power spectra appeared in Eqs.
(196) and (197) are given by Eq. (163) with $(\sigma_{1},\sigma_{2})=(\pm
s_{1},\pm s_{2})$ and $(\pm s_{1},\mp s_{2})$, respectively. Substituting the
expression of the linear theory into Eqs. (192)–(193), we have

$\displaystyle C^{\mathrm{EE}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2})$
$\displaystyle=\sum_{\begin{subarray}{c}l_{1},l_{2}\\\
l_{1}-s_{1}+l=\mathrm{even}\\\
l_{2}-s_{2}+l=\mathrm{even}\end{subarray}}(-1)^{(l_{1}+l_{2}-s_{1}-s_{2})/2}E^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(r_{1},r_{2}),$
(212) $\displaystyle C^{\mathrm{BB}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2})$
$\displaystyle=\sum_{\begin{subarray}{c}l_{1},l_{2}\\\
l_{1}-s_{1}+l=\mathrm{odd}\\\
l_{2}-s_{2}+l=\mathrm{odd}\end{subarray}}(-1)^{(l_{1}+l_{2}-s_{1}-s_{2}-2)/2}E^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(r_{1},r_{2}),$
(213) $\displaystyle C^{\mathrm{EB}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2})$
$\displaystyle=\sum_{\begin{subarray}{c}l_{1},l_{2}\\\
l_{1}-s_{1}+l=\mathrm{even}\\\
l_{2}-s_{2}+l=\mathrm{odd}\end{subarray}}(-1)^{(l_{1}+l_{2}-s_{1}-s_{2}-1)/2}E^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(r_{1},r_{2}),$
(214) $\displaystyle C^{\mathrm{BE}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2})$
$\displaystyle=\sum_{\begin{subarray}{c}l_{1},l_{2}\\\
l_{1}-s_{1}+l=\mathrm{odd}\\\
l_{2}-s_{2}+l=\mathrm{even}\end{subarray}}(-1)^{(l_{1}+l_{2}-s_{1}-s_{2}-1)/2}E^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(r_{1},r_{2}),$
(215)

where

$E^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(r_{1},r_{2})\equiv
A_{s_{1}}A_{s_{2}}(-1)^{s_{1}+s_{2}+l}(2l+1)(2l_{1}+1)(2l_{2}+1)\\\
\times\sum_{l_{z1},l_{z2}}\begin{bmatrix}l&l_{1}&l_{z1}+s_{1}\\\
s_{1}&0&-s_{1}\end{bmatrix}\begin{bmatrix}l&l_{2}&l_{z2}+s_{2}\\\
s_{2}&0&-s_{2}\end{bmatrix}\\\
\times\begin{pmatrix}s_{1}&l_{z1}&l_{z1}+s_{1}\\\
s_{1}&0&-s_{1}\end{pmatrix}\begin{pmatrix}s_{2}&l_{z2}&l_{z2}+s_{2}\\\
s_{2}&0&-s_{2}\end{pmatrix}\\\
\times\int\frac{k^{2}dk}{2\pi^{2}}j_{l_{1}}(kr_{1})j_{l_{2}}(kr_{2})G_{X_{1}s_{1}l_{z1}}(k)G_{X_{2}s_{2}l_{z2}}(k)P_{\mathrm{L}}(k),$
(216)

and $G_{Xsl_{z}}(k)$ is given by Eq. (160). In deriving the above results, we
use the properties that $l_{z1}$ and $l_{z2}$ are even numbers in Eq. (160).
While the expressions of Eqs. (212)–(215) have the same forms with Eqs.
(205)–(208), the summands $E^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}$ are not
the counterparts of $\hat{C}^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}$ in linear
theory. They give the same results only after the summations over $l_{1}$ and
$l_{2}$ are taken. The above results can also be derived by Eqs. (205)–(209),
where the invariant power spectrum
$C^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(r_{1},r_{2})$ is given by Eq. (162)
in linear theory, applying appropriate sum rules involving $3j$\- and
$6j$-symbols [36].

The summand in the expression of Eq. (216) survives only when
$l_{1}-s_{1}+l=\mathrm{even}$ and $l_{2}-s_{2}+l=\mathrm{even}$, because
$G_{Xsl_{z}}(k)$ is non-zero only when $l_{z}=\mathrm{even}$ and the symmetry
of $3j$-symbols ensures that $l+l_{1}+l_{z1}+s_{1}$ and $l+l_{2}+l_{z2}+s_{2}$
are even numbers in order to have non-zero values. Therefore, BB/EB/BE power
spectra vanish in linear theory:

$C^{\mathrm{BB}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2})=C^{\mathrm{EB}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2})=C^{\mathrm{BE}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2})=0.$
(217)

This property is the general prediction of the iPT in the lowest-order
approximation with renormalized bias functions, and is consistent with the
results derived in Paper III with the distant-observer approximation, in which
the BB/EB/BE power spectra vanish:
$P^{\mathrm{BB}(s_{1}s_{2})}_{X_{1}X_{2}}(\bm{k})=P^{\mathrm{EB}(s_{1}s_{2})}_{X_{1}X_{2}}(\bm{k})=P^{\mathrm{BE}(s_{1}s_{2})}_{X_{1}X_{2}}(\bm{k})=0$,
and only EE power spectrum $P^{\mathrm{BB}(s_{1}s_{2})}_{X_{1}X_{2}}(\bm{k})$
survives in the lowest-order approximation.444When the parity symmetry holds
both in initial conditions and dynamical evolutions, the EB power spectra
always vanish. The results here states more than that without assuming the
parity symmetry, and hold only in linear theory.

Substituting Eqs. (160) and (216) into Eq. (212), the EE power spectrum in
linear theory is explicitly given. When both $X_{1}$ and $X_{2}$ are scalar
fields, $s_{1}=s_{2}=0$, we have

$(2l+1)^{-1}C^{\mathrm{EE}(00)}_{X_{1}X_{2}l}(r_{1},r_{2})=I^{(00)}_{X_{1}X_{2}ll}(r_{1},r_{2})\\\
+\left(1+\frac{f}{3}\right)\left[c^{(0)}_{X_{1}}I^{(0)}_{X_{2}ll}(r_{1},r_{2})+c^{(0)}_{X_{2}}I^{(0)}_{X_{1}ll}(r_{1},r_{2})\right]\\\
+\left(1+\frac{f}{3}\right)^{2}c^{(0)}_{X_{1}}c^{(0)}_{X_{2}}I_{ll}(r_{1},r_{2})\\\
+\frac{2f}{3}\sum_{l^{\prime}}(-1)^{(l^{\prime}-l)/2}(2l^{\prime}+1)\begin{pmatrix}l&l^{\prime}&2\\\
0&0&0\end{pmatrix}^{2}\\\
\times\Biggl{\\{}c^{(0)}_{X_{1}}I^{(0)}_{X_{2}l^{\prime}l}(r_{1},r_{2})+c^{(0)}_{X_{2}}I^{(0)}_{X_{1}ll^{\prime}}(r_{1},r_{2})\\\
\hskip
60.0pt+\left(1+\frac{f}{3}\right)c^{(0)}_{X_{1}}c^{(0)}_{X_{2}}\left[I_{l^{\prime}l}(r_{1},r_{2})+I_{ll^{\prime}}(r_{1},r_{2})\right]\Biggr{\\}}\\\
+\frac{4f^{2}}{9}c^{(0)}_{X_{1}}c^{(0)}_{X_{2}}\sum_{l_{1},l_{2}}(-1)^{(l_{1}+l_{2}-2l)/2}(2l_{1}+1)(2l_{2}+1)\\\
\times\begin{pmatrix}l&l_{1}&2\\\
0&0&0\end{pmatrix}^{2}\begin{pmatrix}l&l_{2}&2\\\
0&0&0\end{pmatrix}^{2}I_{l_{1}l_{2}}(r_{1},r_{2}),$ (218)

where the integrals defined in Eqs. (166)–(168) are employed. When $X_{1}$ is
a non-scalar field and $X_{2}$ is a scalar field, we have $s_{1}=s$ and
$s_{2}=0$ where $s\geq 1$ in Eq. (216). The explicit form in this case is
given by

$C^{\mathrm{EE}(s0)}_{X_{1}X_{2}l}(r_{1},r_{2})=\frac{(2l+1)A_{s}}{\sqrt{2s+1}}\sum_{l_{1}}(-1)^{(l_{1}+s-l)/2}(2l_{1}+1)\\\
\times\begin{bmatrix}s&l_{1}&l\\\
s&0&-s\end{bmatrix}\left[I^{(s0)}_{X_{1}X_{2}l_{1}l}(r_{1},r_{2})+\left(1+\frac{f}{3}\right)c^{(0)}_{X_{2}}I^{(s)}_{X_{1}l_{1}l}(r_{1},r_{2})\right.\\\
\left.+\frac{2f}{3}c^{(0)}_{X_{2}}\sum_{l_{2}}(-1)^{(l_{2}-l)/2}(2l_{2}+1)\begin{pmatrix}l&l_{2}&2\\\
0&0&0\end{pmatrix}^{2}I^{(s)}_{X_{1}l_{1}l_{2}}(r_{1},r_{2})\right].$ (219)

When both $X_{1}$ and $X_{2}$ are non-scalar fields, we have $s_{1},s_{2}\geq
1$ in Eq. (216). The explicit form in this case is given by

$C^{\mathrm{EE}(s_{1}s_{2})}_{X_{1}X_{2}l}(r_{1},r_{2})=\frac{(2l+1)A_{s_{1}}A_{s_{2}}}{\sqrt{(2s_{1}+1)(2s_{2}+1)}}\\\
\times\sum_{l_{1},l_{2}}(-1)^{(l_{1}+l_{2}+s_{1}+s_{2}-2l)/2}(2l_{1}+1)(2l_{2}+1)\\\
\times\begin{bmatrix}s_{1}&l_{1}&l\\\
s_{1}&0&-s_{1}\end{bmatrix}\begin{bmatrix}s_{2}&l_{2}&l\\\
s_{2}&0&-s_{2}\end{bmatrix}I^{(s_{1}s_{2})}_{X_{1}X_{2}l_{1}l_{2}}(r_{1},r_{2}).$
(220)

For given values of $l$, $s_{1}$ and $s_{2}$, the number of terms in the
summations over $l_{1},l_{2},l^{\prime}$ appeared in Eqs. (218)–(220) are
finite because of the triangular inequality in $3j$-symbols. Therefore
numerical evaluations of the above expressions are straightforward with one-
dimensional integrations of finite number.

###  IV.3 The correlation function of projected tensors

####  IV.3.1 General formulas

We define the tangential/cross ($+/\times$) modes of the projected tensors in
spherical coordinates by

$\tilde{f}^{(\pm s)}_{X}(\bm{r})=(\pm
1)^{s}\left[\tilde{f}^{+(s)}_{X}(\bm{r})\pm
i\tilde{f}^{\times(s)}_{X}(\bm{r})\right]$ (221)

for non-negative integers $s$. Equivalently, we have

$\displaystyle\tilde{f}^{+(s)}_{X}(\bm{r})$
$\displaystyle=\frac{1}{2}\left[\tilde{f}^{(+s)}_{X}(\bm{r})+(-1)^{s}\tilde{f}^{(-s)}_{X}(\bm{r})\right],$
(222) $\displaystyle\tilde{f}^{\times(s)}_{X}(\bm{r})$
$\displaystyle=\frac{1}{2i}\left[\tilde{f}^{(+s)}_{X}(\bm{r})-(-1)^{s}\tilde{f}^{(-s)}_{X}(\bm{r})\right].$
(223)

These definitions for $+/\times$ modes are consistent with those defined in
Paper III in the distant-observer limit.555As explicitly shown in Paper III,
the $+/\times$ modes in Cartesian coordinates are given by $f^{(\pm
s)}_{X}(\bm{x})=e^{\pm is\phi}\tilde{f}^{(\pm s)}_{X}(\bm{r})$. Due to Eq.
(180), complex conjugates of Eqs. (222) and (223) are given by

$\tilde{f}^{+(s)*}_{X}(\bm{r})=\tilde{f}^{+(s)}_{X}(\bm{r}),\quad\tilde{f}^{\times(s)*}_{X}(\bm{r})=\tilde{f}^{\times(s)}_{X}(\bm{r}),$
(224)

i.e., they are real functions. Due to Eq. (181), parity transformations of
Eqs. (222) and (223) are given by

$\displaystyle\tilde{f}^{+(s)}_{X}(\bm{r})$
$\displaystyle\xrightarrow{\mathbb{P}}\tilde{f}^{+(s)\prime}_{X}(\bm{r})=(-1)^{p_{X}+s}\tilde{f}^{+(s)}_{X}(-\bm{r}),$
(225) $\displaystyle\tilde{f}^{\times(s)}_{X}(\bm{r})$
$\displaystyle\xrightarrow{\mathbb{P}}\tilde{f}^{\times(s)\prime}_{X}(\bm{r})=(-1)^{p_{X}+s+1}\tilde{f}^{\times(s)}_{X}(-\bm{r}).$
(226)

The two-point correlation functions of $+/\times$ modes in spherical
coordinates can be defined by

$\displaystyle\xi^{++(s_{1}s_{2})}_{X_{1}X_{2}}(\bm{r}_{1},\bm{r}_{2})$
$\displaystyle\equiv\left\langle\tilde{f}^{+(s_{1})}_{X_{1}}(\bm{r}_{1})\tilde{f}^{+(s_{2})}_{X_{2}}(\bm{r}_{2})\right\rangle,$
(227)
$\displaystyle\xi^{\times\times(s_{1}s_{2})}_{X_{1}X_{2}}(\bm{r}_{1},\bm{r}_{2})$
$\displaystyle\equiv\left\langle\tilde{f}^{\times(s_{1})}_{X_{1}}(\bm{r}_{1})\tilde{f}^{\times(s_{2})}_{X_{2}}(\bm{r}_{2})\right\rangle,$
(228)
$\displaystyle\xi^{+\times(s_{1}s_{2})}_{X_{1}X_{2}}(\bm{r}_{1},\bm{r}_{2})$
$\displaystyle\equiv\left\langle\tilde{f}^{+(s_{1})}_{X_{1}}(\bm{r}_{1})\tilde{f}^{\times(s_{2})}_{X_{2}}(\bm{r}_{2})\right\rangle,$
(229)
$\displaystyle\xi^{\times+(s_{1}s_{2})}_{X_{1}X_{2}}(\bm{r}_{1},\bm{r}_{2})$
$\displaystyle\equiv\left\langle\tilde{f}^{\times(s_{1})}_{X_{1}}(\bm{r}_{1})\tilde{f}^{+(s_{2})}_{X_{2}}(\bm{r}_{2})\right\rangle.$
(230)

Due to the properties of Eq. (224), the correlation functions of $+/\times$
modes defined above are all real functions. These functions are linearly
related to the correlation functions of three-dimensional tensors of Eq. (89)
through Eqs. (179), (221)–(223). Explicit relations are straightforwardly
derived, and the results are given by

$\displaystyle\xi^{++}$
$\displaystyle=\frac{1}{4}\left[\left(\tilde{X}^{++}+\tilde{X}^{--}\right)+\left(\tilde{X}^{+-}+\tilde{X}^{-+}\right)\right],$
(231) $\displaystyle\xi^{\times\times}$
$\displaystyle=-\frac{1}{4}\left[\left(\tilde{X}^{++}+\tilde{X}^{--}\right)-\left(\tilde{X}^{+-}+\tilde{X}^{-+}\right)\right],$
(232) $\displaystyle\xi^{+\times}$
$\displaystyle=\frac{1}{4i}\left[\left(\tilde{X}^{++}-\tilde{X}^{--}\right)-\left(\tilde{X}^{+-}-\tilde{X}^{-+}\right)\right],$
(233) $\displaystyle\xi^{\times+}$
$\displaystyle=\frac{1}{4i}\left[\left(\tilde{X}^{++}-\tilde{X}^{--}\right)+\left(\tilde{X}^{+-}-\tilde{X}^{-+}\right)\right],$
(234)

where the abbreviations,

$\displaystyle\tilde{X}^{\pm\pm}=(\pm
i)^{s_{1}+s_{2}}A_{s_{1}}A_{s_{2}}\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2},\pm
s_{1},\pm s_{2}}(\bm{r}_{1},\bm{r}_{2}),$ (235)
$\displaystyle\tilde{X}^{\pm\mp}=(\pm
i)^{s_{1}-s_{2}}A_{s_{1}}A_{s_{2}}\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2},\pm
s_{1},\mp s_{2}}(\bm{r}_{1},\bm{r}_{2}),$ (236)
$\displaystyle\xi^{++}=\xi^{++(s_{1}s_{2})}_{X_{1}X_{2}}(\bm{r}_{1},\bm{r}_{2}),\quad\xi^{+\times}=\xi^{+\times(s_{1}s_{2})}_{X_{1}X_{2}}(\bm{r}_{1},\bm{r}_{2}),$
(237)
$\displaystyle\xi^{\times+}=\xi^{\times+(s_{1}s_{2})}_{X_{1}X_{2}}(\bm{r}_{1},\bm{r}_{2}),\quad\xi^{\times\times}=\xi^{\times\times(s_{1}s_{2})}_{X_{1}X_{2}}(\bm{r}_{1},\bm{r}_{2}),$
(238)

are adopted just for simplicity of representation. The correlation functions
in Eqs. (235) and (236) are given by
$\tilde{\xi}^{(s_{1}s_{2})}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}$ of Eq. (89) with
$(\sigma_{1},\sigma_{2})=(\pm s_{1},\pm s_{2})$ and $(\pm s_{1},\mp s_{2})$,
respectively. Substituting Eq. (91) into Eqs. (235) and (236), Eqs.
(231)–(234) give the correlation functions of $+/\times$ modes of the
projected tensors. Due to Eq. (99), the property of complex conjugates of Eqs.
(235) and (236) are simply given by $\tilde{X}^{\pm\pm*}=\tilde{X}^{\mp\mp}$
and $\tilde{X}^{\pm\mp*}=\tilde{X}^{\mp\pm}$. Therefore, the correlation
functions of $+/\times$ modes, Eqs. (231)–(234) are all real functions, as
noted above.

Instead of the expression of Eq. (91), however, it is practically convenient
to substitute the alternative expression of Eq. (121) for numerical
evaluations of the correlation function of projected tensors with a special
choice of the coordinates system for each pair of tensors,
$(\theta_{1},\phi_{1})=(\theta,0)$ and $(\theta_{2},\phi_{2})=(0,0)$. For
convenience, we give explicit expressions of the ingredients on the LHS’s of
Eqs. (231)–(234):

$\tilde{X}^{++}\pm\tilde{X}^{--}=(-i)^{s_{1}-s_{2}}A_{s_{1}}A_{s_{2}}\sum_{l,m}\left[1\pm(-1)^{l-s_{1}-s_{2}}\right]i^{l}\tilde{P}_{l}^{m}(\mu_{2})\\\
\times
u^{(l_{1})}_{s_{1},s_{2}-m}(\theta)\sum_{l_{1},l_{2}}(-1)^{l_{2}}\begin{pmatrix}l_{1}&l_{2}&l\\\
m-s_{2}&s_{2}&-m\end{pmatrix}\tilde{\xi}^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(x),$
(239)

and

$\tilde{X}^{+-}\pm\tilde{X}^{-+}=(-i)^{s_{1}-s_{2}}A_{s_{1}}A_{s_{2}}\sum_{l,m}\left[1\pm(-1)^{l-s_{1}-s_{2}}\right]i^{l}\tilde{P}_{l}^{m}(\mu_{2})\\\
\times
u^{(l_{1})}_{s_{1},-s_{2}-m}(\theta)\sum_{l_{1},l_{2}}\begin{pmatrix}l_{1}&l_{2}&l\\\
m+s_{2}&-s_{2}&-m\end{pmatrix}\tilde{\xi}^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(x),$
(240)

where $\mu_{2}$ is given by Eq. (122), and
$\tilde{\xi}^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(x)$ is the function
$\tilde{\Xi}^{s_{1}s_{2};l_{1}l_{2}l}_{X_{1}X_{2}\sigma_{1}\sigma_{2}}(x)$ of
Eq. (120) substituted by $\sigma_{1}=s_{1}$ and $\sigma_{2}=s_{2}$, i.e.,

$\tilde{\xi}^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(x)\equiv\sum_{l_{z1},l_{z2}}\begin{pmatrix}s_{1}&l_{z1}&l_{1}\\\
s_{1}&0&-s_{1}\end{pmatrix}\begin{pmatrix}s_{2}&l_{z2}&l_{2}\\\
s_{2}&0&-s_{2}\end{pmatrix}\Xi^{s_{1}s_{2};l_{1}l_{2}l}_{X_{1}X_{2}l_{z1}l_{z2}}(x).$
(241)

The flip symmetry in redshift space discussed in sentences below Eq. (116)
indicates that the summations of $l_{z1}$ and $l_{z2}$ are taken only over
even integers. In the cases of plus sign of the double sign, i.e.,
$\tilde{X}^{+\pm}+\tilde{X}^{-\mp}$, the values are real numbers, because the
summations are non-zero only when $l-s_{1}-s_{2}=\mathrm{even}$. Similarly, in
the cases of minus sign, i.e., $\tilde{X}^{+\pm}-\tilde{X}^{-\mp}$, the values
are pure imaginary numbers, because the summations are non-zero only when
$l-s_{1}-s_{2}=\mathrm{odd}$. These properties are consistent with the
property that the correlation functions given in Eqs. (231)–(234) are all real
functions.

Substituting Eqs. (239) and (240) into Eqs. (231)–(234), we have expressions
of the correlation functions of $+/\times$ modes of the specific coordinates
system for each pairs of tensors. The results are given by

$\displaystyle\xi^{++(s_{1}s_{2})}_{X_{1}X_{2}}(r_{1},r_{2},\theta)$
$\displaystyle=\sum_{\begin{subarray}{c}l\\\
l+s_{1}-s_{2}=\mathrm{even}\end{subarray}}(-1)^{(l+s_{1}-s_{2})/2}Z^{s_{1}s_{2};l}_{X_{1}X_{2}+}(r_{1},r_{2},\theta),$
(242)
$\displaystyle\xi^{\times\times(s_{1}s_{2})}_{X_{1}X_{2}}(r_{1},r_{2},\theta)$
$\displaystyle=\sum_{\begin{subarray}{c}l\\\
l+s_{1}-s_{2}=\mathrm{even}\end{subarray}}(-1)^{(l+s_{1}-s_{2}+2)/2}Z^{s_{1}s_{2};l}_{X_{1}X_{2}-}(r_{1},r_{2},\theta),,$
(243)
$\displaystyle\xi^{+\times(s_{1}s_{2})}_{X_{1}X_{2}}(r_{1},r_{2},\theta)$
$\displaystyle=\sum_{\begin{subarray}{c}l\\\
l+s_{1}-s_{2}=\mathrm{odd}\end{subarray}}(-1)^{(l+s_{1}-s_{2}+1)/2}Z^{s_{1}s_{2};l}_{X_{1}X_{2}-}(r_{1},r_{2},\theta),$
(244)
$\displaystyle\xi^{\times+(s_{1}s_{2})}_{X_{1}X_{2}}(r_{1},r_{2},\theta)$
$\displaystyle=\sum_{\begin{subarray}{c}l\\\
l+s_{1}-s_{2}=\mathrm{odd}\end{subarray}}(-1)^{(l+s_{1}-s_{2}+1)/2}Z^{s_{1}s_{2};l}_{X_{1}X_{2}+}(r_{1},r_{2},\theta),$
(245)

where

$Z^{s_{1}s_{2};l}_{X_{1}X_{2}\pm}(r_{1},r_{2},\theta)=\frac{1}{2}A_{s_{1}}A_{s_{2}}\sum_{l_{1},l_{2},m}\tilde{P}_{l}^{m}(\mu_{2})\begin{pmatrix}l_{1}&l_{2}&l\\\
s_{2}-m&-s_{2}&m\end{pmatrix}\\\
\times\left[(-1)^{l_{1}}u^{(l_{1})}_{s_{1},s_{2}-m}(\theta)\pm
u^{(l_{1})}_{-s_{1},s_{2}-m}(\theta)\right]\tilde{\xi}^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(x),$
(246)

and the variables $(x,\mu_{2})$ in the last expression are related to
$(r_{1},r_{2},\theta)$ by Eq. (122).

If we take the distant-observer limit in the above expressions, Eqs.
(241)–(246), one can confirm that the corresponding results derived in Paper
III in the same limit from the first place are reproduced. The outline of the
proof is given as follows: first, we note that
$u^{(l)}_{m^{\prime}m}(\theta)\rightarrow
g^{(l)}_{m^{\prime}m}=(-1)^{m}\delta_{m^{\prime},-m}$ in the distant-observer
limit with $\theta\rightarrow 0$. In this limit, the summation over $m$ in Eq.
(246) is trivially taken. Substituting Eq. (241) into the result, and applying
recoupling formula with the $9j$-symbol (given in Appendix C of Paper I), they
are represented by a function $\xi^{s_{1}s_{2};l\,l_{z};s}_{X_{1}X_{2}}(x)$
given in Eq. (145). The relation between the last function and a normalized
function $\hat{\xi}^{s_{1}s_{2};l\,l_{z};s}_{X_{1}X_{2}}(x)$ introduced in
Paper III is given by
$\hat{\xi}^{s_{1}s_{2};l\,l_{z};s}_{X_{1}X_{2}}(x)=A_{s_{1}}A_{s_{2}}\sqrt{4\pi/(2l+1)}\,\xi^{s_{1}s_{2};l\,l_{z};s}_{X_{1}X_{2}}(x)$.
After some calculations described above, the corresponding formulas derived in
Paper III in the distant-observer limit are exactly reproduced.

####  IV.3.2 The linear correlation function

As derived above, the correlation functions of $+/\times$ modes are given by
the invariant function $\tilde{\xi}^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(x)$.
In the linear theory, substituting Eq. (174) into Eq. (241), we have

$\tilde{\xi}^{s_{1}s_{2};l_{1}l_{2}}_{X_{1}X_{2}l}(x)=(2l+1)\begin{pmatrix}l_{1}&l_{2}&l\\\
0&0&0\end{pmatrix}\\\ \times\begin{pmatrix}s_{1}&l_{1}-s_{1}&l_{1}\\\
s_{1}&0&-s_{1}\end{pmatrix}\begin{pmatrix}s_{2}&l_{2}-s_{2}&l_{2}\\\
s_{2}&0&-s_{2}\end{pmatrix}\\\
\times\int\frac{k^{2}dk}{2\pi}j_{l}(kx)\tilde{G}_{X_{1}s_{1}l_{1}}(k)\tilde{G}_{X_{2}s_{2}l_{2}}(k)P_{\mathrm{L}}(k),$
(247)

where the function $\tilde{G}_{Xsl}(k)$ is given by Eq. (176).

As in the previous subsections, we give explicit expressions depending on
whether the spins are zero or not. In order to represent the results, we
define the following integrals of the linear power spectrum:

$\displaystyle\xi_{l}(x)$
$\displaystyle\equiv\int\frac{k^{2}dk}{2\pi^{2}}j_{l}(kx)P_{\mathrm{L}}(k),$
(248) $\displaystyle\xi^{(s)}_{Xl}(x)$
$\displaystyle\equiv\int\frac{k^{2}dk}{2\pi^{2}}j_{l}(kx)c^{(1)}_{Xs}(k)P_{\mathrm{L}}(k),$
(249) $\displaystyle\xi^{(s_{1}s_{2})}_{X_{1}X_{2}l}(x)$
$\displaystyle\equiv\int\frac{k^{2}dk}{2\pi^{2}}j_{l}(kx)c^{(1)}_{X_{1}s_{1}}(k)c^{(1)}_{X_{2}s_{2}}(k)P_{\mathrm{L}}(k).$
(250)

When both $X_{1}$ and $X_{2}$ are scalar fields, $s_{1}=s_{2}=0$, we have

$\tilde{\xi}^{00;l_{1}l_{2}}_{X_{1}X_{2}l}(x)=\delta_{l_{1}0}\delta_{l_{2}0}\delta_{l0}\left[\xi^{(00)}_{X_{1}X_{2}0}(x)+\left(1+\frac{f}{3}\right)c^{(0)}_{X_{1}}\xi^{(0)}_{X_{2}0}(x)\right.\\\
\left.+(1\leftrightarrow
2)+\left(1+\frac{f}{3}\right)^{2}c^{(0)}_{X_{1}}c^{(0)}_{X_{2}}\xi_{0}(x)\right]\\\
+\delta_{l_{1}2}\delta_{l_{2}0}\delta_{l2}\frac{2\sqrt{5}}{3}fc^{(0)}_{X_{1}}\left[\xi^{(0)}_{X_{2}2}+\left(1+\frac{f}{3}\right)c^{(0)}_{X_{2}}\xi_{2}(x)\right]+(1\leftrightarrow
2)\\\