# Information content in anisotropic cosmological fields: Impact of different
multipole expansion scheme for galaxy density and ellipticity correlations

Takuya Inoue<EMAIL_ADDRESS>Department of Physics, National
Taiwan University, No. 1, Section 4, Roosevelt Road, Taipei 106216, Taiwan
Academia Sinica Institute of Astronomy and Astrophysics (ASIAA), No. 1,
Section 4, Roosevelt Road, Taipei 106216, Taiwan  Teppei Okumura  Academia
Sinica Institute of Astronomy and Astrophysics (ASIAA), No. 1, Section 4,
Roosevelt Road, Taipei 106216, Taiwan  Kavli Institute for the Physics and
Mathematics of the Universe (WPI), UTIAS, The University of Tokyo, 5-1-5
Kashiwanoha, Kashiwa, Chiba 277-8583, Japan  Shohei Saga  Institute for
Advanced Research, Nagoya University, Furo-cho Chikusa-ku, Nagoya 464-8601,
Japan  Kobayashi-Maskawa Institute for the Origin of Particles and the
Universe, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan  Atsushi
Taruya  Kavli Institute for the Physics and Mathematics of the Universe (WPI),
UTIAS, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583,
Japan  Center for Gravitational Physics and Quantum Information, Yukawa
Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

###### Abstract

Multipole expansions have been often used for extracting cosmological
information from anisotropic quantities in observation. However, which basis
of the expansion is best suited to quantify the anisotropies is not a trivial
question in any summary statistics. In this paper, using the Fisher matrix
formalism, we investigate the information content in multipole moments of the
power spectra of galaxy density and intrinsic ellipticity fields in linear
theory from the Alcock-Paczynski effect and redshift-space distortions (RSD).
We consider two expansion schemes, the associated Legendre basis as well as
the standard Legendre basis conventionally used in literature. We find that
the information in the multipoles of the intrinsic alignment (IA) power
spectra in the associated Legendre basis converges more slowly to that in the
full 2D power spectra than in the Legendre basis. This trend is particularly
significant when we consider a high number density sample. In this high number
density case, we show that the errors on the Hubble parameter obtained from
the multipoles of the IA cross- and auto-power spectra in the associated
Legendre basis are respectively about $6\%$ and $10\%$ larger than the full 2D
case even when we use multipoles up to $\ell=6$. Our results demonstrate that
the choice of basis is arbitrary but changes the information content encoded
in multipoles depending on the sample and statistics under consideration.

##  I Introduction

In galaxy redshift surveys, probing the nature of dark energy or testing
modifications of gravity is a key scientific goal to explain the current
accelerated expansion of the Universe in cosmology. Measurements of baryon
acoustic oscillations (BAO) Peebles and Yu (1970); Eisenstein and Hu (1998);
Meiksin et al. (1999); Seo and Eisenstein (2003) and redshift-space
distortions (RSD) Sargent and Turner (1977); Peebles (1980); Kaiser (1987);
Hamilton (1998); Linder (2008) in galaxy redshift surveys have simultaneously
constrained the history of cosmic expansion and structure formation Taruya et
al. (2011); Oka et al. (2014); Reid et al. (2012). In doing so, the two-point
statistics of the tracer fields containing meaningful cosmological
information, such as galaxy density fields, have played a major role as one of
the representative summary statistics.

It has been recently advocated that the intrinsic alignments (IAs) of galaxy
shapes, which have been long considered as a contamination of weak lensing
analysis Croft and Metzler (2000); Catelan et al. (2001); Heavens et al.
(2000); Hirata and Seljak (2004) (see Refs. Troxel and Ishak (2015); Joachimi
et al. (2015); Kiessling et al. (2015); Kirk et al. (2015); Lamman et al.
(2024) for reviews), can be used as an additional cosmological probe Schmidt
and Jeong (2012); Schmidt et al. (2015); Chisari et al. (2016); Chuang et al.
(2022). It has been shown by Refs. Taruya and Okumura (2020); Okumura and
Taruya (2022) that combining the galaxy clustering with the IA improves
dynamical and geometric constraints. Larger contributions from the IA are
expected in upcoming surveys with the higher number density and higher quality
galaxy shapes.

The IA statistically traces tidal fields of the large-scale matter
distribution. The simplest model to describe the statistics of IA is the
linear alignment (LA) model Catelan et al. (2001); Hirata and Seljak (2004);
Blazek et al. (2011, 2015); Okumura et al. (2020); Okumura and Taruya (2020),
which linearly relates the ellipticity fields observed by each galaxy image
with matter density fields. Note that similar to galaxy clustering in redshift
space, the statistics of IA exhibit anisotropies along the line-of-sight
direction, but the IA appears distorted even without RSD, providing an
alternative route to improve cosmological constraints. To maximize the
cosmological information in the anisotropic nature of the IA statistics, Refs.
Taruya and Okumura (2020); Okumura and Taruya (2022) used the LA model for the
power spectra in full 2D characterized by the wave number $k$ and the
directional cosine $\mu$ between the line-of-sight direction and wavevector,
namely $P(k,\mu)$, for the forecast studies.

As known in studies of the galaxy clustering, since the actual analysis in
full 2D is complicated by a large amount of data (e.g., Ref. Okumura et al.
(2008)), the multipole expansion, $P_{\ell}(k)$ Hamilton (1992); Cole et al.
(1994), has been frequently used instead of the full 2D spectrum. As shown in
Ref. Taruya et al. (2011), measuring multipole moments up to the hexadecapole
moment (i.e., $\ell=4$) provides the information comparable to the full 2D
power spectrum. Their finding was based on the galaxy density auto-power
spectrum, but it is not obvious how well multipoles of the IA power spectra
have the information relative to the full 2D spectra.

In order to obtain the multipoles of the IA correlations, the Legendre
polynomials have been conventionally used as a basis to expand the IA
correlations Singh and Mandelbaum (2016); Okumura and Taruya (2020); Chuang et
al. (2022); Zwetsloot and Chisari (2022). However, Ref. Kurita and Takada
(2022) revealed that the associated Legendre polynomials provides a more
natural basis for IA to characterize the anisotropies originated from the
projection of the galaxy shapes onto the sky (see also Refs. Saga et al.
(2023); Kurita and Takada (2023); Shi et al. (2024); Singh et al. (2023)).
Furthermore, Ref. Okumura et al. (2024) has recently derived new analytic
formulas of multipoles in terms of the associated Legendre basis for both
linear and nonlinear IA correlations. In terms of the information content for
the cosmological statistical analysis, the choice of the optimal basis for the
multipole expansions of anisotropic fields can be a general issue, but has not
been well discussed.

Focusing on the specific case, IA power spectra, in this paper, we
quantitatively investigate the information content obtained from multipoles in
the different bases, especially through the effects of geometric distortions
and RSD. Considering the three-dimensional galaxy density and ellipticity
fields, we provide linear theory expressions of the IA power spectra and
analytic expressions for the multipole covariances in the associated Legendre
basis, which are needed for the forecasts. Ref. Grieb et al. (2016) mentioned
that even if the power spectrum multipoles vanish above a certain multipole,
the higher-order multipole covariances do not vanish. Therefore, we show the
impact of the higher-order multipole contributions in the covariances on
cosmological constraints. Moreover, using multipoles instead of the full 2D
spectrum, we simultaneously obtain the geometric and dynamical constraints
combining the galaxy clustering with IA and then show the convergence of
multipole contributions to the full 2D case.

The rest of this paper is organized as follows. In Sec. II, we introduce the
multipoles for the IA power spectra in the linear theory limit and present the
analytic formulas of the multipole covariance in the associated Legendre
basis. In Sec. III, we describe the forecast formalisms used for our numerical
computations. We investigate the impact of the basis on the signal-to-noise
ratios and the parameter constraints in Sec. IV with further discussions in
Sec. V. Our conclusions are given in Sec. VI. In Appendix A, we present useful
relations between the IA power spectrum and its multipoles in the associated
Legendre basis used in Sec. IV.3. In Appendix B, we provide the multipole
covariances for the IA power spectra in terms of the associated Legendre
basis. In Appendix C, we provide the derivatives of the power spectra and
multipoles with respect to parameters.

In this paper, we assume a flat $\Lambda{\rm CDM}$ model, in which our
fiducial values of the cosmological parameters are taken from the 7 year
Wilkinson Microwave Anisotropy Probe results Komatsu et al. (2011).

##  II Multipole moments in terms of the associated Legendre basis in linear
theory

In this section we present the formalism to deal with IA statistics in the
linear theory limit. In Sec. II.1, adopting the linear alignment (LA) model
which linearly relates the ellipticity field with the tidal field of large-
scale structure (Catelan et al., 2001; Hirata and Seljak, 2004), we provide
analytic formulas of the multipole moments of the galaxy density-ellipticity
cross-power spectrum and ellipticity-ellpticity auto-power spectrum in
redshift space expanded in terms of the associated Legendre basis These
expressions are also summarized in Appendix C of Ref. Okumura et al. (2024).
In Sec. II.2, we consider the geometric distortions by Alcock-Paczynski
effect, which arises from the apparent mismatch of the underlying cosmology
when we convert the redshift and angular position of each galaxy to the
comoving radial and transverse distances. We then provide analytic expressions
of their Gaussian covariance matrices in Sec. II.3.

Throughout this paper, we works with the plane-parallel approximation in which
the line-of-sight direction is fixed to a specific direction. We omit the
redshift dependence in all the equations.

###  II.1 Power spectrum multipoles

In galaxy imaging surveys, we observe ellipticities of galaxies projected onto
the sky. The ellipticity field is characterized by two independent degrees of
freedom ($\gamma_{+},\gamma_{\times}$) and measured from the traceless part of
the second moments of the observed intensity. The observed ellipticity field
$\gamma_{(+/\times)}$ receives two contributions, the intrinsic ellitpicity
$\gamma^{\rm I}_{(+/\times)}$ and gravitational shear induced by the lensing
effect $\gamma^{\rm G}_{(+/\times)}$. In this paper, we ignore the latter
because it is subdominant when taking correlations in a specific redshift
slice Okumura and Taruya (2022).

In the LA model, the ellipticity field, which is primarily given as a
traceless rank-2 tensor sourced by the scalar gravitational potential, is
characterized by the two-component fields in Fourier space as Okumura and
Taruya (2020); Kurita et al. (2021)

$\gamma_{(+,\times)}(\bm{k})=b_{\rm
K}\Bigl{(}1-\mu^{2}\Bigr{)}\Bigl{(}\cos{2\phi},\sin{2\phi}\Bigr{)}\delta_{\rm
m}(\bm{k}),$ (1)

where $\delta_{\rm m}(\bm{k})$ is the linear matter density field, which is
related to the gravitational potential via the Poisson equation, $\mu$ is the
directional cosine between the wavevector and the line of sight direction, and
$\phi$ is the azimuthal angle of the wavevector projected on the celestial
sphere. The quantity $b_{\rm K}$ characterizes the strength of galaxy
alignment, and is regarded as a shape bias parameter. Conventionally, the
parameter $A_{\rm IA}$ has been also used in the literature. These are related
to each other through Schmidt et al. (2015); Kurita et al. (2021); Okumura and
Taruya (2022); Saga et al. (2023),

$b_{\rm K}=-0.0134A_{\rm IA}\Omega_{\rm m,0}/D_{+}(a),$ (2)

with $D_{+}$ and $a$ being, respectively, the linear growth factor and the
scale factor of the universe. In Ref. Kurita et al. (2021), the redshift
dependence of the amplitude $A_{\rm IA}$ was found fairly weak for dark matter
halos. This is irrespective of whether halos are selected for a certain mass
range or a fixed number density.

Under the plane-parallel approximation, the ellipticity field is naturally
decomposed into E/B-modes to be rotational invariance through

$\gamma_{\rm E}(\bm{k})+i\gamma_{\rm
B}(\bm{k})=e^{-2i\phi}\Bigl{\\{}\gamma_{+}(\bm{k})+i\gamma_{\times}(\bm{k})\Bigr{\\}}.$
(3)

Substituting Eq. (1) into Eq. (3) leads to $\gamma_{\rm E}(\bm{k})=b_{\rm
K}(1-\mu^{2})\delta_{\rm m}(\bm{k})$ and $\gamma_{\rm B}(\bm{k})=0$. As long
as we consider the linear theory limit, the power spectra associated with the
$\rm B$-mode vanish Blazek et al. (2015); Kurita et al. (2021). To extract the
cosmological information from the galaxy density and ellipticity fields, we
calculate the auto- and cross-power spectra of these fields. In the linear
theory limit, the galaxy density field $\delta_{\rm g}$ in redshift space is
given as Kaiser (1987); Hamilton (1992); Cole et al. (1994), $\delta_{{\rm
g}}(\bm{k})=\left(b+f\mu^{2}\right)\delta_{\rm m}(\bm{k})$, where $b$ and $f$
are, respectively, the linear galaxy bias and the linear growth rate defined
by $f\equiv d{\rm ln}D_{+}(a)/d{\rm ln}a$. Note that unlike the galaxy density
field, the observed ellipticity field is not affected by RSD at linear order,
and this is the reason why it does not depend on the growth rate in Eq. (1)
Singh and Mandelbaum (2016); Okumura and Taruya (2020).

Given the galaxy density and ellipticity fields, we have the auto-power
spectra of the density and $\rm E$-mode and their cross-power spectrum,
defined by

$\displaystyle\Braket{\delta_{\rm g}(\bm{k})\delta_{\rm g}(\bm{k}^{\prime})}$
$\displaystyle=(2\pi)^{3}\delta_{\rm D}(\bm{k}+\bm{k}^{\prime})P^{\rm
gg}(\bm{k}),$ (4) $\displaystyle\Braket{\gamma_{\rm E}(\bm{k})\gamma_{\rm
E}(\bm{k}^{\prime})}$ $\displaystyle=(2\pi)^{3}\delta_{\rm
D}(\bm{k}+\bm{k}^{\prime})P^{\rm EE}(\bm{k}),$ (5)
$\displaystyle\Braket{\delta_{\rm g}(\bm{k})\gamma_{\rm E}(\bm{k}^{\prime})}$
$\displaystyle=(2\pi)^{3}\delta_{\rm D}(\bm{k}+\bm{k}^{\prime})P^{\rm
gE}(\bm{k}).$ (6)

The resulting expressions are then given as Taruya and Okumura (2020); Kurita
et al. (2021); Okumura and Taruya (2022),

$\displaystyle P^{\rm gg}(k,\mu)$ $\displaystyle=(b+f\mu^{2})^{2}P_{\rm
L}(k),$ (7) $\displaystyle P^{\rm EE}(k,\mu)$ $\displaystyle=b^{2}_{{\rm
K}}(1-\mu^{2})^{2}P_{\rm L}(k),$ (8) $\displaystyle P^{\rm gE}(k,\mu)$
$\displaystyle=b_{{\rm K}}(b+f\mu^{2})(1-\mu^{2})P_{\rm L}(k),$ (9)

where $P_{\rm L}$ is the linear matter power spectrum defined by
$\Braket{\delta_{\rm m}(\bm{k})\delta_{\rm
m}(\bm{k}^{\prime})}=(2\pi)^{3}\delta_{\rm D}(\bm{k}+\bm{k}^{\prime})P_{\rm
L}(k)$. As seen in Eqs. (9) and (8), they are, respectively, proportional to
$(1-\mu^{2})$ and $(1-\mu^{2})^{2}$ due to the projection of the ellipticity
field along the line of sight.

To characterize anisotropies in the power spectra given above, the multipole
expansion via the Legendre polynomials, denoted by $\mathcal{L}_{\ell}(\mu)$,
has been frequently used in the literature. However, for the IA power spectra
involving the projection factors, the expansion in terms of the associated
Legendre polynomials $\mathcal{L}^{m}_{\ell}(\mu)$ with $m>0$ is convenient
and they could provide a more natural basis. The IA power spectra are thus
expressed as

$\displaystyle P^{\rm XY}(k,\mu)=\sum_{\ell\geq m}\widetilde{P}^{\rm
XY}_{\ell,m}(k)\Theta_{\ell,m}(\mu),$ (10)

where $\rm XY=\\{gE,EE\\}$ and $\Theta_{\ell,m}(\mu)$ is the normalized
associate Legendre polynomials as a basis function in the multipole expansions
Matsubara (2023); Okumura et al. (2024), defined by

$\displaystyle\Theta_{\ell,m}(\mu)\equiv\sqrt{2\pi}Y_{\ell,m}(\theta,0)=\sqrt{\frac{2\ell+1}{2}\frac{(\ell-m)!}{(\ell+m)!}}\mathcal{L}^{m}_{\ell}(\mu).$
(11)

It satisfies $\Theta_{\ell,-m}(\mu)=(-1)^{m}\Theta_{\ell,m}(\mu)$ and
$\Theta_{\ell,m}(-\mu)=(-1)^{\ell+m}\Theta_{\ell,m}(\mu)$, as well as a simple
orthonormal relation,
$\int^{1}_{-1}{d}\mu\Theta_{\ell,m}(\mu)\Theta_{\ell^{\prime},m}(\mu)=\delta^{\rm
K}_{\ell,\ell^{\prime}}$, with $\delta^{\rm K}_{\ell,\ell^{\prime}}$ being the
Kronecker delta. Eq. (10) holds for the arbitrary choice of $m$. Here, the
quantities with tilde represent those expanded in terms of the normalized
associated Legendre polynomials, not the (conventional) Legendre polynomials.
Making use of the orthonormality, the coefficients are obtained by

$\displaystyle\widetilde{P}^{\rm XY}_{\ell,m}(k)$
$\displaystyle=\int^{1}_{-1}{d}\mu\Theta_{\ell,m}(\mu)P^{\rm XY}(k,\mu).$ (12)

In this paper, we call these coefficients the multipoles expanded in terms of
the associated Legendre polynomials. Accordingly, those with $m=0$ are
referred to as multipoles in terms of the conventional Legendre polynomials,
though
$\mathcal{L}_{\ell}(\mu)=\mathcal{L}_{\ell}^{0}(\mu)=\sqrt{2/(2\ell+1)}\Theta_{\ell,0}(\mu)$.
Some useful relations hold for the coefficients, as shown in Appendix A.

Substituting Eq. (9) into Eq. (12), the multipole moments of the gE power
spectrum are concisely expressed as

$\widetilde{P}^{\rm gE}_{\ell,m}(k)=A^{\rm gE}_{\ell,m}b_{\rm K}P_{\rm L}(k).$
(13)

When choosing the Legendre basis, $m=0$, the non-vanishing coefficients
$A^{\rm gE}_{\ell,0}$ are given by

$\displaystyle\left(A^{\rm gE}_{0,0},A^{\rm gE}_{2,0},A^{\rm gE}_{4,0}\right)$
$\displaystyle=\left(\frac{2\sqrt{2}}{15}(5b+f),\frac{2}{21}\sqrt{\frac{2}{5}}(f-7b),-\frac{8\sqrt{2}}{105}f\right).$
(14)

If we choose the associated Legendre basis with $m=2$, we have two non-zero
multipoles and the coefficients $A^{\rm gE}_{\ell,2}$ ($\ell=2,4$) are given
by

$\displaystyle\left(A^{\rm gE}_{2,2},A^{\rm
gE}_{4,2}\right)=\left(\frac{4}{7\sqrt{15}}(7b+f),\frac{8}{21\sqrt{5}}f\right).$
(15)

Similarly to the gE power spectrum, substituting Eq. (8) into Eq. (12) leads
to the multipole moments of the EE power spectrum:

$\widetilde{P}^{\rm EE}_{\ell,m}(k)=A^{\rm EE}_{\ell,m}b^{2}_{\rm K}P_{\rm
L}(k)~{}$ (16)

with the non-vanishing moments up to $m=4$ summarized as follows:

$\displaystyle\left(A^{\rm EE}_{0,0},A^{\rm EE}_{2,0},A^{\rm EE}_{4,0}\right)$
$\displaystyle=\left(\frac{8\sqrt{2}}{15},-\frac{16}{21}\sqrt{\frac{2}{5}},\frac{8\sqrt{2}}{105}\right),$
(17) $\displaystyle\left(A^{\rm EE}_{2,2},A^{\rm EE}_{4,2}\right)$
$\displaystyle=\left(\frac{8}{7}\sqrt{\frac{3}{5}},-\frac{8}{21\sqrt{5}}\right),$
(18) $\displaystyle A^{\rm EE}_{4,4}$ $\displaystyle=\frac{16}{3\sqrt{35}}.$
(19)

As increasing $m$, the linear-order contributions become compressed into a
fewer numbers of multipoles in the associated Legendre basis. For the EE power
spectrum, we have only a single multipole for $m=4$. Therefore, in the
following analysis, we will treat $m=4$ as the representative associated
Legendre basis for the EE power spectrum. It is, however, not obvious whether
the choice of this basis is optimal to extract the cosmological information or
not. Quantitative understanding of it is therefore the goal of this paper, and
we will investigate it with the Fisher matrix analysis in Sec. IV.

###  II.2 Alcock-Paczynski effect

The observed IA power spectra of galaxies contain anisotropies induced not
only from RSD and projection of the shape described in Sec. II.1 but also from
the Alcock-Paczynski (AP) effect Alcock and Paczynski (1979). The AP effect is
caused by the apparent mismatch between underlying cosmology and fiducial
model to convert the observed galaxy position, i.e., redshift and angular
position on the sky, into the comoving space. As a result, the measured wave
number perpendicular and parallel to the line of sight, $k_{\perp}$ and
$k_{\parallel}$, are respectively modulated from true ones, $q_{\perp}$ and
$q_{\parallel}$, as $k_{\perp}=(d_{\rm A}/d_{{\rm A}}^{{\rm fid}})q_{\perp}$
and $k_{\parallel}=(H/H^{{\rm fid}})^{-1}q_{\parallel}$, with $d_{\rm A}$ and
$H$ being the angular diameter distance and Hubble parameter, respectively.
The superscript “fid” implies quantities computed assuming the fiducial
cosmology. As a result, the observed power spectrum with isotropic and
anisotropic shifts due to the AP effect is given by Seo and Eisenstein (2003);
Taruya and Okumura (2020); Kobayashi et al. (2020); Okumura and Taruya (2022),

$P^{\rm XY,obs}(k_{\perp},k_{\parallel})=\frac{H(z)}{H^{{\rm
fid}}(z)}\left\\{\frac{d_{{\rm A}}^{{\rm fid}}(z)}{d_{{\rm
A}}(z)}\right\\}^{2}P^{\rm XY}(q_{\perp},q_{\parallel}),$ (20)

where the superscript “obs” denotes observed quantities. Given the RSD and AP
effect, we can simultaneously constrain the cosmologically informative
parameters: the linear growth rate $f(z)$, the Hubble parameter $H(z)$ and the
angular diameter distance $d_{\rm A}(z)$, using the power spectra Taruya et
al. (2011); Oka et al. (2014); Reid et al. (2012).

###  II.3 Gaussian covariance for the power spectrum multipoles

Assuming that our observables, galaxy density and ellipticity fields, follow
the Gaussian distribution, we present the analytic covariance matrices of
power spectrum multipoles expanded in terms of the normalized associated
Legendre basis. This Gaussian assumption is reasonable as we consider the
correlations at the large scales where linear theory is applied. The Gaussian
covariance matrices for the multipoles of the auto-power spectra of the galaxy
density and ellipticity fields, $P^{\rm gg}$ and $P^{\rm EE}$, and their
cross-power spectrum, $P^{\rm gE}$, are described as

$\displaystyle{\rm COV}^{\rm XY,X^{\prime}Y^{\prime}}_{\ell
m,\ell^{\prime}m^{\prime}}(k)=\frac{2}{N_{k}}\int^{1}_{-1}{\rm
d}\mu\Theta_{\ell,m}(\mu)\Theta_{\ell^{\prime},m^{\prime}}(\mu)$
$\displaystyle\times\Bigl{\\{}\widehat{P}^{\rm
XX^{\prime}}(k,\mu)\widehat{P}^{\rm YY^{\prime}}(k,\mu)+\widehat{P}^{\rm
XY^{\prime}}(k,\mu)\widehat{P}^{\rm YX^{\prime}}(k,\mu)\Bigr{\\}}~{},$ (21)

where the superscripts ${\rm X}$, ${\rm Y}$, $\rm X^{\prime}$, and $\rm
Y^{\prime}$ are the galaxy density field or the E-mode ellipticity field,
respectively denoted as $\rm g$ and $\rm E$. The quantity $N_{k}=4\pi
k^{2}\Delta{k}V_{\rm s}/(2\pi)^{3}$ is the number of independent Fourier modes
at the $k$ bin with width $\Delta k$ with $V_{\rm s}$ being the survey volume.
We have added a hat on $P^{\rm XX^{\prime}}$ to emphasize that the auto-power
spectra contain Poisson shot/shape noise contribution, namely
$\widehat{P}^{\rm gg}=P^{\rm gg}+1/n_{\rm g}$ and $\widehat{P}^{\rm EE}=P^{\rm
EE}+\sigma^{2}_{\rm\gamma}/n_{\rm g}$, where $n_{\rm g}$ and
$\sigma_{\rm\gamma}$ are respectively the mean number density of galaxies and
the scatter of the intrinsic shape. Since the cross power spectrum does not
have it, we have, simply, $\widehat{P}^{\rm gE}=P^{\rm gE}$.

The analytic expressions of the covariance matrix for the $\rm gE$ cross- and
$\rm EE$ auto-power spectrum expanded in terms of the Legendre basis up to
$\ell=4$ were already derived in Ref. Taruya and Akitsu (2021). As for the
expansion in terms of the associated Legendre basis, Ref. Kurita and Takada
(2023) derived the covariance matrix for the $\rm gE$ power spectrum including
the window and other observational effects. Here we newly derive the
expression for the Gaussian covariance of the EE power spectrum expanded in
terms of the normalized associated Legendre basis with $m>0$. Rewriting $\mu$
dependence of the full 2D power spectrum in Eq. (21) to the normalized
associated Legendre polynomials $\Theta_{\ell,m}(\mu)$, we perform the angular
integral with the relation between three normalized associated Legendre
polynomials and the Wigner 3-j symbols (e.g., Mavromatis and Alassar (1999)),

$\displaystyle\Theta_{\ell_{1},m_{1}}(\mu)\Theta_{\ell_{2},m_{2}}(\mu)\Theta_{\ell_{3},m_{3}}(\mu)$
$\displaystyle=\sum_{L}\sum_{M}\sqrt{\frac{(2\ell_{1}+1)(2\ell_{2}+1)(2L+1)}{2}}\left(\begin{array}[]{ccc}\ell_{1}&\ell_{2}&L\\\
m_{1}&m_{2}&M\end{array}\right)$ (24)
$\displaystyle\times\left(\begin{array}[]{ccc}\ell_{1}&\ell_{2}&L\\\
0&0&0\end{array}\right)\Theta_{L,M}(\mu)\Theta_{\ell_{3},m_{3}}(\mu)~{},$ (27)

where $L$ and $M$ are integers restricted by $L\geq|M|$ and the selection
rules of the Wigner 3-j symbols. Using the orthonormal property of
$\Theta_{\ell,m}$ in Eq. (27), we obtain the covariance for the $\rm gE$
cross- and $\rm EE$ auto-power spectrum multipoles. In Appendix B, starting
from Eq. (21), we provide the analytic expressions of the auto- and cross-
covariance matrices for the gE and EE power spectrum multipoles. The first,
second and third terms of Eqs. (61)–(86) correspond to the pure cosmic
variance (CV $\times$ CV), the cross talk between the cosmic variance and
Poisson noise (CV $\times$ P) and pure Poisson noise (P $\times$ P),
respectively. The explicit forms up to $\ell=4$ are given in Eqs. (88)–(93).

While Ref. Taruya and Akitsu (2021) claimed that the multipole covariance
matrices at $\ell>4$ vanish, each of the multipole covariances has all non-
vanishing multipole contributions as Ref. Grieb et al. (2016) pointed out (see
also Ref. Shiraishi et al. (2020)). As long as the selection rules in Wigner
3-j symbols in Eqs. (61)–(86) are satisfied, the CV $\times$ CV and CV
$\times$ P terms can take non-zero values and thus have the cosmological
information even at higher-order multipoles $\ell>4$ where the power spectrum
multipoles themselves vanish. The P $\times$ P term also takes non-zero values
at the higher-order multipoles although it is an isotropic quantity (see e.g.,
Refs. Taruya et al. (2010); Taruya and Akitsu (2021); Saga et al. (2023)). The
higher-order multipole contributions ultimately have non-negligible impacts on
the forecast results as we will discuss in Sec. IV.

##  III Forecast formalism

As mentioned in the previous section, even though a power spectrum multipole
vanishes at a given order, its covariance does not necessarily vanish,
indicating that the non-vanishing higher-order multipoles of the covariance
contains the cosmological information. In this section, we introduce the
formalisms of the cumulative signal-to-noise ratio in Sec. III.1 and the
Fisher matrix in Sec. III.2, which will be used to quantify how much the
cosmological information is contained in the higher-order multipoles of the
covariance and how it depends on the choice of the basis for the multipole
expansion in the next section.

###  III.1 Signal-to-noise ratio

Given the power spectrum multipoles $\widetilde{P}^{\rm XY}_{\ell,m}$ ($\rm
X,Y=g\ or\ E$) and their covariance matrices, we can numerically calculate the
cumulative signal-to-noise ratio via the following equation:

$\displaystyle\left(\frac{{S}}{{N}}\right)^{2}_{\ell_{\rm max}}$
$\displaystyle=\frac{V_{\rm s}}{4\pi^{2}}\int^{k_{{\rm max}}}_{k_{{\rm
min}}}k^{2}dk$ $\displaystyle\times\sum^{\ell_{{\rm
max}}}_{\ell,\ell^{\prime}}\widetilde{P}^{\rm
XY}_{\ell,m}(k)\\{\widetilde{{\rm COV}}_{\ell m,\ell^{\prime}m^{\prime}}^{\rm
XY,XY}(k)\\}^{-1}\widetilde{P}^{\rm XY}_{\ell^{\prime},m^{\prime}}(k),$ (28)

where $k_{{\rm min}}$ and $k_{{\rm max}}$ are the minimum and maximum wave
numbers used for the analysis, which will be specified later (see Sec. IV.1),
and $\widetilde{{\rm COV}}^{\rm XY,XY}_{\ell m,\ell^{\prime}m^{\prime}}(k)$ is
related to Eq. (21) through

$\widetilde{{\rm COV}}^{\rm XY,XY}_{\ell
m,\ell^{\prime}m^{\prime}}(k)=\frac{N_{k}}{2}\times{\rm COV}^{\rm XY,XY}_{\ell
m,\ell^{\prime}m^{\prime}}(k).$ (29)

Even if the power spectrum multipole is zero at a given high $\ell$ ($\ell>4$
for our case at the linear theory limit), its covariance is not. Hence the
vanishing multipoles would contribute to the non-vanishing ones at lower
$\ell$ via the off-diagonal components of the covariance matrix. As a result,
the sum of the multipoles that contain linear-order contributions, up to
$\ell=4$, does not necessarily coincide with the result obtained from the full
2D power spectrum. Regardless of whether the standard or associated Legendre
basis is adopted, as $\ell_{{\rm max}}$ increases, the signal-to-noise ratio
computed by Eq. (28) gradually approaches that obtained from the full 2D
spectra given by

$\displaystyle\left(\frac{{S}}{{N}}\right)^{2}_{\rm 2D}=\frac{V_{\rm
s}}{4\pi^{2}}\int^{k_{{\rm max}}}_{k_{{\rm min}}}k^{2}dk\int^{1}_{-1}d\mu$
$\displaystyle\times\Biggl{[}\frac{\\{P^{\rm
XY}(k,\mu)\\}^{2}}{\widehat{P}^{\rm XX}(k,\mu)\widehat{P}^{\rm
YY}(k,\mu)+\widehat{P}^{\rm XY}(k,\mu)\widehat{P}^{\rm YX}(k,\mu)}\Biggr{]}.$
(30)

When we consider only the shot/shape noise term (${\rm P}\times{\rm P}$) in
the covariance matrix, which is the case where the cosmic variance is
subdominant compared to the shot noise due to the limited number of samples,
the signal-to-noise ratio for the multipole contributions [Eq. (28)] has a
simple form as

$\displaystyle\left(\frac{S}{N}\right)^{2}_{\ell_{\rm max}}=\frac{V_{\rm
s}}{4\pi^{2}}\int^{k_{{\rm max}}}_{k_{{\rm min}}}k^{2}dk\frac{1}{N^{\rm
X}N^{\rm Y}}\sum^{\ell_{\rm max}}_{\ell\geq m}\\{\widetilde{P}^{\rm
XY}_{\ell,m}(k)\\}^{2},$ (31)

where $\rm X,Y=g\ or\ E$ and we used the orthonormal relation of
$\Theta_{\ell,m}$. The shot/shape noise contribution, $N^{\rm X}$, indicates
$N^{\rm g}=1/n_{\rm g}$ or $N^{\rm E}=\sigma^{2}_{\gamma}/n_{\rm g}$.

###  III.2 Fisher matrix

Ref. Taruya and Okumura (2020) used the full 2D power spectra, $P(k,\mu)$, to
demonstrate that combining the conventional galaxy clustering with the IA
leads to improved constraints on the parameters $f(z),\ H(z)$ and $d_{\rm
A}(z)$. Since the cosmological analysis of the observed $P(k,\mu)$ is
complicated due to the large degree of freedom, it is more convenient to
analyze the multipole moments. Thus, we extend the work of Ref. Taruya and
Okumura (2020) and perform the joint analysis of the multipoles of the galaxy
density and ellipticity power spectra, taking into account the different
expansion schemes. In this subsection, we provide the Fisher matrix formalism
Tegmark (1997); Tegmark et al. (1997) for the joint analysis.

Given the free parameters $\bm{\theta}=(b,f,b_{\rm K},H/H^{{\rm fid}},d_{\rm
A}/d^{\rm fid}_{\rm A})$ introduced in Sec. II and provided a set of observed
power spectra ($P^{\rm gg},\ P^{\rm gE},\ P^{\rm EE}$), the Fisher matrix is
evaluated with

$\displaystyle F_{ij}^{\ell_{\rm max}}=\frac{V_{\rm s}}{4\pi^{2}}\int^{k_{{\rm
max}}}_{k_{{\rm min}}}k^{2}dk$ $\displaystyle\times\sum_{a,b}^{N_{\rm
P}}\sum^{\ell_{{\rm
max}}}_{\ell,\ell^{\prime}}\frac{\partial\widetilde{P}^{a}_{\ell,(m,m^{\prime},m^{\prime\prime})}(k)}{\partial\theta_{i}}\\{\widetilde{{\rm
COV}}^{-1}\\}^{a,b}_{\ell,\ell^{\prime}}\frac{\partial\widetilde{P}^{b}_{\ell^{\prime},(m,m^{\prime},m^{\prime\prime})}(k)}{\partial\theta_{j}},$
(32)

where $N_{\rm P}$ is the number of different power spectra and
$\widetilde{P}^{a}_{\ell,{(m,m^{\prime},m^{\prime\prime})}}(k)=$
$\\{\widetilde{P}^{\rm gg}_{\ell,m}(k),\ \widetilde{P}^{\rm
gE}_{\ell,m^{\prime}}(k),\ \widetilde{P}^{\rm
EE}_{\ell,m^{\prime\prime}}(k)\\}$ for $a=1,2,3$, respectively, where the
lower limit for the summation of $\ell$ for each power spectrum should be a
chosen value of $m,m^{\prime}$, and $m^{\prime\prime}$. As shown in Appendix
C, derivatives of the power spectra become non zero up to $\ell=6$ at the
linear theory limit due to the AP effect.

When combining all the statistics, the covariance matrix is given by

$\widetilde{{\rm
COV}}_{\ell,\ell^{\prime}}^{a,b}=\begin{pmatrix}{\widetilde{\rm{COV}}^{\rm
gg,gg}_{\ell m,\ell^{\prime}m}}&{\widetilde{\rm{COV}}^{\rm gg,gE}_{\ell
m,\ell^{\prime}m^{\prime}}}&{\widetilde{\rm{COV}}^{\rm gg,EE}_{\ell
m,\ell^{\prime}m^{\prime\prime}}}\\\ {\widetilde{\rm{COV}}^{\rm gE,gg}_{\ell
m^{\prime},\ell^{\prime}m}}&{\widetilde{\rm{COV}}^{\rm gE,gE}_{\ell
m^{\prime},\ell^{\prime}m^{\prime}}}&{\widetilde{\rm{COV}}^{\rm gE,EE}_{\ell
m^{\prime},\ell^{\prime}m^{\prime\prime}}}\\\ {\widetilde{\rm{COV}}^{\rm
EE,gg}_{\ell m^{\prime\prime},\ell^{\prime}m}}&{\widetilde{\rm{COV}}^{\rm
EE,gE}_{\ell
m^{\prime\prime},\ell^{\prime}m^{\prime}}}&{\widetilde{\rm{COV}}^{\rm
EE,EE}_{\ell m^{\prime\prime},\ell^{\prime}m^{\prime\prime}}}\end{pmatrix},$
(33)

where the elements in the parentheses are calculated through Eq. (29). On the
other hand, the Fisher matrix for the full 2D case is given by Refs. Taruya
and Okumura (2020); Okumura and Taruya (2022),

$\displaystyle F_{ij}^{\rm 2D}$ $\displaystyle=\frac{V_{\rm
s}}{4\pi^{2}}\int^{k_{{\rm max}}}_{k_{{\rm min}}}k^{2}dk\int^{1}_{-1}d\mu$
$\displaystyle\times\sum_{a,b}^{N_{\rm P}}\frac{\partial
P^{a}(k,\mu)}{\partial\theta_{i}}\\{\widetilde{{\rm
COV}}^{-1}\\}^{a,b}\frac{\partial P^{b}(k,\mu)}{\partial\theta_{j}}.$ (34)

Here, we construct the covariance matrix $\widetilde{{\rm COV}}^{a,b}$ in the
same way as Eq. (33) but with the elements using the full 2D spectra,
${\widetilde{\rm{COV}}^{\rm XY,X^{\prime}Y^{\prime}}}=\widehat{P}^{\rm
XX^{\prime}}\widehat{P}^{\rm YY^{\prime}}+\widehat{P}^{\rm
XY^{\prime}}\widehat{P}^{\rm YX^{\prime}}$. As in the signal-to-noise ratio,
the parameter constraints obtained from the multipoles get tighter and
approach the full 2D result as $\ell_{\rm max}$ increases.

Given the Fisher matrix, we can obtain the two-dimensional error ellipses for
a specific pair of parameters from $\bm{\theta}=(b,f,b_{\rm K},H/H^{{\rm
fid}},d_{\rm A}/d^{\rm fid}_{\rm A})$ by constructing the $2\times 2$
submatrix from the inverse Fisher matrix, $\left(F^{-1}\right)_{ij}$, while
the other parameters are marginalized over. Also the one-dimensional
marginalized error on the $i$th parameter $\theta_{i}$ is obtained by the
$i$th diagonal component, $\sigma_{\theta}=\sqrt{\left(F^{-1}\right)_{ij}}$.

Figure 1: Diagonal components of the covariance matrix normalized by the
survey volume, divided by the square of each power spectrum multipole, ${\rm
COV}^{\rm XY,XY}_{\ell m,\ell m}(k,k)(V_{\rm s}/[{h^{-3}{\rm
Gpc}^{3}}])/\\{\widetilde{P}^{\rm XY}_{\ell,m}(k)\\}^{2}$, where ${\rm
XY}=\\{{\rm gg},{\rm gE},{\rm EE}\\}$ from the left to right panels,
respectively. The results for the high- and low-number density samples are
shown in the upper and lower rows, respectively. The $\ell=0$, $2$, and $4$
moments are colored by red, blue, and black, and the moments with $m=0$, $2$,
and $4$ are shown by the dotted, solid, and dashed lines, respectively.

##  IV Results

In this section, we quantitatively show the information content encoded in the
multipoles of the IA power spectra in the application of the different
multipole expansion scheme. After briefly providing our setup for the
numerical calculation in Sec. IV.1, we first show the fractional errors of the
power spectrum multipoles in Sec. IV.2 using the formulas presented in Sec.
II. We then show the signal-to-noise ratios in Sec. IV.3 and the geometric and
dynamical constraints expected from the multipoles of the individual power
spectra in Sec. IV.4. We also present well-known results of the galaxy auto-
power spectrum, $P^{\rm gg}$, as a reference.

###  IV.1 Setup

We consider two hypothetical galaxy surveys with the different number
densities, $n_{\rm g}=10^{-2}\ h^{3}{\rm Mpc}^{-3}$ and $n_{\rm g}=10^{-4}\
h^{3}{\rm Mpc}^{-3}$ but the other parameters being the same. The former one
roughly corresponds to the Bright Galaxy Survey (BGS) samples observed by Dark
Energy Spectroscopic Instrument (DESI) DESI Collaboration et al. (2016), while
the latter to galaxy samples such as the Luminous Red Galaxies (LRG) and the
Emission Line Galaxies (ELG). For each sample, we fix the redshift and the
galaxy bias to $z=0.5$, and $b=1$, respectively. Following the work of Ref.
Okumura and Taruya (2022), we set the amplitude of IA, $A_{\rm IA}$, to
$A_{\rm IA}=18$, assuming that we can estimate the IA of the dark matter halos
that host the above galaxies Shi et al. (2021). We adopt a value of the
scatter of the intrinsic shape considered in up-coming galaxy surveys,
$\sigma_{\gamma}=0.2$. Since our formalism is based on the linear theory, to
be conservative, we restrict our analysis to large scales and set $k_{{\rm
min}}=2\pi/V_{\rm s}^{1/3}$ and $k_{{\rm max}}=0.1\ h{\rm Mpc}^{-1}$. All
numerical results in following subsections are normalized by the survey
volume. Except for the results in Sec. IV.2, in this section, we focus on the
results expected from the high number density sample and discuss the results
for the low number density sample in Sec. V.

###  IV.2 Error of power spectrum multipoles

To compare the information content of a multipole moment of the IA power
spectra in the associated Legendre basis with those in the conventional
Legendre one, we first look at the fractional errors, which are defined by the
diagonal components of the covariance matrix divided by the square of the
power spectrum multipoles.

The left to right panels of Fig.1 show the fractional errors of the multipoles
of $P^{\rm gg}$, $P^{\rm gE}$ and $P^{\rm EE}$, respectively. The top and
bottom panels show the results for the high and low number density samples,
respectively. We find that the fractional errors on $\ell=2$ and $4$ moments
of the associated Legendre basis ($m>0$) are lower than those of the standard
Legendre one ($m=0$) for the IA power spectra, $P^{\rm gE}$ and $P^{\rm EE}$
in both of the two different number densities. Comparing the lowest-order
multipoles of the IA power spectra between the two different bases, the
results with $m>0$ have slightly lower fractional errors in the low number
density samples and vice versa in the high number density ones. That is, a
beneficial basis to extract more information from a single multipole moment
depends on the sample under consideration. Although the multipole of $P^{\rm
EE}$ in the associated Legendre basis is expressed as the single multipole,
namely $\widetilde{P}^{\rm EE}_{4,4}$, it does not necessarily give the lowest
fractional errors as shown in the upper-right panel. This would reflect the
fact that the information from the higher-order multipoles in the covariance
matrix is not negligible, as we discuss in detail in Sec. IV.3.

Figure 2: Top panels: signal-to-noise ratios normalized by the survey volume,
$S/N(V_{\rm s}/[h^{-3}{\rm Gpc^{3}}])^{-1/2}$, for different choices of $m$ in
$\widetilde{P}^{\rm XY}_{\ell,m}(k)$ expected from the high number density
sample. From the left to right panels, the results are obtained from $P^{\rm
gg}$, $P^{\rm gE}$, and $P^{\rm EE}$, respectively. The circles show the
multipoles in the standard Legendre basis as a function of $\ell_{\rm max}$.
The squares and triangles show those in the associated Legendre basis. They
are compared with the results from the full 2D spectra shown by the dashed
horizontal lines. The red and blue colors indicate the results given with the
noise from the cosmic variance only (CV $\times$ CV) and total noise
contributions (CV $\times$ CV + CV $\times$ P + P $\times$ P), respectively.
Bottom panels: ratio of the signal-to-noise ratios obtained from the multipole
and full 2D spectra, namely $(S/N)_{\ell_{\rm max}}/(S/N)_{\rm 2D}$. Note that
some symbols are slightly shifted horizontally from integer values for
visibility.

###  IV.3 Signal-to-noise ratio

Here we estimate the signal-to-noise ratio obtained from the multipoles and
full 2D spectra and see how the contributions of the multipoles in the two
different bases reach the full 2D information by increasing $\ell_{\rm max}$.
The top panels in Fig. 2 show the signal-to-noise ratios of each of power
spectra, $P^{\rm gg}$, $P^{\rm gE}$, and $P^{\rm EE}$. We show the results
obtained from the multipole spectra as a function of $\ell_{\rm max}$ as well
as those from the full 2D spectra. The results including all the noise
contributions, namely both the cosmic variance and the Poisson noise and their
crosstalk (CV $\times$ CV $+$ CV $\times$ P $+$ P$\times$ P), are colored by
blue. Those with the noise from the cosmic variance only are colored by red.
The bottom panels show the ratio of the signal-to-noise ratios between the
multipole and 2D spectra, $(S/N)_{\ell_{\rm max}}/(S/N)_{\rm 2D}$. The
information encoded in the multipoles of $P^{\rm gg}$ becomes almost
equivalent to that in the full 2D case when summing up contributions up to
$\ell=4$ in the Legendre basis, consistent with the previous work Taruya et
al. (2011). When we consider only the ${\rm CV}\times{\rm CV}$ term in the
covariance, the signal-to-noise ratios of multipoles for the IA spectra in
both of the two different bases slowly converge to those in the full 2D case
compared with $P^{\rm gg}$, indicating the non-negligible contributions from
higher-order multipoles. This behavior is more significant in the auto-power
spectrum, $P^{\rm EE}$, than the cross-power spectrum $P^{\rm gE}$. We also
find an interesting feature that the results for the associated Legendre basis
converge more slowly than those for the conventional Legendre one. Note that
all the full 2D spectra (i.e., $P^{\rm gg}$, $P^{\rm gE}$ and $P^{\rm EE}$)
have the same signal-to-noise ratios in the case of ${\rm CV}\times{\rm CV}$
since their information is originated from the same underlying matter density
field.

As described in Sec. III.1, when we consider only the shot/shape noise (${\rm
P}\times{\rm P}$), the signal-to-noise ratio for the power spectra can have a
simple form [see Eq. (31)]. In this specific case, the ratio of the signal-to-
noise ratios obtained from the multipoles at a given $\ell_{\rm max}$ and the
full 2D spectra, $\left[\left(S/N\right)^{2}_{\ell_{\rm
max}}/\left(S/N\right)^{2}_{\rm 2D}\right]$, is further simplified using the
relation derived in Appendix A as follows:

$\displaystyle\frac{\left(S/N\right)^{2}_{\ell_{\rm
max}}}{\left(S/N\right)^{2}_{\rm 2D}}=\frac{\sum^{\ell_{\rm max}}_{\ell\geq
m}\left(Q^{\rm XY}_{\ell,m}\right)^{2}}{\int^{1}_{-1}d\mu\\{Q^{\rm
XY}(\mu)\\}^{2}}=\frac{\sum^{\ell_{\rm max}}_{\ell\geq m}\left(Q^{\rm
XY}_{\ell,m}\right)^{2}}{\sum^{\bar{\ell}_{\rm max}}_{\ell\geq m}\left(Q^{\rm
XY}_{\ell,m}\right)^{2}},$ (35)

where $Q^{\rm XY}(\mu)\equiv P^{\rm XY}(k,\mu)/P_{\rm L}(k)$ and $Q^{\rm
XY}_{\ell,m}\equiv\widetilde{P}^{\rm XY}_{\ell,m}(k)/P_{\rm L}(k)$. The
integer $\bar{\ell}_{\rm max}$ is the highest power of $\mu$ in the function
$P^{\rm XY}(k,\mu)$. Namely, $\bar{\ell}_{\rm max}=4$ for our case at the
linear theory limit. Eq. (35) leads to $\left(S/N\right)_{\ell_{\rm
max}}=\left(S/N\right)_{\rm 2D}$ when $\ell_{\rm max}\geq\bar{\ell}_{\rm max}$
for arbitrary $m$. Thus, irrespective of the basis we choose, we can extract
the full 2D information using the multipoles only up to $\ell=4$ for the ${\rm
P}\times{\rm P}$ only case, unlike the ${\rm CV}\times{\rm CV}$ only case.

Figure 3:  Top panels: two-dimensional error contours (68 % C.L.) on the
geometric distances, $d_{\rm A}$ and $H$, expected from the high number
density sample. The solid contours from the left to right panels respectively
show the results obtained from multipole contributions of $P^{\rm
gg}_{\ell,m}$, $P^{\rm gE}_{\ell,m}$, and $P^{\rm EE}_{\ell,m}$ in the
Legendre basis ($m=0$). The red, blue, black, and green contours show the
results from individual multipoles, while the cyan, yellow, and orange ones
shows the results of the combinations up to $\ell_{\rm max}$. For reference,
results from the full 2D spectra are shown by the dashed contours. bottom
panels: Same as the top panels but the results of $P^{\rm gE}$ and $P^{\rm
EE}$ in the associated Legendre basis, $m=2$ and $m=4$, respectively.  Figure
4: Top panels: one-dimensional fractional errors on the growth rate, $f$ or
$fb_{\rm K}$ (magenta), and the geometric distances, $d_{\rm A}$ (green) and
$H$ (black), normalized by the square root of the survey volume,
$\sigma_{\theta}(V_{\rm s}/[h^{-3}{\rm Gpc^{3}}])^{1/2}/\theta^{\rm fid}$,
expected from the high number density sample. From the left to right panels,
the results are obtained from $P^{\rm gg}$, $P^{\rm gE}$, and $P^{\rm EE}$,
respectively. The symbols and dashed horizontal lines are the same as in
Fig.2. Bottom panels: ratio of the one-dimensional errors from the multipoles
and full 2D spectra, $\sigma_{\theta}^{\ell_{\rm max}}/\sigma_{\theta}^{\rm
2D}$. Note that the vertical axis mixes different linear scalings.

Hence, considering the total noise contributions in the signal-to-noise ratio
for the high number density sample where the cosmic variance dominates,
higher-order multipoles such as $\ell=6$ or $8$ must be used to obtain the
information comparable to the full 2D case particularly when the associated
Legendre basis is adopted.

Figure 5:  Same as Fig. 2 and 4 but the results expected from the low number
density sample. Note that some symbols are slightly shifted horizontally from
integer values for visibility.

Figure 6: Left set: two-dimensional marginalized errors (68 % C.L.) on the
growth rate $f$, the angular diameter distance $d_{\rm A}$ and Hubble
parameter $H$, expected from the high number density sample. The blue contours
indicate the results obtained from the full 2D spectra of $P^{\rm gg}$ alone,
while the black, yellow, and red contours from its combinations with the IA
power spectra, $P^{\rm gg}+P^{\rm gE}$, $P^{\rm gg}+P^{\rm EE}$, and $P^{\rm
gg}+P^{\rm gE}+P^{\rm EE}$, respectively. Right set: Same as the left panels
but the results expected from the low number density sample.

###  IV.4 Parameter constraints

As mentioned in the previous section, we have information in vanishing higher-
order multipoles via the off-diagonal components of the covariance matrix, and
therefore it should affect the parameter constraints as well. In this
subsection, we estimate the errors on the parameters obtained from the
multipoles of $P^{\rm gg}$ in the Legendre basis and the IA power spectra,
$P^{\rm gE}$ and $P^{\rm EE}$, in the two different bases. Here we consider
the anisotropies induced not only from the RSD and projection of the galaxy
shapes, but also from the AP effect described in Sec. II.2.

For the Fisher matrix calculation, free parameters considered here are
$\bm{\theta}=(b,f,d_{\rm A}/d^{\rm fid}_{\rm A},H/H^{\rm fid})$ for $P^{\rm
gg}$, $\bm{\theta}=(bb_{\rm K},fb_{\rm K},d_{\rm A}/d^{\rm fid}_{\rm
A},H/H^{\rm fid})$ for $P^{\rm gE}$, and $\bm{\theta}=(b_{\rm K},d_{\rm
A}/d^{\rm fid}_{\rm A},H/H^{\rm fid})$ for $P^{\rm EE}$. Note that with
$P^{\rm gE}$ alone, we cannot determine $f$ because it degenerates with
$b_{\rm K}$. We are interested in the parameters that carry cosmological
information, i.e., $f$, $fb_{\rm K}$, $H$, and $d_{\rm A}$. Hence, we below
marginalize over the others as nuisance parameters, and present the results
for those four parameters.

The top panels of Fig. 3 show the two-dimensional marginalized error ellipses
(68 % C.L.) on the geometric distances ($d_{\rm A}$, $H$) forecasted from the
multipoles up to $\ell=6$ in the Legendre basis and full 2D spectra of $P^{\rm
gg}$, $P^{\rm gE}$, and $P^{\rm EE}$. Although the higher-order multipoles
$\ell\geq 4$ are noisy, if measured accurately, they have non-negligible
impacts on the parameter constraints thanks to their different degeneracy
directions from the other lower-order multipoles as discussed in the
following. Obviously, a single multipole moment in each power spectrum alone
cannot determine $d_{\rm A}$ and $H$ due to the degeneracy between them.
However, combining the $\ell=0$ and $2$ moments, namely choosing $\ell_{\rm
max}=2$, can provide the simultaneous constraints on the geometric distances
and adding further the $\ell=4$ moment, $\ell_{\rm max}=4$, yields a tighter
constraint as clearly shown in the results of $P^{\rm gg}$ and $P^{\rm gE}$
(Padmanabhan and White, 2008; Taruya et al., 2011).

Next, the bottom panels of Fig. 3 show the results in the associated Legendre
basis. An interesting aspect in the results is the contribution from the
additional non-vanishing multipole moment, namely $\ell=6$, caused by the AP
effect. This multipole plays an important role in obtaining the tightest
constraints when the associated Legendre basis is adopted. Particularly,
$\widetilde{P}^{\rm EE}_{4,4}$ alone cannot determine $d_{\rm A}$ and $H$,
while $\widetilde{P}^{\rm EE}_{6,4}$ can help to break the degeneracy between
those parameters. Another important finding is that the $\ell=6$ moment of the
IA power spectra gives tighter geometrical constraints than the $\ell=4$
moment.

We further investigate how well multipole moments of each power spectrum,
$P^{\rm gg}$, $P^{\rm gE}$ and $P^{\rm EE}$, can constrain the parameters as a
function of $\ell_{\rm max}$. The top panels in Fig. 4 show the one-
dimensional fractional errors on $f$, $fb_{\rm K}$, $d_{\rm A}$, and $H$. The
bottom panels show the ratio of the errors from the multipole and full 2D
spectra, $\sigma_{\theta}^{\ell_{\rm max}}/\sigma_{\theta}^{\rm 2D}$. We do
not show the results obtained only from the lowest-order multipole
contribution in all the bases because the degeneracy between $d_{\rm A}$ and
$H$ cannot be broken as shown in Fig.3.

For $P^{\rm gg}$, the use of multipoles up to $\ell=4$ or $6$ can provide the
geometrical and dynamical constraints comparable to the full 2D case, in good
agreement with Ref. Taruya et al. (2011). Similarly to Fig. 2, the results of
the multipole spectra for the high number density sample in the associated
Legendre basis converge more slowly to the full 2D information than those in
the conventional Legendre one. Notably, the one-dimensional errors on $H$
obtained from $P^{\rm gE}$ and $P^{\rm EE}$ are respectively about $6\%$ and
$10\%$ larger than the full 2D case even at $\ell_{\rm max}=6$, although the
derivatives of the signal vanish at $\ell>6$.

Figure 7: Ratios of the one-dimensional marginalized errors on the growth rate
$f$ (left), the angular diameter distance $d_{\rm A}$ (middle), and the Hubble
parameter $H$ (right) obtained from the multipoles and full 2D spectra. From
the top to bottom panels, we show the results for the combinations of
different statistics, $P^{\rm gg}+P^{\rm gE}$, $P^{\rm gg}+P^{\rm EE}$, and
$P^{\rm gg}+P^{\rm gE}+P^{\rm EE}$. For reference, we show the results from
the individual statistics by the circles, squares, and triangles. Note that
the vertical axis mixes different linear scalings.

##  V Discussion

###  V.1 Number density dependence

In Sec. IV we discussed the results expected from the high number density
sample. Here we present the same analysis for the low number density sample
and compare the constraining power as a function of $\ell_{\rm max}$. Fig. 5
shows the results of the same analysis as in Figs. 2 and 4 but for the low
number density. In this case the multipoles of the IA power spectra up to
$\ell=4$ contain contributions comparable to the full 2D case. This is because
the observed IA spectra are more affected by the shot noise than by the cosmic
variance for the low number density sample. Similar to the signal-to-noise
ratios, the constraints obtained from the multipoles of the IA power spectra
quickly converge to the full 2D case.

###  V.2 Combinations of the different statistics

Here we present the parameter constraints from combinations of the different
statistics. We here examine three combinations: $P^{\rm gg}+P^{\rm gE}$,
$P^{\rm gg}+P^{\rm EE}$, and $P^{\rm gg}+P^{\rm gE}+P^{\rm EE}$. To jointly
analyze multiple statistics, we fully take into account the cross-covariance
part (Eq. (33)). We compute the Fisher matrix, taking the five parameters
$\bm{\theta}=(b,f,b_{\rm K},d_{\rm A}/d^{\rm fid}_{\rm A},H/H^{\rm fid})$ to
be free. Note that the covariance matrix of the multiple statistics becomes
ill-defined if we consider only the CV $\times$ CV term. It gives the
vanishing determinant because the different statistics have the same
information and are completely correlated.

Fig. 6 presents the two-dimensional marginalized errors on $f$, $d_{\rm A}$
and $H$. No matter how we combine the galaxy clustering statistics with IA
statistics, the constraints are improved compared to the galaxy clustering
alone Taruya and Okumura (2020). The improvement for the high number density
sample is evident regardless of the choice for the combinations of IA
statistics. On the other hand, for the low number density sample, the observed
IA auto-power spectrum, $P^{\rm EE}$, is more severely affected by the shot
noise, resulting in little improvement for combination of $P^{\rm gg}+P^{\rm
EE}$, as shown in the yellow dash-dotted contours of the right panels.

Finally, considering the combinations of the multiple statistics, we
investigate how fast the information content of the finite multipoles
converges to the full 2D case. Fig. 7 shows the ratio of the one-dimensional
errors on $f$, $d_{\rm A}$, and $H$ between the multipole and full 2D spectra,
$\sigma_{\theta}^{\ell_{\rm max}}/\sigma_{\theta}^{\rm 2D}$, obtained by the
individual statistics as well as the three cases of the combinations. We find
that the use of multipoles up to $\ell=4$ or $6$ gives the information
comparable to the full 2D case even when we adopt the associated Legendre
basis to expand the IA correlations.

##  VI Conclusions

In statistical data analysis of galaxy redshift surveys, one often invokes the
multipole expansion in order to characterize the anisotropies along the line-
of-sight direction. In this paper, based on the forecast studies, signal-to-
noise ratios and Fisher matrix, we have shown the information content of
multipole moments for the IA power spectra, considering the two different
bases in the multipole expansions.

In our analysis, we have used the multipole moments of the IA power spectra
and their Gaussian covariance matrices in the linear theory limit, expanded by
the normalized associated Legend polynomials. As a representative example, we
considered a sample with the high number density of $10^{-2}[h^{3}{\rm
Mpc}^{-3}]$ at $z=0.5$. In this work, we have newly found that the multipole
contributions of the IA power spectra in the associated Legendre basis ($m>0$)
slowly converge to the full 2D cases. It occurs because even if the higher-
order multipole vanishes, its covariance does not and the vanishing multipole
would contribute to the non-vanishing ones at lower order. We have shown that
the errors on the Hubble parameter, $H$, obtained from multipoles of the IA
cross- and auto-power spectra expanded in terms of the associated Legendre
basis are about $6\%$ and $10\%$ larger than the full 2D case even when using
multipoles up to $\ell=6$, although the multipole with $\ell=6$ is zero. The
convergence was found faster for the lower-number density sample. Our results
show that the choice of the basis for multipole expansions is important for
precise parameter constraints, since it changes the information encoded in the
cosmological observables.

Finally we comment on the modelling and assumptions used in this study. We
adopted the linear theory for the power spectra and their analytic covariance
matrices. Beyond the linear theory, i.e., when considering the non-linear
effects, the multipole moments of the IA spectra are non-zero even at $\ell>4$
Taruya et al. (2011); Scoccimarro (2004); Okumura et al. (2024). Thus, we will
further investigate how the non-linear effects change the information
contained in the multipoles of the IA spectra in our future work. In this
work, we have also assumed the plane-parallel approximation, which caused the
contributions from the higher-order multipoles in the covariance. As pointed
out by Ref. Shiraishi et al. (2020) (see also Ref. Shiraishi et al. (2021)),
one can avoid these contributions by adopting the Tripolar spherical harmonics
(TripoSH) as a basis in the multipole expansions. Hence, it is necessary to
investigate the information content encoded in the multipoles of the IA power
spectra using a more general basis such as the TripoSH and we leave its detail
investigation to our future work.

###### Acknowledgements.

TO acknowledges support from the Ministry of Science and Technology of Taiwan
under Grants No. MOST 111-2112-M-001-061- and No. NSTC 112-2112-M-001-034- and
the Career Development Award, Academia Sinica (AS-CDA-108-M02) for the period
of 2019-2023. This work is supported by the Japan Society for the Promotion of
Science (JSPS) KAKENHI Grant No. JP23K19050 and No. 24K17043 (S. S.) and in
part by MEXT/JSPS KAKENHI Grants No. JP20H05861 and No. JP21H01081 (A. T.).

##  Appendix A Useful relations

In this appendix, we present the derivation of useful relations between the IA
power spectrum and its multipoles, expanded in terms of the normalized
associated Legendre basis. The expansion coefficients, $\widetilde{P}^{\rm
gE}_{\ell,m}$ and $\widetilde{P}^{\rm EE}_{\ell,m}$, satisfy the following
relations

$\displaystyle\sum^{4}_{\ell=0}\sqrt{\frac{2\ell+1}{2}}\widetilde{P}^{\rm
gE}_{\ell,0}(k)$
$\displaystyle=\sum^{4}_{\ell=0}\sqrt{\frac{2\ell+1}{2}}\widetilde{P}^{\rm
EE}_{\ell,0}(k)=0,$ (36) $\displaystyle\int^{1}_{-1}d\mu\left\\{P^{\rm
gE}(k,\mu)\right\\}^{2}$
$\displaystyle=\sum^{4}_{\ell=0}\left\\{\widetilde{P}^{\rm
gE}_{\ell,0}(k)\right\\}^{2}$
$\displaystyle=\sum^{4}_{\ell=2}\left\\{\widetilde{P}^{\rm
gE}_{\ell,2}(k)\right\\}^{2},$ (37)
$\displaystyle\int^{1}_{-1}d\mu\left\\{P^{\rm EE}(k,\mu)\right\\}^{2}$
$\displaystyle=\sum^{4}_{\ell=0}\left\\{\widetilde{P}^{\rm
EE}_{\ell,0}(k)\right\\}^{2}$ $\displaystyle=\left\\{\widetilde{P}^{\rm
EE}_{4,4}(k)\right\\}^{2}.$ (38)

Similar relations also hold for the correlation function. These relations
simplify the ratio of the signal-to-noise ratios obtained from the multipoles
at a given $\ell_{\rm max}$ and the full 2D spectra, discussed in Sec. IV.3.
We note that Ref. Saga et al. (2023) derived Eq. (36) for the multipoles of
the galaxy density-ellipticity cross-correlation function in terms of the
conventional Legendre basis $\mathcal{L}_{\ell}(\mu)$ (see Appendix A in their
paper). The derivation of Eqs. (37) and (38), known as the Parseval’s
identity, is given below (see e.g., Ref. Varshalovich et al. (1988) for the
spherical harmonics case).

Let us start by expanding an arbitrary function $f(\mu)$ by the normalized
associated Legendre polynomials $\Theta_{\ell,m}(\mu)$,

$f(\mu)=\sum_{\ell\geq m}f_{\ell,m}\Theta_{\ell,m}(\mu),$ (39)

where $\ell$ and $m$ are integers with $\ell\geq|m|$ and the latter is
arbitrarily chosen. In our case, this function $f$ is either
$\widetilde{P}^{\rm gE}_{\ell,m}$ or $\widetilde{P}^{\rm EE}_{\ell,m}$.
Squaring both sides of this equation yields

$\left\\{f(\mu)\right\\}^{2}=\sum_{\ell\geq m}\sum_{\ell^{\prime}\geq
m}f_{\ell,m}f_{\ell^{\prime},m}\Theta_{\ell,m}(\mu)\Theta_{\ell^{\prime},m}(\mu).$
(40)

Integrating both sides from -1 to 1 over $\mu$, we obtain

$\displaystyle\int^{1}_{-1}d\mu\left\\{f(\mu)\right\\}^{2}$
$\displaystyle=\sum_{\ell\geq m}\sum_{\ell^{\prime}\geq
m}f_{\ell,m}f_{\ell^{\prime},m}$
$\displaystyle\qquad\times\int^{1}_{-1}d\mu\Theta_{\ell,m}(\mu)\Theta_{\ell^{\prime},m}(\mu)$
$\displaystyle=\sum_{\ell\geq m}\sum_{\ell^{\prime}\geq
m}f_{\ell,m}f_{\ell^{\prime},m}\delta^{\rm K}_{\ell,\ell^{\prime}}$
$\displaystyle=\sum_{\ell\geq m}\left(f_{\ell,m}\right)^{2},$ (41)

where we used the orthonormal condition of $\Theta_{\ell,m}$. Obviously, this
equation holds for arbitrary $m$.

##  Appendix B Multipole covariance

In Sec. II.3, we provided the formula of the multipole covariance in terms of
the normalized associated Legendre polynomials $\Theta_{\ell,m}$. In this
appendix, we show the explicit expressions of the auto- and cross-covariance
matrices for the $\rm gE$ and $\rm EE$ power spectrum multipoles.

By integrating Eq. (21) over $\mu$ using Eq. (27), we obtain the multipole
coefficients satisfying the selection rules of the Wigner 3-j symbols and
orthonormality condition of $\Theta_{\ell,m}$, which are given as

$\displaystyle{\rm COV}_{\ell 2,\ell^{\prime}2}^{\rm gE,gE}(k)$
$\displaystyle=\frac{2}{N_{k}}\Biggl{[}\frac{32}{15015}\sqrt{\frac{7(2\ell+1)(2\ell^{\prime}+1)}{2}}\Biggl{\\{}3\sqrt{5}\left(143b^{2}+26bf+3f^{2}\right)\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&4\\\
2&2&-4\end{array}\right)\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&4\\\
0&0&0\end{array}\right)$ (46)
$\displaystyle+52f\left(5b+f\right)\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&6\\\
2&2&-4\end{array}\right)\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&6\\\
0&0&0\end{array}\right)+8\sqrt{11}f^{2}\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&8\\\
2&2&-4\end{array}\right)\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&8\\\
0&0&0\end{array}\right)\Biggr{\\}}\left\\{b_{\rm K}P_{\rm L}(k)\right\\}^{2}$
(55)
$\displaystyle+\Biggl{\\{}\sigma^{2}_{\gamma}\left(b^{2}+\frac{2}{3}bf+\frac{1}{5}f^{2}\right)\delta^{\rm
K}_{\ell,\ell^{\prime}}+\frac{4}{105}\sqrt{\left(2\ell+1\right)\left(2\ell^{\prime}+1\right)}\sum_{L}\sqrt{2L+1}\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&L\\\
2&2&-4\end{array}\right)\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&L\\\
0&0&0\end{array}\right)$ (60)
$\displaystyle\times\Bigl{\\{}f\sigma_{\gamma}^{2}\Bigl{(}\sqrt{5}\left(7b+3f\right)\mathcal{I}^{L,-4}_{2,0}+2f\mathcal{I}^{L,-4}_{4,0}\Bigr{)}+2\sqrt{70}b_{\rm
K}^{2}\mathcal{I}^{L,-4}_{4,4}\Bigr{\\}}\Biggr{\\}}\frac{P_{\rm L}(k)}{n_{\rm
g}}+\frac{\sigma_{\gamma}^{2}}{n_{\rm g}^{2}}\delta^{\rm
K}_{{\ell},{\ell^{\prime}}}\Biggr{]},$ (61)

for the auto-covariance between the $\rm gE$ power spectrum multipoles with
$m=2$,

$\displaystyle{\rm COV}_{\ell 2,\ell^{\prime}4}^{\rm gE,EE}(k)$
$\displaystyle=\frac{2}{N_{k}}\Biggl{[}\frac{64}{15}\sqrt{\left(2\ell+1\right)\left(2\ell^{\prime}+1\right)}\Biggl{\\{}\frac{1}{\sqrt{231}}\left(15b+f\right)\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&6\\\
2&4&-6\end{array}\right)\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&6\\\
0&0&0\end{array}\right)+\frac{2}{\sqrt{429}}f\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&8\\\
2&4&-6\end{array}\right)$ (68)
$\displaystyle\times\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&8\\\
0&0&0\end{array}\right)\Biggr{\\}}b_{{\rm K}}^{3}\left\\{P_{\rm
L}(k)\right\\}^{2}+\Biggl{\\{}\frac{8}{7}\sqrt{\frac{\left(2\ell+1\right)\left(2\ell^{\prime}+1\right)}{30}}\sum_{L}\sqrt{2L+1}\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&L\\\
2&4&-6\end{array}\right)\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&L\\\
0&0&0\end{array}\right)$ (75)
$\displaystyle\times\Bigl{\\{}\left(7b+f\right)\mathcal{I}^{L,-6}_{2,2}+\frac{2\sqrt{3}}{3}f\mathcal{I}^{L,-6}_{4,2}\Bigr{\\}}\Biggr{\\}}b_{\rm
K}P_{\rm L}(k)\frac{\sigma_{\gamma}^{2}}{n_{\rm g}}\Biggr{]},$ (76)

for the cross-covariance between the $\rm gE$ and $\rm EE$ power spectrum
multipoles with $m=2$ and $4$, respectively, and

$\displaystyle{\rm COV}_{\ell 4,\ell^{\prime}4}^{\rm EE,EE}(k)$
$\displaystyle=\frac{4}{N_{k}}\Biggl{[}\frac{256}{3}\sqrt{\frac{(2\ell+1)(2\ell^{\prime}+1)}{1430}}\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&8\\\
4&4&-8\end{array}\right)\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&8\\\
0&0&0\end{array}\right)\left\\{b_{{\rm K}}^{2}P_{\rm
L}(k)\right\\}^{2}+\frac{32}{3}\sqrt{\frac{\left(2\ell+1\right)\left(2\ell^{\prime}+1\right)}{70}}$
(81)
$\displaystyle\times\sum_{L}\sqrt{2L+1}\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&L\\\
4&4&-8\end{array}\right)\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&L\\\
0&0&0\end{array}\right)\mathcal{I}^{L,-8}_{4,4}b_{{\rm K}}^{2}P_{\rm
L}(k)\frac{\sigma_{\gamma}^{2}}{n_{\rm g}}+\frac{\sigma_{\gamma}^{4}}{n_{\rm
g}^{2}}\delta^{\rm K}_{{\ell},{\ell^{\prime}}}\Biggr{]},$ (86)

for the auto-covariance between the $\rm EE$ spectrum multipoles with $m=4$.
In the above expressions, we define the quantity $\mathcal{I}^{L,M}_{\ell,m}$
as follows:

$\displaystyle\mathcal{I}^{L,M}_{\ell,m}=\int_{-1}^{1}d\mu\Theta_{L,M}(\mu)\Theta_{\ell,m}(\mu),$
(87)

where the condition $M=m$ leads to the orthonormality relation,
$\mathcal{I}^{L,m}_{\ell,m}=\delta^{\rm K}_{L,\ell}$.

For reference, we provide the multipole covariance matrices up to $\ell=4$ in
terms of the associated Legendre basis as follows:

$\displaystyle{\rm COV}_{22,22}^{\rm gE,gE}(k)$
$\displaystyle=\frac{2}{3003N_{k}}\Biggl{[}32\left(143b^{2}+26bf+3f^{2}\right)\left\\{b_{\rm
K}P_{\rm L}(k)\right\\}^{2}+143\Biggl{\\{}\frac{16}{{n_{\rm g}}}b_{\rm
K}^{2}+\left(21b^{2}+6bf+f^{2}\right)\frac{\sigma^{2}_{\rm\gamma}}{n_{\rm
g}}\Biggr{\\}}P_{\rm L}(k)$
$\displaystyle\qquad\qquad\qquad+\frac{3003\sigma^{2}_{\rm\gamma}}{n^{2}_{\rm
g}}\Biggr{]},$ (88) $\displaystyle{\rm COV}_{22,42}^{\rm gE,gE}(k)$
$\displaystyle=\frac{8}{1001\sqrt{3}N_{k}}\Biggl{[}-16\left(13b^{2}-4bf-f^{2}\right)\left\\{b_{\rm
K}P_{\rm L}(k)\right\\}^{2}+13\Biggl{\\{}-\frac{8}{n_{\rm g}}b_{\rm
K}^{2}+f\left(11b+3f\right)\frac{\sigma^{2}_{\rm\gamma}}{n_{\rm
g}}\Biggr{\\}}P_{\rm L}(k)\Biggr{]},$ (89) $\displaystyle{\rm
COV}_{42,42}^{\rm gE,gE}(k)$
$\displaystyle=\frac{2}{17017N_{k}}\Biggl{[}32\left(459b^{2}+170bf+39f^{2}\right)\left\\{b_{\rm
K}P_{\rm L}(k)\right\\}^{2}+17\Biggl{\\{}\frac{432}{n_{\rm g}}b_{{\rm
K}}^{2}+\left(1001b^{2}+806bf+237f^{2}\right)\frac{\sigma^{2}_{\rm\gamma}}{n_{\rm
g}}\Biggr{\\}}$ $\displaystyle\qquad\qquad\qquad\times P_{\rm
L}(k)+\frac{17017\sigma^{2}_{\rm\gamma}}{n^{2}_{\rm g}}\Biggr{]},$ (90)

for the auto-covariance of the $\rm gE$ power spectra,

$\displaystyle{\rm COV}_{22,44}^{\rm gE,EE}(k)$
$\displaystyle=\frac{16}{143\sqrt{21}N_{k}}\Biggl{[}8\left(15b+f\right)b_{\rm
K}^{3}\left\\{P_{\rm L}(k)\right\\}^{2}+13\left(11b+f\right)b_{\rm K}P_{\rm
L}(k)\frac{\sigma^{2}_{\rm\gamma}}{n_{\rm g}}\Biggr{]},$ (91)
$\displaystyle{\rm COV}_{42,44}^{\rm gE,EE}(k)$
$\displaystyle=\frac{32}{2431\sqrt{7}N_{k}}\Biggl{[}8\left(-34b+f\right)b_{\rm
K}^{3}\left\\{P_{\rm L}(k)\right\\}^{2}+17\left(-13b+2f\right)b_{\rm K}P_{\rm
L}(k)\frac{\sigma^{2}_{\rm\gamma}}{n_{\rm g}}\Biggr{]},$ (92)

for the cross-covariance of the $\rm gE$ and $\rm EE$ power spectra.

$\displaystyle{\rm COV}_{44,44}^{\rm EE,EE}(k)$
$\displaystyle=\frac{4}{2431N_{k}}\Biggl{[}1792\left\\{b_{\rm K}^{2}P_{\rm
L}(k)\right\\}^{2}+4080b_{\rm K}^{2}P_{\rm
L}(k)\frac{\sigma^{2}_{\rm\gamma}}{n_{\rm
g}}+2431\left(\frac{\sigma^{2}_{\rm\gamma}}{n_{\rm g}}\right)^{2}\Biggr{]},$
(93)

for the auto-covariance of the $\rm EE$ power spectra. As we mentioned in Sec.
II.3, the multipole coefficients of the covariances generally appear even at
higher-order moments where the power spectrum multipoles themselves vanish.

##  Appendix C Derivatives of power spectra

In this appendix, we present the derivatives of the full 2D power spectra and
power spectrum multipoles with respect to their free parameters, which we used
in the Fisher analysis. The observed power spectra $P^{\rm XY}(k,\mu)$ (${\rm
X,Y={g\ or\ E}}$) distorted by the linear RSD and AP effects are generally
expressed as $P^{\rm
XY,obs}(k,\mu)=(\alpha_{\parallel}/\alpha^{2}_{\perp})P^{\rm XY}(q,\nu)$,
where $\alpha_{\perp}\equiv d_{\rm A}/d^{\rm fid}_{\rm A}$ and
$\alpha_{\parallel}\equiv H/H^{\rm fid}$. The superscripts “obs” and “fid”
respectively denote observed and fiducial quantities. The new variables $q$
and $\nu$ are respectively defined by
$q\equiv\sqrt{q^{2}_{\parallel}+q^{2}_{\perp}}=k\sqrt{\alpha^{-2}_{\perp}+(\alpha^{2}_{\parallel}-\alpha^{-2}_{\perp})\mu^{2}}$
and $\nu\equiv
q_{\parallel}/q=\alpha_{\parallel}\mu/\sqrt{\alpha^{-2}_{\perp}+(\alpha^{2}_{\parallel}-\alpha^{-2}_{\perp})\mu^{2}},$
where $k_{\perp}$ and $k_{\parallel}$ are, respectively, related to the true
ones ($q_{\perp}$, $q_{\parallel}$) by
$k_{\perp}=k\sqrt{1-\mu^{2}}=\alpha_{\perp}q_{\perp}$ and
$k_{\parallel}=k\mu=q_{\parallel}/\alpha_{\parallel}$.

By differentiating each power spectrum with respect to their free parameters,
namely $\partial P^{\rm gg,obs}/\partial\theta$ with
$\theta=(b,f,\alpha_{\perp},\alpha_{\parallel})$, $\partial P^{\rm
gE,obs}/\partial\theta$ with $\theta=(bb_{\rm K},fb_{\rm
K},\alpha_{\perp},\alpha_{\parallel})$, and $\partial P^{\rm
EE,obs}/\partial\theta$ with $\theta=(b_{\rm
K},\alpha_{\perp},\alpha_{\parallel})$, at the fiducial values $(b^{\rm
fid},f^{\rm fid},b_{\rm K}^{\rm fid},\alpha^{\rm fid}_{\perp},\alpha^{\rm
fid}_{\parallel})$ , we have

$\displaystyle\left.\frac{\partial P^{\rm gg,obs}}{\partial b}\right|_{\rm
fid}$ $\displaystyle=2\left(b+f\mu^{2}\right)P_{\rm L}(k),$ (94)
$\displaystyle\left.\frac{\partial P^{\rm gg,obs}}{\partial f}\right|_{\rm
fid}$ $\displaystyle=2\mu^{2}\left(b+f\mu^{2}\right)P_{\rm L}(k),$ (95)
$\displaystyle\left.\frac{\partial P^{\rm
gg,obs}}{\partial\alpha_{\perp}}\right|_{\rm fid}$
$\displaystyle=\left(b+f\mu^{2}\right)\Bigl{[}-b\left(n+2\right)+\left\\{bn+f\left(2-n\right)\right\\}$
$\displaystyle\times\mu^{2}+f\left(n-4\right)\mu^{4}\Bigr{]}P_{\rm L}(k),$
(96) $\displaystyle\left.\frac{\partial P^{\rm
gg,obs}}{\partial\alpha_{\parallel}}\right|_{\rm fid}$
$\displaystyle=\left(b+f\mu^{2}\right)\left\\{b+\left(5f+bn\right)\mu^{2}\right.$
$\displaystyle+\left.f\left(n-4\right)\mu^{4}\right\\}P_{\rm L}(k),$ (97)

for the derivatives of the $\rm gg$ auto-power spectrum,

$\displaystyle\left.\frac{\partial P^{\rm gE,obs}}{\partial bb_{\rm
K}}\right|_{\rm fid}$ $\displaystyle=\left(1-\mu^{2}\right)P_{\rm L}(k),$ (98)
$\displaystyle\left.\frac{\partial P^{\rm gE,obs}}{\partial fb_{\rm
K}}\right|_{\rm fid}$ $\displaystyle=\mu^{2}\left(1-\mu^{2}\right)P_{\rm
L}(k),$ (99) $\displaystyle\left.\frac{\partial P^{\rm
gE,obs}}{\partial\alpha_{\perp}}\right|_{\rm fid}$ $\displaystyle=b_{\rm
K}\left(1-\mu^{2}\right)\Bigl{[}-b\left(n+2\right)$
$\displaystyle+\left\\{b\left(n-2\right)-fn\right\\}\mu^{2}+f\left(n-4\right)\mu^{4}\Bigr{]}P_{\rm
L}(k),$ (100) $\displaystyle\left.\frac{\partial P^{\rm
gE,obs}}{\partial\alpha_{\parallel}}\right|_{\rm fid}$ $\displaystyle=b_{\rm
K}\left(1-\mu^{2}\right)\Bigl{[}b+\left\\{b\left(n-2\right)+3f\right\\}\mu^{2}$
$\displaystyle+f\left(n-4\right)\mu^{4}\Bigr{]}P_{\rm L}(k),$ (101)

for the derivatives of the $\rm gE$ cross-power spectrum, and

$\displaystyle\left.\frac{\partial P^{\rm EE,obs}}{\partial b_{\rm
K}}\right|_{\rm fid}$ $\displaystyle=2b_{\rm
K}\left(1-\mu^{2}\right)^{2}P_{\rm L}(k),$ (102)
$\displaystyle\left.\frac{\partial P^{\rm
EE,obs}}{\partial\alpha_{\perp}}\right|_{\rm fid}$ $\displaystyle=b_{\rm
K}^{2}\left(1-\mu^{2}\right)^{2}\left\\{-n-2+\left(n-4\right)\mu^{2}\right\\}P_{\rm
L}(k),$ (103) $\displaystyle\left.\frac{\partial P^{\rm
EE,obs}}{\partial\alpha_{\parallel}}\right|_{\rm fid}$ $\displaystyle=b_{\rm
K}^{2}\left(1-\mu^{2}\right)^{2}\left\\{1+\left(n-4\right)\mu^{2}\right\\}P_{\rm
L}(k),$ (104)

for the derivatives of the $\rm EE$ auto-power spectrum. Here, we define the
logarithmic derivatives of $P_{\rm L}(k)$ with respect to $k$,
$n\equiv\partial\ln P_{\rm L}(k)/\partial\ln k$. In the above expressions, we
already substituted $\alpha^{\rm fid}_{\perp}=\alpha^{\rm fid}_{\parallel}=1$
and omitted the superscript “fid” on the other parameters, $b$, $f$, and
$b_{\rm K}$.

With the above expressions, we compute the derivatives of the multipole
moments with respect to the free parameters in terms of the normalized
associated Legendre polynomials $\Theta_{\ell,m}$ by

$\displaystyle\left.\frac{\partial\widetilde{P}^{\rm
XY}_{\ell,m}}{\partial\theta}\right|_{\rm fid}$
$\displaystyle=\int^{1}_{-1}{d}\mu\Theta_{\ell,m}(\mu)\left.\frac{\partial
P^{\rm XY}}{\partial\theta}\right|_{\rm fid}.$ (105)

In the following, we present the derivatives of the non-vanishing gg, gE, and
EE power spectrum multipoles for $m=0$, $m=0,2$, and $m=0,4$, respectively
(see Sec. II.1 for the corresponding multipoles).

First, we show the derivatives of the non-vanishing $\rm gg$ auto-power
spectrum multipoles with respect to the parameters
$(b,f,\alpha_{\perp},\alpha_{\parallel})$ for $m=0$:

$\displaystyle\left(\left.\frac{\partial\widetilde{P}^{\rm
gg,obs}_{0,0}}{\partial b}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm gg,obs}_{2,0}}{\partial
b}\right|_{\rm fid}\right)$
$\displaystyle=\left(\frac{2}{3}\sqrt{2}\left(3b+f\right)P_{\rm
L}(k),\,\frac{4}{3}\sqrt{\frac{2}{5}}fP_{\rm L}(k)\right),$ (106)
$\displaystyle\left(\left.\frac{\partial\widetilde{P}^{\rm
gg,obs}_{0,0}}{\partial f}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm gg,obs}_{2,0}}{\partial
f}\right|_{\rm fid},\left.\frac{\partial\widetilde{P}^{\rm
gg,obs}_{4,0}}{\partial f}\right|_{\rm fid}\right)$
$\displaystyle=\left(\frac{2}{15}\sqrt{2}\left(5b+3f\right)P_{\rm
L}(k),\right.\frac{4}{21}\sqrt{\frac{2}{5}}\left(7b+6f\right)P_{\rm L}(k),$
$\displaystyle\qquad\left.\frac{16}{105}\sqrt{2}fP_{\rm L}(k)\right),$ (107)
$\displaystyle\left(\left.\frac{\partial\widetilde{P}^{\rm
gg,obs}_{0,0}}{\partial\alpha_{\perp}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
gg,obs}_{2,0}}{\partial\alpha_{\perp}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
gg,obs}_{4,0}}{\partial\alpha_{\perp}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
gg,obs}_{6,0}}{\partial\alpha_{\perp}}\right|_{\rm fid}\right)$
$\displaystyle=\left(-\frac{2\sqrt{2}}{105}\left(n+3\right)\left(35b^{2}+14bf+3f^{2}\right)P_{\rm
L}(k),\right.$
$\displaystyle\qquad\frac{2}{21}\sqrt{\frac{2}{5}}\left\\{7b^{2}n-2bf\left(n+12\right)-f^{2}\left(n+8\right)\right\\}P_{\rm
L}(k),$
$\displaystyle\qquad\left.\frac{16\sqrt{2}f}{1155}\left\\{11b\left(n-2\right)+f\left(2n-19\right)\right\\}P_{\rm
L}(k),\right.$
$\displaystyle\qquad\left.\frac{16}{231}\sqrt{\frac{2}{13}}f^{2}\left(n-4\right)P_{\rm
L}(k)\right),$ (108)
$\displaystyle\left(\left.\frac{\partial\widetilde{P}^{\rm
gg,obs}_{0,0}}{\partial\alpha_{\parallel}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
gg,obs}_{2,0}}{\partial\alpha_{\parallel}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
gg,obs}_{4,0}}{\partial\alpha_{\parallel}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
gg,obs}_{6,0}}{\partial\alpha_{\parallel}}\right|_{\rm fid}\right)$
$\displaystyle=\left(\frac{\sqrt{2}}{105}\left(n+3\right)\left(35b^{2}+42bf+15f^{2}\right)P_{\rm
L}(k),\right.$
$\displaystyle\qquad\frac{2}{21}\sqrt{\frac{2}{5}}\left\\{7b^{2}n+6bf\left(2n+3\right)+5f^{2}\left(n+2\right)\right\\}P_{\rm
L}(k),$
$\displaystyle\qquad\left.\frac{8\sqrt{2}f}{1155}\left\\{22b\left(n-2\right)+5f\left(3n-1\right)\right\\}P_{\rm
L}(k),\right.$
$\displaystyle\qquad\left.\frac{16}{231}\sqrt{\frac{2}{13}}f^{2}\left(n-4\right)P_{\rm
L}(k)\right).$ (109)

Next, we show the derivatives of non-vanishing $\rm gE$ cross-power spectrum
multipoles with respect to the parameters $(bb_{\rm K},fb_{\rm
K},\alpha_{\perp},\alpha_{\parallel})$:

$\displaystyle\left(\left.\frac{\partial\widetilde{P}^{\rm
gE,obs}_{0,0}}{\partial bb_{\rm K}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm gE,obs}_{2,0}}{\partial bb_{\rm
K}}\right|_{\rm fid}\right)$ $\displaystyle=\left(\frac{2}{3}\sqrt{2}P_{\rm
L}(k),-\frac{2}{3}\sqrt{\frac{2}{5}}P_{\rm L}(k)\right),$ (110)
$\displaystyle\left(\left.\frac{\partial\widetilde{P}^{\rm
gE,obs}_{0,0}}{\partial fb_{\rm K}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm gE,obs}_{2,0}}{\partial fb_{\rm
K}}\right|_{\rm fid},\left.\frac{\partial\widetilde{P}^{\rm
gE,obs}_{4,0}}{\partial fb_{\rm K}}\right|_{\rm fid}\right)$
$\displaystyle=\left(\frac{2}{15}\sqrt{2}P_{\rm
L}(k),\frac{2}{21}\sqrt{\frac{2}{5}}P_{\rm L}(k),-\frac{8}{105}\sqrt{2}P_{\rm
L}(k)\right),$ (111)
$\displaystyle\left(\left.\frac{\partial\widetilde{P}^{\rm
gE,obs}_{0,0}}{\partial\alpha_{\perp}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
gE,obs}_{2,0}}{\partial\alpha_{\perp}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
gE,obs}_{4,0}}{\partial\alpha_{\perp}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
gE,obs}_{6,0}}{\partial\alpha_{\perp}}\right|_{\rm fid}\right)$
$\displaystyle=\left(-\frac{8\sqrt{2}}{105}\left(n+3\right)(7b+f)b_{\rm
K}P_{\rm L}(k),\right.$
$\displaystyle\qquad\frac{8}{21}\sqrt{\frac{2}{5}}\left\\{b\left(2n+3\right)-f\right\\}b_{\rm
K}P_{\rm L}(k),$
$\displaystyle\qquad\left.\frac{8\sqrt{2}}{1155}\left\\{-11b\left(n-2\right)+f\left(7n+16\right)\right\\}b_{\rm
K}P_{\rm L}(k),\right.$
$\displaystyle\qquad\left.-\frac{16}{231}\sqrt{\frac{2}{13}}f\left(n-4\right)b_{\rm
K}P_{\rm L}(k)\right),$ (112)
$\displaystyle\left(\left.\frac{\partial\widetilde{P}^{\rm
gE,obs}_{0,0}}{\partial\alpha_{\parallel}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
gE,obs}_{2,0}}{\partial\alpha_{\parallel}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
gE,obs}_{4,0}}{\partial\alpha_{\parallel}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
gE,obs}_{6,0}}{\partial\alpha_{\parallel}}\right|_{\rm fid}\right)$
$\displaystyle=\left(\frac{2\sqrt{2}}{105}\left(n+3\right)(7b+3f)b_{\rm
K}P_{\rm L}(k),\right.$
$\displaystyle\qquad\frac{2}{21}\sqrt{\frac{2}{5}}\left\\{b\left(n-9\right)+f\left(n-1\right)\right\\}b_{\rm
K}P_{\rm L}(k),$
$\displaystyle\qquad\left.-\frac{8\sqrt{2}}{1155}\left\\{11b\left(n-2\right)+f\left(4n+17\right)\right\\}b_{\rm
K}P_{\rm L}(k),\right.$
$\displaystyle\qquad\left.-\frac{16}{231}\sqrt{\frac{2}{13}}f\left(n-4\right)b_{\rm
K}P_{\rm L}(k)\right),$ (113)

for $m=0$, and

$\displaystyle\left.\frac{\partial\widetilde{P}^{\rm gE,obs}_{2,2}}{\partial
bb_{\rm K}}\right|_{\rm fid}=\frac{4P_{\rm L}(k)}{\sqrt{15}},$ (114)
$\displaystyle\left(\left.\frac{\partial\widetilde{P}^{\rm
gE,obs}_{2,2}}{\partial fb_{\rm K}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm gE,obs}_{4,2}}{\partial fb_{\rm
K}}\right|_{\rm fid}\right)=\left(\frac{4P_{\rm
L}(k)}{7\sqrt{15}},\frac{8P_{\rm L}(k)}{21\sqrt{5}}\right),$ (115)
$\displaystyle\left(\left.\frac{\partial\widetilde{P}^{\rm
gE,obs}_{2,2}}{\partial\alpha_{\perp}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
gE,obs}_{4,2}}{\partial\alpha_{\perp}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
gE,obs}_{6,2}}{\partial\alpha_{\perp}}\right|_{\rm fid}\right)$
$\displaystyle=\left(-\frac{8}{21\sqrt{15}}\left\\{3b\left(3n+8\right)+f\left(n+2\right)\right\\}b_{\rm
K}P_{\rm L}(k),\right.$
$\displaystyle\qquad\frac{8}{231\sqrt{5}}\left\\{11b\left(n-2\right)-f\left(5n+24\right)\right\\}b_{\rm
K}P_{\rm L}(k),$
$\displaystyle\qquad\left.\frac{32}{33}\sqrt{\frac{2}{1365}}f\left(n-4\right)b_{\rm
K}P_{\rm L}(k)\right),$ (116)
$\displaystyle\left(\left.\frac{\partial\widetilde{P}^{\rm
gE,obs}_{2,2}}{\partial\alpha_{\parallel}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
gE,obs}_{4,2}}{\partial\alpha_{\parallel}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
gE,obs}_{6,2}}{\partial\alpha_{\parallel}}\right|_{\rm fid}\right)$
$\displaystyle=\left(\frac{4}{21\sqrt{15}}\left(3b+f\right)\left(n+5\right)b_{\rm
K}P_{\rm L}(k),\right.$
$\displaystyle\qquad\frac{8}{231\sqrt{5}}\left\\{11b\left(n-2\right)+3f\left(2n+3\right)\right\\}b_{\rm
K}P_{\rm L}(k),$
$\displaystyle\qquad\left.\frac{32}{33}\sqrt{\frac{2}{1365}}f\left(n-4\right)b_{\rm
K}P_{\rm L}(k)\right),$ (117)

for $m=2$. Finally, we show the derivatives of non-vanishing $\rm EE$ auto-
power spectrum multipoles with respect to the parameters $(b_{\rm
K},\alpha_{\perp},\alpha_{\parallel})$:

$\displaystyle\left(\left.\frac{\partial\widetilde{P}^{\rm
EE,obs}_{0,0}}{\partial b_{\rm K}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm EE,obs}_{2,0}}{\partial b_{\rm
K}}\right|_{\rm fid},\left.\frac{\partial\widetilde{P}^{\rm
EE,obs}_{4,0}}{\partial b_{\rm K}}\right|_{\rm fid}\right)$
$\displaystyle=\left(\frac{16}{15}\sqrt{2}b_{\rm K}P_{\rm
L}(k),-\frac{32}{21}\sqrt{\frac{2}{5}}b_{\rm K}P_{\rm
L}(k),\frac{16}{105}\sqrt{2}b_{\rm K}P_{\rm L}(k)\right),$ (118)
$\displaystyle\left(\left.\frac{\partial\widetilde{P}^{\rm
EE,obs}_{0,0}}{\partial\alpha_{\perp}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
EE,obs}_{2,0}}{\partial\alpha_{\perp}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
EE,obs}_{4,0}}{\partial\alpha_{\perp}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
EE,obs}_{6,0}}{\partial\alpha_{\perp}}\right|_{\rm fid}\right)$
$\displaystyle=\left(-\frac{16\sqrt{2}}{35}\left(n+3\right)b_{\rm K}^{2}P_{\rm
L}(k),\right.\frac{16}{21}\sqrt{\frac{2}{5}}\left(n+2\right)b_{\rm
K}^{2}P_{\rm L}(k),$
$\displaystyle\quad\left.-\frac{16\sqrt{2}}{385}\left(3n-1\right)b_{\rm
K}^{2}P_{\rm
L}(k),\right.\left.\frac{16}{231}\sqrt{\frac{2}{13}}\left(n-4\right)b_{\rm
K}^{2}P_{\rm L}(k)\right),$ (119)
$\displaystyle\left(\left.\frac{\partial\widetilde{P}^{\rm
EE,obs}_{0,0}}{\partial\alpha_{\parallel}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
EE,obs}_{2,0}}{\partial\alpha_{\parallel}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
EE,obs}_{4,0}}{\partial\alpha_{\parallel}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
EE,obs}_{6,0}}{\partial\alpha_{\parallel}}\right|_{\rm fid}\right)$
$\displaystyle=\left(\frac{8\sqrt{2}}{105}\left(n+3\right)b_{\rm K}^{2}P_{\rm
L}(k),\right.-\frac{16}{21}\sqrt{\frac{2}{5}}b_{\rm K}^{2}P_{\rm L}(k),$
$\displaystyle\quad\left.-\frac{8\sqrt{2}}{1155}\left(7n-39\right)b_{\rm
K}^{2}P_{\rm
L}(k),\right.\left.\frac{16}{231}\sqrt{\frac{2}{13}}\left(n-4\right)b_{\rm
K}^{2}P_{\rm L}(k)\right),$ (120)

for $m=0$, and

$\displaystyle\left.\frac{\partial\widetilde{P}^{\rm EE,obs}_{4,4}}{\partial
b_{\rm K}}\right|_{\rm fid}=\frac{32}{3\sqrt{35}}b_{\rm K}P_{\rm L}(k),$ (121)
$\displaystyle\left(\left.\frac{\partial\widetilde{P}^{\rm
EE,obs}_{4,4}}{\partial\alpha_{\perp}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
EE,obs}_{6,4}}{\partial\alpha_{\perp}}\right|_{\rm fid}\right)$
$\displaystyle=\left(-\frac{32}{33\sqrt{35}}\left(5n+13\right)b_{\rm
K}^{2}P_{\rm L}(k),\right.$
$\displaystyle\qquad\left.\frac{32}{33\sqrt{91}}\left(n-4\right)b_{\rm
K}^{2}P_{\rm L}(k)\right),$ (122)
$\displaystyle\left(\left.\frac{\partial\widetilde{P}^{\rm
EE,obs}_{4,4}}{\partial\alpha_{\parallel}}\right|_{\rm
fid},\left.\frac{\partial\widetilde{P}^{\rm
EE,obs}_{6,4}}{\partial\alpha_{\parallel}}\right|_{\rm fid}\right)$
$\displaystyle=\left(\frac{16}{33\sqrt{35}}\left(n+7\right)b_{\rm K}^{2}P_{\rm
L}(k),\right.$
$\displaystyle\qquad\left.\frac{32}{33\sqrt{91}}\left(n-4\right)b_{\rm
K}^{2}P_{\rm L}(k)\right),$ (123)

for $m=4$. We note that, due to the AP effect, the power spectrum multipoles
and their derivatives with respect to the parameters are non-zero for $\ell>6$
even in the linear theory limit. However, after substituting the fiducial
values into their expressions, the multipoles $\ell>4$ for the power spectrum
and $\ell>6$ for their derivatives vanish.

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