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

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)

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),

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

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

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

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

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 (2024); Okumura et al. (2024), defined by

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

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

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

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

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

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

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.

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),

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).

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

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)),

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. (66)–(91) 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. (93)–(98).

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. (66)–(91) 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.

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:

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

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

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

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 $N^{\rm E}=\sigma^{2}_{\gamma}/n_{\rm g}$.

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

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

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),

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, $\sigma_{\theta}=\sqrt{\left(F^{-1}\right)_{ij}}$.

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 $k_{{\rm max}}=0.1\ h{\rm Mpc}^{-1}$. Our conservative choice of $k_{\rm max}$ can justify the use of linear expressions and Gaussian covariance Takahashi et al. (2009, 2010); Blot et al. (2014, 2019); Chan and Blot (2017). 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.

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. In the low number density case, turnover scales arise due to the shot noise contribution. This effect is particularly evident in the EE power spectrum, since the P $\times$ P contribution dominates the noise. Consequently, the shapes of the fractional errors are influenced by the shapes of the power spectrum multipoles. 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. This is because the information is compressed into higher-order multipole moments, leading to the minimum set of multipoles in the associated Legendre basis. 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.

To see the correlations between the power spectrum multipoles, we show the correlation matrix, ${\rm COV}^{\rm XY,X^{\prime}Y^{\prime}}_{\ell m,\ell^{\prime}m^{\prime}}/\sqrt{{\rm COV}^{\rm XY,XY}_{\ell m,\ell m}\times{\rm COV}^{\rm X^{\prime}Y^{\prime},X^{\prime}Y^{\prime}}_{\ell^{\prime}m^{\prime},\ell^{\prime}m^{\prime}}}$, as a function of $\ell$ and $\ell^{\prime}$ fixing the wavenumber to $k=0.1h{\rm Mpc}^{-1}$ in Fig. 2. The left and right panels show the results for the high and low number density samples, respectively. We show the results obtained from the multipoles of the IA power spectra for $m=0$ and $m>0$, which are presented in the upper and lower panels, respectively. Note that the correlations still appear even at $\ell>4$ where the power spectrum multipoles vanish, as discussed above. For the high number density case, the contribution of the shot noise has less impact, i.e., cosmic variance dominates, leading to the stronger correlations between statistics that trace the same underlying matter density field. Since the sign of the multipoles in the IA power spectra at a given order varies with the basis choice, the sign of the covariance matrix varies accordingly. Interestingly, we find that the cross-correlation between IA power spectra, $P^{\rm gE}$ and $P^{\rm EE}$, shows a stronger correlation compared to the correlation between $P^{\rm gg}$ and IA power spectra, as clearly shown in the high number density case.

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. 3 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:

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.

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.

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. 4 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. 4 show the results in the associated Legendre basis. When comparing the results of the same-order multipoles between the two bases, the degeneracy directions are different. This is because the multipole moment in the associated Legendre basis can be expressed by the several multipole moments in the Legendre basis, as shown in Appendix A. 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. 5 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.4.

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. 3, 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 $\ell>6$.

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. 8 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, since the IA statistics tightly constrain the geometric distances, $d_{\rm A}$ and $H$, thereby breaking the degeneracy between the geometric distances and $f$ 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. 9 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.